Introduction
The study of motivic invariants such as motivic cohomology, algebraic K K K K K K [3 ] .
The development of motivic stable homotopy theory by Morel and Voevodsky [28 ] established a framework needed to translate questions in algebraic geometry into questions in homotopy theory, making it possible to apply familiar tools—such as spectral sequences and generalized cohomology theories—to motivic spaces. In this sense, these motivic invariants often refine their classical counterparts, encoding additional structures that become visible only in the motivic setting. This perspective was instrumental in Voevodsky’s proof of the Milnor and Bloch–Kato conjectures, which established profound connections between motivic cohomology, étale cohomology, and Milnor K K K
From the topological perspective, generalized (co)homology theories of classifying spaces have been studied extensively. In particular, Bruner and Greenlees [11 , 12 ] computed connective complex and real topological K K K
Over the past decade, computational motivic homotopy theory has undergone rapid development. Isaksen and Shkembi [26 ] introduced C \mathbb{C} C K K K [34 ] , a pivotal result for the present paper. Further advances by Hill [23 ] , Dugger and Isaksen [20 ] , and others developed computational tools for the motivic Adams spectral sequence over R \mathbb{R} R [27 ] introduced powerful comparison techniques between classical and motivic Adams spectral sequences, leading to substantial new computations of motivic stable stems and significantly expanding the computational reach of motivic stable homotopy theory.
Building on these developments, in [DFN+25] we, together with Field, Nigam, Quigley, and Yang, applied motivic Adams and Atiyah–Hirzebruch spectral sequence techniques to compute the 2 2 2 K K K K K K 2 2 2
Extending this line of investigation, the present paper computes the classifying spaces of cyclic groups of order 2 n 2^n 2 n n > 1 n>1 n > 1 F F F 2 2 2
Theorem 1.1. Let n > 1 n>1 n > 1 2 2 2 F F F 2 n 2^n 2 n
Integral motivic homology (see Theorem 5.1 and Theorem 5.4 );
Effective algebraic K K K Theorem 6.2 and Theorem 6.5);
Very effective Hermitian K K K Theorem 7.3 and Theorem 7.6 ).
These computations are complete in all stems for integral motivic homology and through stem 14 for the K K K n = 2 n=2 n = 2
Although our computations are carried out over the complex numbers C \mathbb{C} C R \mathbb{R} R C \mathbb{C} C R \mathbb{R} R
We begin with the classical computations and then adapt the arguments to the C \mathbb{C} C R \mathbb{R} R B C 2 BC_2 B C 2 C \mathbb{C} C τ \tau τ R \mathbb{R} R ρ \rho ρ C \mathbb{C} C
H Z ∗ ∗ m o t i v i c ( B μ 2 n ) ↠ H Z ∗ c l a s s i c a l ( B C 2 n ) HZ^{\mathrm{motivic}}_{**}(B\mu_{2^n}) \twoheadrightarrow HZ^{\mathrm{classical}}_*(BC_{2^n}) H Z ∗∗ motivic ( B μ 2 n ) ↠ H Z ∗ classical ( B C 2 n ) k g l ∗ ∗ m o t i v i c ( B μ 2 n ) ↠ k u ∗ c l a s s i c a l ( B C 2 n ) kgl^{\mathrm{motivic}}_{**}(B\mu_{2^n}) \twoheadrightarrow ku^{\mathrm{classical}}_*(BC_{2^n}) k g l ∗∗ motivic ( B μ 2 n ) ↠ k u ∗ classical ( B C 2 n ) k q ∗ ∗ m o t i v i c ( B μ 2 n ) ↠ k o ∗ c l a s s i c a l ( B C 2 n ) . kq^{\mathrm{motivic}}_{**}(B\mu_{2^n}) \twoheadrightarrow ko^{\mathrm{classical}}_*(BC_{2^n}). k q ∗∗ motivic ( B μ 2 n ) ↠ k o ∗ classical ( B C 2 n ) . For n > 1 n>1 n > 1 B μ 2 n B\mu_{2^n} B μ 2 n n = 1 n=1 n = 1 S q 1 \mathrm{Sq}^1 Sq 1 B μ 2 n B\mu_{2^n} B μ 2 n n > 1 n>1 n > 1 E 2 E_2 E 2 n = 1 n=1 n = 1 S q 1 \mathrm{Sq}^1 Sq 1
Our computations proceed in two stages. First, we determine the order of the groups via the Atiyah–Hirzebruch spectral sequence associated to the cellular filtration of B μ 2 n B\mu_{2^n} B μ 2 n
E 2 s , ( t , w ) = Ext A F s , ( t , w ) ( H ∗ ∗ ( X ) , M 2 F ) ⇒ π t − s , w ( X ) ( 2 , η ) ∧ E_2^{s,(t,w)}=\operatorname{Ext}_{A_F}^{s,(t,w)}\left(H^{**}(X),\mathbb{M}_2^F\right)\Rightarrow\pi_{t-s,w}(X)^{\wedge}_{(2,\eta)} E 2 s , ( t , w ) = Ext A F s , ( t , w ) ( H ∗∗ ( X ) , M 2 F ) ⇒ π t − s , w ( X ) ( 2 , η ) ∧ Over F = C F=\mathbb{C} F = C R \mathbb{R} R ( 2 , η ) (2,\eta) ( 2 , η ) § 3 .
A key feature of our approach is that we first compute the k o ko k o B C 2 n BC_{2^n} B C 2 n k u ku k u k o ko k o
p : k o ∗ ( B C 2 n ) ⟶ K O ∗ ( B C 2 n ) , p:ko_*(BC_{2^n})\longrightarrow KO_*(BC_{2^n}), p : k o ∗ ( B C 2 n ) ⟶ K O ∗ ( B C 2 n ) , and studying the eta invariant on periodic K O KO K O k o ko k o E 2 E_2 E 2 k o ko k o k u ku k u
Beyond the explicit calculations, we highlight the framework combining Atiyah–Hirzebruch and motivic Adams spectral sequences to analyze these motivic homology theories.
Future Work
First, the author is currently working toward completing the computation of higher differentials in the Adams spectral sequence for n > 2 n>2 n > 2 h 0 h_0 h 0 theorem. Bruner and others have used a similar approach for quaternion groups in their work in [7 ] .
Then, we plan to extend these computations to finite fields F q \mathbb{F}_q F q q > 1 q > 1 q > 1
Another natural direction is to investigate equivariant refinements of these motivic homology theories. The equivariant Betti realization functor connects the motivic stable homotopy category H o ( S H ( R ) ) Ho(SH(\mathbb{R})) H o ( S H ( R )) C 2 C_2 C 2 H o ( S p C 2 ) Ho(Sp^{C_2}) H o ( S p C 2 ) R O ( C 2 ) RO(C_2) R O ( C 2 ) R \mathbb{R} R C 2 C_2 C 2
The C 2 C_2 C 2 K K K k R k\mathbb{R} k R k o C 2 ko_{C_2} k o C 2 R \mathbb{R} R k R ∗ ∗ ( B C 2 μ 2 n ) k\mathbb{R}_{**}(BC_2\mu_{2^n}) k R ∗∗ ( B C 2 μ 2 n ) ( k o C 2 ) ∗ ∗ ( B C 2 μ 2 n ) (ko_{C_2})_{**}(BC_2\mu_{2^n}) ( k o C 2 ) ∗∗ ( B C 2 μ 2 n )
Organization
Section 2 gathers the essential background, beginning with the motivic analogs of cohomology of a point, steenrod algebra and connective K K K B μ 2 n B\mu_{2^n} B μ 2 n n > 1 n > 1 n > 1
In Section 3 we describe the general framework that guides all of our computations, emphasizing how the Atiyah–Hirzebruch and Adams spectral sequences fit together in this setting. Section 4 begins with the classical calculation, allowing us to highlight the familiar patterns before turning to the richer C \mathbb{C} C R \mathbb{R} R K K K K K K
Acknowledgments
This project grew out of a summer collaborative workshop with Rebecca Field, Arjun Nigam, J. D. Quigley, and Albert Jinghui Yang. I am also grateful for many insightful conversations with Bert Guillou and Arun Debray, which clarified and strengthened several parts of this work. I also want to thank Jackson Morris for his thoughtful feedback on earlier drafts.
Background
In this section, we recall some definitions and basic facts from motivic homotopy theory that will be used throughout this paper.
Throughout this section, let F F F C \mathbb{C} C R \mathbb{R} R C \mathbb{C} C R \mathbb{R} R
Cohomology of a point
Define M 2 F : = H ∗ ∗ ( p t ; F 2 ) \mathbb{M}_2^F := H^{**}(\mathrm{pt};\mathbb{F}_2) M 2 F := H ∗∗ ( pt ; F 2 ) F F F
By Voevodsky ([33 ] , [34 ] ),
M 2 F ≅ { F 2 [ τ ] , F = C , F 2 [ τ , ρ ] , F = R \mathbb{M}_2^F \cong
\begin{cases}
\mathbb{F}_2[\tau], & F = \mathbb{C}, \\
\mathbb{F}_2[\tau,\rho], & F = \mathbb{R}
\end{cases} M 2 F ≅ { F 2 [ τ ] , F 2 [ τ , ρ ] , F = C , F = R where τ \tau τ ∣ τ ∣ = ( 0 , 1 ) |\tau| = (0,1) ∣ τ ∣ = ( 0 , 1 ) ρ \rho ρ ∣ ρ ∣ = ( 1 , 1 ) |\rho| = (1,1) ∣ ρ ∣ = ( 1 , 1 )
The Motivic Steenrod Algebra
The mod two motivic Steenrod algebra over F F F A F \mathcal{A}^{F} A F M 2 F \mathbb{M}^{F}_{2} M 2 F
Sq i : H p , q ( − ; F 2 ) ⟶ H p + i , q + ⌊ i / 2 ⌋ ( − ; F 2 ) , \operatorname{Sq}^{i}:H^{p,q}(-;\mathbb{F}_{2})\longrightarrow H^{p+i,q+\lfloor i/2\rfloor}(-;\mathbb{F}_{2}), Sq i : H p , q ( − ; F 2 ) ⟶ H p + i , q + ⌊ i /2 ⌋ ( − ; F 2 ) , subject to motivic Adem relations; see [33 ] .
A C \mathcal{A}^{\mathbb{C}} A C A \mathcal{A} A τ \tau τ
A C ⊗ M 2 M 2 [ τ − 1 ] ⟶ A ⊗ M 2 M 2 [ τ − 1 ] \mathcal{A}^{\mathbb{C}}\otimes_{\mathbb{M}_{2}}\mathbb{M}_{2}[\tau^{-1}]\longrightarrow\mathcal{A}\otimes_{\mathbb{M}_{2}}\mathbb{M}_{2}[\tau^{-1}] A C ⊗ M 2 M 2 [ τ − 1 ] ⟶ A ⊗ M 2 M 2 [ τ − 1 ] defined by sending
Sq 2 k ⟼ τ − k Sq 2 k and Sq 2 k − 1 ⟼ τ − k Sq 2 k − 1 , \operatorname{Sq}^{2k}\longmapsto\tau^{-k}\operatorname{Sq}^{2k}\qquad\text{and}\qquad\operatorname{Sq}^{2k-1}\longmapsto\tau^{-k}\operatorname{Sq}^{2k-1}, Sq 2 k ⟼ τ − k Sq 2 k and Sq 2 k − 1 ⟼ τ − k Sq 2 k − 1 , which is an isomorphism of bigraded rings, see [18 Theorem 2.9] for details.
A R \mathcal{A}^{\mathbb{R}} A R ρ ∈ H 1 , 1 ( p t ; F 2 ) \rho\in H^{1,1}(\mathrm{pt};\mathbb{F}_{2}) ρ ∈ H 1 , 1 ( pt ; F 2 ) ρ \rho ρ ρ \rho ρ A R \mathcal{A}^{\mathbb{R}} A R [20 ] .
Remark 2.1. A F \mathcal{A}^{F} A F H ∗ ∗ ( X ; F 2 ) H^{**}(X;\mathbb{F}_{2}) H ∗∗ ( X ; F 2 ) X X X A F \mathcal{A}^{F} A F
In particular, M 2 F \mathbb{M}^{F}_{2} M 2 F A F \mathcal{A}^{F} A F A C \mathcal{A}^{\mathbb{C}} A C M 2 C ≅ F 2 [ τ ] \mathbb{M}^{\mathbb{C}}_{2}\cong\mathbb{F}_{2}[\tau] M 2 C ≅ F 2 [ τ ] M 2 C \mathbb{M}^{\mathbb{C}}_{2} M 2 C Sq i \operatorname{Sq}^{i} Sq i A R \mathcal{A}^{\mathbb{R}} A R M 2 R ≅ F 2 [ τ , ρ ] \mathbb{M}^{\mathbb{R}}_{2}\cong\mathbb{F}_{2}[\tau,\rho] M 2 R ≅ F 2 [ τ , ρ ] Sq 1 ( τ ) = ρ \operatorname{Sq}^{1}(\tau)=\rho Sq 1 ( τ ) = ρ Sq 2 ( τ 2 ) = τ ρ 2 \operatorname{Sq}^{2}(\tau^{2})=\tau\rho^{2} Sq 2 ( τ 2 ) = τ ρ 2
We now introduce motivic analogs of some classical Steenrod subalgebras.
Definition 2.2. Let
A F ( 0 ) \mathcal{A}^{F}(0) A F ( 0 ) M 2 F \mathbb{M}^{F}_{2} M 2 F A F \mathcal{A}^{F} A F Q 0 = Sq 1 Q_{0}=\operatorname{Sq}^{1} Q 0 = Sq 1
E F ( 1 ) \mathcal{E}^{F}(1) E F ( 1 ) M 2 F \mathbb{M}^{F}_{2} M 2 F A F \mathcal{A}^{F} A F Q 0 Q_{0} Q 0 Q 1 = Sq 1 Sq 2 + Sq 2 Sq 1 Q_{1}=\operatorname{Sq}^{1}\operatorname{Sq}^{2}+\operatorname{Sq}^{2}\operatorname{Sq}^{1} Q 1 = Sq 1 Sq 2 + Sq 2 Sq 1
A F ( 1 ) \mathcal{A}^{F}(1) A F ( 1 ) M 2 F \mathbb{M}^{F}_{2} M 2 F A F \mathcal{A}^{F} A F Sq 1 \operatorname{Sq}^{1} Sq 1 Sq 2 \operatorname{Sq}^{2} Sq 2
Definition 2.3. For a subalgebra B B B A \mathcal{A} A
A / / B : = A ⊗ B M 2 F , \mathcal{A}//B:=\mathcal{A}\otimes_{B}\mathbb{M}^{F}_{2}, A // B := A ⊗ B M 2 F , where M 2 F \mathbb{M}^{F}_{2} M 2 F B B B B B B A F ( 0 ) \mathcal{A}^{F}(0) A F ( 0 ) E F ( 1 ) \mathcal{E}^{F}(1) E F ( 1 ) A F ( 1 ) \mathcal{A}^{F}(1) A F ( 1 )
Motivic “Connective” K K K
In the classical stable homotopy category, the connective K K K k u ku k u k o ko k o K K K K U KU K U K O KO K O
Motivic homotopy theory admits closely analogous constructions. In the F F F S H ( F ) \mathrm{SH}(F) SH ( F ) S H ( F ) e f f \mathrm{SH}(F)^{\mathrm{eff}} SH ( F ) eff F F F t t t S H ( F ) v e f f \mathrm{SH}(F)^{\mathrm{veff}} SH ( F ) veff
The effective algebraic K K K k g l F kgl^F k g l F K K K k u ku k u K K K k q F kq^F k q F K K K k o ko k o K K K k o ko k o k u ku k u [32 ] , while its very effective refinement was introduced by Spitzweck and Østvær [30 ] and further developed by Bachmann [5 ] .
A key feature of very effective Hermitian K K K K K K [26 , 4 ] and recalled in the following proposition.
Proposition 2.4. [4 Proposition 2.11] Over a field F F F
Σ 1 , 1 k q F → η k q F ⟶ k g l F , \Sigma^{1,1} kq^F \xrightarrow{\eta} kq^F \longrightarrow kgl^F, Σ 1 , 1 k q F η k q F ⟶ k g l F , where the map is induced by multiplication by η \eta η K K K η \eta η K K K k o ko k o k u ku k u
Let us recall the classical descriptions of the mod 2 cohomology of H Z HZ H Z k u ku k u k o ko k o A \mathcal{A} A [2 ] . These descriptions play a fundamental role in computations with the classical Adams spectral sequence.
Proposition 2.5. There are isomorphisms of A \mathcal{A} A H ∗ ( H Z ; F 2 ) ≅ A / / A ( 0 ) H^*(HZ;\mathbb{F}_2) \cong\mathcal{A}//\mathcal{A}(0) H ∗ ( H Z ; F 2 ) ≅ A // A ( 0 ) H ∗ ( k u ; F 2 ) ≅ A / / E ( 1 ) H^*(ku;\mathbb{F}_2) \cong\mathcal{A}//\mathcal{E}(1) H ∗ ( k u ; F 2 ) ≅ A // E ( 1 ) H ∗ ( k o ; F 2 ) ≅ A / / A ( 1 ) H^*(ko;\mathbb{F}_2) \cong\mathcal{A}//\mathcal{A}(1) H ∗ ( k o ; F 2 ) ≅ A // A ( 1 )
In the motivic setting, analogous descriptions hold for the motivic spectra H Z F HZ^F H Z F k g l F kgl^F k g l F k q F kq^F k q F F = C F=\mathbb{C} F = C F = R F=\mathbb{R} F = R
Proposition 2.6. Let F = C F=\mathbb{C} F = C R \mathbb{R} R A F \mathcal{A}^F A F H ∗ ∗ ( H Z F ; M 2 F ) ≅ A F / / A F ( 0 ) H^{**}(HZ^F;M_2^F) \cong\mathcal{A}^F//\mathcal{A}^F(0) H ∗∗ ( H Z F ; M 2 F ) ≅ A F // A F ( 0 ) H ∗ ∗ ( k g l F ; M 2 F ) ≅ A F / / E F ( 1 ) H^{**}(kgl^F;M_2^F) \cong\mathcal{A}^F//\mathcal{E}^F(1) H ∗∗ ( k g l F ; M 2 F ) ≅ A F // E F ( 1 ) H ∗ ∗ ( k q F ; M 2 F ) ≅ A F / / A F ( 1 ) H^{**}(kq^F;M_2^F) \cong\mathcal{A}^F//\mathcal{A}^F(1) H ∗∗ ( k q F ; M 2 F ) ≅ A F // A F ( 1 )
The case F = C F=\mathbb{C} F = C [26 Thm. 5.7] , while the case F = R F=\mathbb{R} F = R [4 Section 2] .
Motivic classifying space and its mod two motivic cohomology
As our goal is to compute motivic homology theories of motivic analogues of classifying spaces, we briefly recall the notion of the motivic classifying space used throughout this paper. We work with the geometric motivic classifying spaces B g m G B_{\mathrm{gm}}G B gm G [28 ] . For simplicity, we write B G BG B G B g m G B_{\mathrm{gm}}G B gm G B G = E G / G BG=EG/G B G = E G / G G G G
Let μ 2 n \mu_{2^n} μ 2 n 2 n 2^n 2 n
μ 2 n : = ker ( G m → ( ⋅ ) 2 n G m ) , \mu_{2^n}:=\ker\left(\mathbb{G}_m \xrightarrow{(\cdot)^{2^n}} \mathbb{G}_m\right), μ 2 n := ker ( G m ( ⋅ ) 2 n G m ) , see for example, [33 §6] .
Over algebraically closed fields, this coincides with the cyclic group of order 2 n 2^n 2 n B C 2 n B C_{2^n} B C 2 n [31 ] .
Moreover, there is a realization functor from C \mathbb{C} C X X X X ( C ) X(\mathbb{C}) X ( C ) B μ 2 n B\mu_{2^n} B μ 2 n B C 2 n B C_{2^n} B C 2 n
This realization functor is symmetric monoidal [25 ] and induces natural maps
H p , q ( X ) ⟶ H p ( X ( C ) ) , H^{p,q}(X)\longrightarrow H^p(X(\mathbb{C})), H p , q ( X ) ⟶ H p ( X ( C )) , where the target is viewed as a bigraded object concentrated in weight 0 0 0 [18 ] . After standard technical adjustments, these maps extend to a comparison between the C \mathbb{C} C
For our computations, we care about the motivic cellular structure of these spaces. Analogous to the classical CW-structure, B μ 2 n B\mu_{2^n} B μ 2 n x i x_i x i
( i , ⌊ i + 1 2 ⌋ ) , i ≥ 0. \left(i,\left\lfloor\frac{i+1}{2}\right\rfloor\right),\qquad i\geq0. ( i , ⌊ 2 i + 1 ⌋ ) , i ≥ 0. The attaching maps are similarly structured: each even-dimensional cell is attached to the preceding odd-dimensional cell via a map of degree h n h^n h n h ∈ π 0 , 0 h\in\pi_{0,0} h ∈ π 0 , 0 h 0 h_0 h 0 h n h^n h n h 0 n h_0^n h 0 n C \mathbb{C} C h h h R \mathbb{R} R 2 − ρ η 2-\rho\eta 2 − ρ η [20 Definition 8.1] . Each cell in dimension i ≡ 0 , 1 ( m o d 4 ) i\equiv0,1\pmod4 i ≡ 0 , 1 ( mod 4 ) i ≡ 2 , 3 ( m o d 4 ) i\equiv2,3\pmod4 i ≡ 2 , 3 ( mod 4 ) η \eta η Sq 2 \operatorname{Sq}^2 Sq 2 A ( 0 ) \mathcal{A}(0) A ( 0 ) E ( 1 ) \mathcal{E}(1) E ( 1 ) A ( 1 ) \mathcal{A}(1) A ( 1 ) A \mathcal{A} A
Figure 27.
Recall,
Proposition 2.7. The mod–2 cohomology of the classifying space of C 2 n C_{2^n} C 2 n n ≥ 2 n\geq2 n ≥ 2
H ∗ ( B C 2 n ; F 2 ) ≅ F 2 [ x , y ] / ( x 2 ) , H^*(B C_{2^n};\mathbb{F}_2)\cong\mathbb{F}_2[x,y]/(x^2), H ∗ ( B C 2 n ; F 2 ) ≅ F 2 [ x , y ] / ( x 2 ) , where the degrees are ∣ x ∣ = 1 |x|=1 ∣ x ∣ = 1 [15 Proposition 4.5.1] ). The non-trivial action of the subalgebra A ( 1 ) \mathcal{A}(1) A ( 1 ) Sq 2 ( y ) = y 2 \operatorname{Sq}^2(y)=y^2 Sq 2 ( y ) = y 2
Remark 2.8. H ∗ ( B C 2 n ) H^*(B C_{2^n}) H ∗ ( B C 2 n ) F 2 \mathbb{F}_2 F 2 0 0 0 reduced mod 2 cohomology which we still denote by H ∗ ( B C 2 n ) H^*(B C_{2^n}) H ∗ ( B C 2 n )
Corollary 2.9. As a module over A ( 0 ) \mathcal{A}(0) A ( 0 ) E ( 1 ) \mathcal{E}(1) E ( 1 )
H ∗ ( B C 2 n ) ≅ ⨁ i ≥ 1 Σ i F 2 ≅ ⨁ i ≥ 1 F 2 { x i } . H^*(B C_{2^n})\cong\bigoplus_{i\geq1}\Sigma^i\mathbb{F}_2\cong\bigoplus_{i\geq1}\mathbb{F}_2\{x_i\}. H ∗ ( B C 2 n ) ≅ i ≥ 1 ⨁ Σ i F 2 ≅ i ≥ 1 ⨁ F 2 { x i } . Proof. Since Sq 1 ( x ) = 0 \operatorname{Sq}^{1}(x)=0 Sq 1 ( x ) = 0 Sq 1 ( y ) = 0 \operatorname{Sq}^{1}(y)=0 Sq 1 ( y ) = 0 A ( 0 ) \mathcal{A}(0) A ( 0 ) E ( 1 ) \mathcal{E}(1) E ( 1 ) x y i = x 2 i + 1 xy^{i}=x_{2i+1} x y i = x 2 i + 1 y i = x 2 i y^{i}=x_{2i} y i = x 2 i i ≥ 1 i\geq1 i ≥ 1
Corollary 2.10. As a module over A ( 1 ) \mathcal{A}(1) A ( 1 )
H ∗ ( B C 2 n ) ≅ F 2 ⊕ ⨁ i ≥ 1 ( Σ 4 i + 2 A ( 1 ) / / E ( 1 ) ⊕ Σ 4 i + 3 A ( 1 ) / / E ( 1 ) ) , H^{*}(BC_{2^{n}})\cong\mathbb{F}_{2}\oplus\bigoplus_{i\geq1}\left(\Sigma^{4i+2}\mathcal{A}(1)//\mathcal{E}(1)\oplus\Sigma^{4i+3}\mathcal{A}(1)//\mathcal{E}(1)\right), H ∗ ( B C 2 n ) ≅ F 2 ⊕ i ≥ 1 ⨁ ( Σ 4 i + 2 A ( 1 ) // E ( 1 ) ⊕ Σ 4 i + 3 A ( 1 ) // E ( 1 ) ) , where A ( 1 ) / / E ( 1 ) \mathcal{A}(1)//\mathcal{E}(1) A ( 1 ) // E ( 1 ) F 2 ⊕ Σ 2 F 2 \mathbb{F}_{2}\oplus\Sigma^{2}\mathbb{F}_{2} F 2 ⊕ Σ 2 F 2 Sq 2 \operatorname{Sq}^{2} Sq 2
Proof. Since Sq 2 ( y ) = y 2 \operatorname{Sq}^{2}(y)=y^{2} Sq 2 ( y ) = y 2 A ( 1 ) \mathcal{A}(1) A ( 1 ) H ∗ ( B C 2 n ) H^{*}(BC_{2^{n}}) H ∗ ( B C 2 n ) A ( 1 ) / / E ( 1 ) \mathcal{A}(1)//\mathcal{E}(1) A ( 1 ) // E ( 1 )
Analogously,
Theorem 2.11. The F F F B μ 2 n B\mu_{2^{n}} B μ 2 n x x x y y y ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) A ( 1 ) \mathcal{A}(1) A ( 1 )
Sq 1 ( x ) = 0 = Sq 1 ( y ) , Sq 2 ( x ) = 0 , Sq 2 ( y ) = y 2 . \operatorname{Sq}^{1}(x)=0=\operatorname{Sq}^{1}(y),\qquad\operatorname{Sq}^{2}(x)=0,\qquad\operatorname{Sq}^{2}(y)=y^{2}. Sq 1 ( x ) = 0 = Sq 1 ( y ) , Sq 2 ( x ) = 0 , Sq 2 ( y ) = y 2 . Together with the Cartan formula and the Adem relations, this determines the H ∗ ∗ ( B μ 2 n ; F 2 ) H^{**}(B\mu_{2^{n}};\mathbb{F}_{2}) H ∗∗ ( B μ 2 n ; F 2 ) A ( 1 ) \mathcal{A}(1) A ( 1 )
Proof. By [33 Proposition 6.6] , the F F F B μ 2 n B\mu_{2^{n}} B μ 2 n Z / 2 n \mathbb{Z}/2^{n} Z / 2 n F 2 F \mathbb{F}_{2}^{F} F 2 F
{ v i , u v i ∣ i ≥ 0 } . \{v^{i},uv^{i}\mid i\geq0\}. { v i , u v i ∣ i ≥ 0 } . Here v ∈ H 2 , 1 ( B μ 2 n ; Z ) v\in H^{2,1}(B\mu_{2^{n}};\mathbb{Z}) v ∈ H 2 , 1 ( B μ 2 n ; Z ) B μ 2 n B\mu_{2^{n}} B μ 2 n μ 2 n \mu_{2^{n}} μ 2 n u ∈ H 1 , 1 ( B μ 2 n ; Z / 2 n ) u\in H^{1,1}(B\mu_{2^{n}};\mathbb{Z}/2^{n}) u ∈ H 1 , 1 ( B μ 2 n ; Z / 2 n ) [33 Lemma 6.5] by the condition
δ 2 n ( u ) = v . \delta_{2^{n}}(u)=v. δ 2 n ( u ) = v . The connecting homomorphism
δ 2 n : H 1 , 1 ( B μ 2 n ; Z / 2 n ) ⟶ H 2 , 1 ( B μ 2 n ; Z ) \delta_{2^{n}}:H^{1,1}(B\mu_{2^{n}};\mathbb{Z}/2^{n})\longrightarrow H^{2,1}(B\mu_{2^{n}};\mathbb{Z}) δ 2 n : H 1 , 1 ( B μ 2 n ; Z / 2 n ) ⟶ H 2 , 1 ( B μ 2 n ; Z ) is induced by the short exact sequence
0 ⟶ Z → 2 n Z ⟶ Z / 2 n ⟶ 0. 0\longrightarrow\mathbb{Z}\xrightarrow{2^{n}}\mathbb{Z}\longrightarrow\mathbb{Z}/2^{n}\longrightarrow0. 0 ⟶ Z 2 n Z ⟶ Z / 2 n ⟶ 0. Let x ∈ H 1 , 1 ( B μ 2 n ; F 2 ) x\in H^{1,1}(B\mu_{2^{n}};\mathbb{F}_{2}) x ∈ H 1 , 1 ( B μ 2 n ; F 2 ) y ∈ H 2 , 1 ( B μ 2 n ; F 2 ) y\in H^{2,1}(B\mu_{2^{n}};\mathbb{F}_{2}) y ∈ H 2 , 1 ( B μ 2 n ; F 2 ) u u u v v v 2 2 2 Sq 1 \operatorname{Sq}^{1} Sq 1 x x x
H 1 , 1 ( B μ 2 n ; Z / 2 n ) → δ 2 n ( u ↦ v ) H 2 , 1 ( B μ 2 n ; Z ) H 1 , 1 ( B μ 2 n ; Z / 2 n ) ↓ mod 2 H 2 , 1 ( B μ 2 n ; Z ) ↓ 2 n − 1 H 1 , 1 ( B μ 2 n ; F 2 ) → δ 2 H 2 , 1 ( B μ 2 n ; Z ) . \begin{aligned}
H^{1,1}(B\mu_{2^{n}};\mathbb{Z}/2^{n})&\xrightarrow{\delta_{2^{n}}\ (u\mapsto v)}H^{2,1}(B\mu_{2^{n}};\mathbb{Z})\\
\mathrel{\phantom{H^{1,1}(B\mu_{2^{n}};\mathbb{Z}/2^{n})}}\downarrow\scriptstyle{\operatorname{mod}\,2}&\mathrel{\phantom{H^{2,1}(B\mu_{2^{n}};\mathbb{Z})}}\downarrow\scriptstyle{2^{n-1}}\\
H^{1,1}(B\mu_{2^{n}};\mathbb{F}_{2})&\xrightarrow{\delta_{2}}H^{2,1}(B\mu_{2^{n}};\mathbb{Z}).
\end{aligned} H 1 , 1 ( B μ 2 n ; Z / 2 n ) H 1 , 1 ( B μ 2 n ; Z / 2 n ) ↓ mod 2 H 1 , 1 ( B μ 2 n ; F 2 ) δ 2 n ( u ↦ v ) H 2 , 1 ( B μ 2 n ; Z ) H 2 , 1 ( B μ 2 n ; Z ) ↓ 2 n − 1 δ 2 H 2 , 1 ( B μ 2 n ; Z ) . Commutativity implies that δ 2 ( x ) = 2 n − 1 v \delta_{2}(x)=2^{n-1}v δ 2 ( x ) = 2 n − 1 v n ≥ 2 n\geq2 n ≥ 2 2 2 2
Sq 1 ( x ) = 0 ∈ H 2 , 1 ( B μ 2 n ; F 2 ) , \operatorname{Sq}^{1}(x)=0\in H^{2,1}(B\mu_{2^{n}};\mathbb{F}_{2}), Sq 1 ( x ) = 0 ∈ H 2 , 1 ( B μ 2 n ; F 2 ) , as claimed.
The remaining relations follow from comparison with the classical topological computation of H ∗ ( B C 2 n ; F 2 ) H^*(BC_{2^n};\mathbb{F}_2) H ∗ ( B C 2 n ; F 2 )
Sq 2 ( y ) = y 2 \operatorname{Sq}^2(y)=y^2 Sq 2 ( y ) = y 2 [33 Lemma 9.8] ,
Sq 2 ( x ) = 0 \operatorname{Sq}^2(x)=0 Sq 2 ( x ) = 0 [33 Lemma 9.9] and Sq 1 ( y ) = 0 \operatorname{Sq}^1(y)=0 Sq 1 ( y ) = 0 v v v A ( 1 ) \mathcal{A}(1) A ( 1 ) □ \square □
Corollary 2.12. As a module over A F ( 0 ) \mathcal{A}^{F}(0) A F ( 0 ) E F ( 1 ) \mathcal{E}^{F}(1) E F ( 1 )
H ∗ ∗ ( B μ 2 n ) ≅ ⨁ i ≥ 1 Σ i , ⌊ i + 1 2 ⌋ M 2 F ≅ ⨁ i ≥ 1 M 2 F x i ⌊ i + 1 2 ⌋ . H^{**}(B\mu_{2^n})\cong\bigoplus_{i\geq1}\Sigma^{i,\left\lfloor\frac{i+1}{2}\right\rfloor}\mathbb{M}^{F}_{2}\cong\bigoplus_{i\geq1}\mathbb{M}^{F}_{2}x_i^{\left\lfloor\frac{i+1}{2}\right\rfloor}. H ∗∗ ( B μ 2 n ) ≅ i ≥ 1 ⨁ Σ i , ⌊ 2 i + 1 ⌋ M 2 F ≅ i ≥ 1 ⨁ M 2 F x i ⌊ 2 i + 1 ⌋ . Corollary 2.13. As a module over A F ( 1 ) \mathcal{A}^{F}(1) A F ( 1 )
H ∗ ∗ ( B μ 2 n ) ≅ Σ M 2 F ⊕ ⨁ i ≥ 1 ( Σ 4 i + 2 A ( 1 ) / / E ( 1 ) ⊕ Σ 4 i + 3 A ( 1 ) / / E ( 1 ) ) H^{**}(B\mu_{2^n})\cong\Sigma\mathbb{M}^{F}_{2}\oplus\bigoplus_{i\geq1}\left(\Sigma^{4i+2}\mathcal{A}(1)//\mathcal{E}(1)\oplus\Sigma^{4i+3}\mathcal{A}(1)//\mathcal{E}(1)\right) H ∗∗ ( B μ 2 n ) ≅ Σ M 2 F ⊕ i ≥ 1 ⨁ ( Σ 4 i + 2 A ( 1 ) // E ( 1 ) ⊕ Σ 4 i + 3 A ( 1 ) // E ( 1 ) ) ≅ M 2 F x 1 ⊕ ⨁ i ≥ 1 ( A ( 1 ) / / E ( 1 ) x 4 i + 2 ⊕ A ( 1 ) / / E ( 1 ) x 4 i + 3 ) . \cong\mathbb{M}^{F}_{2}x_1\oplus\bigoplus_{i\geq1}\left(\mathcal{A}(1)//\mathcal{E}(1)x_{4i+2}\oplus\mathcal{A}(1)//\mathcal{E}(1)x_{4i+3}\right). ≅ M 2 F x 1 ⊕ i ≥ 1 ⨁ ( A ( 1 ) // E ( 1 ) x 4 i + 2 ⊕ A ( 1 ) // E ( 1 ) x 4 i + 3 ) . where A ( 1 ) / / E ( 1 ) \mathcal{A}(1)//\mathcal{E}(1) A ( 1 ) // E ( 1 ) M 2 F ⊕ Σ 2 M 2 F \mathbb{M}^{F}_{2}\oplus\Sigma^2\mathbb{M}^{F}_{2} M 2 F ⊕ Σ 2 M 2 F Sq 2 \operatorname{Sq}^2 Sq 2
Remark 2.14. Again, H ∗ ∗ ( B μ 2 n ) H^{**}(B\mu_{2^n}) H ∗∗ ( B μ 2 n ) M 2 F \mathbb{M}^{F}_{2} M 2 F ( 0 , 0 ) (0,0) ( 0 , 0 ) H ∗ ∗ ( B μ 2 n ) H^{**}(B\mu_{2^n}) H ∗∗ ( B μ 2 n )
Extension to other base fields
We conclude by explaining how the preceding calculations extend beyond the base fields C \mathbb{C} C R \mathbb{R} R K K K [9 Proposition 5.2.1] provides a zigzag of base-change functors between the motivic stable homotopy categories over K K K C \mathbb{C} C
Ext A K ( M 2 K , M 2 K ) ≅ Ext A C ( M 2 C , M 2 C ) . \operatorname{Ext}_{\mathcal{A}_K}(M^K_2,M^K_2)\cong\operatorname{Ext}_{\mathcal{A}_{\mathbb{C}}}(M^{\mathbb{C}}_2,M^{\mathbb{C}}_2). Ext A K ( M 2 K , M 2 K ) ≅ Ext A C ( M 2 C , M 2 C ) . Consequently, the calculations carried out over C \mathbb{C} C
Similarly, if K K K R \mathbb{R} R R \mathbb{R} R
General Idea
The goal of the paper is to compute the 2-completed effective and very effective algebraic K K K B μ 2 n B\mu_{2^n} B μ 2 n n > 1 n>1 n > 1 H Z ∗ B C 2 n H\mathbb{Z}_*BC_{2^n} H Z ∗ B C 2 n k u ∗ B C 2 n ku_*BC_{2^n} k u ∗ B C 2 n k o ∗ B C 2 n ko_*BC_{2^n} k o ∗ B C 2 n F F F [8 ] , [11 ] and [12 ] .
Our approach proceeds in two steps:
we determine some additive structure using the Atiyah–Hirzebruch spectral sequence;
(2) we then verify the results from (1) and resolve potential extensions describing the multiplicative structure via the Adams spectral sequence.
Let R = H Z R = H\mathbb{Z} R = H Z k g l kgl k g l k q kq k q
First Step: The Atiyah–Hirzebruch Spectral Sequence
The Atiyah–Hirzebruch spectral sequence for computing R ∗ ∗ B μ 2 n R_{**}B\mu_{2^n} R ∗∗ B μ 2 n
E 1 = ⨁ i -cells in B μ 2 n Σ i R ∗ ∗ ⇒ g r A H ∗ ( R ∗ ∗ B μ 2 n ) , E_1 = \bigoplus_{\substack{i\text{-cells}\\\text{in }B\mu_{2^n}}} \Sigma^i R_{**} \Rightarrow gr^*_{AH}(R_{**}B\mu_{2^n}), E 1 = i -cells in B μ 2 n ⨁ Σ i R ∗∗ ⇒ g r A H ∗ ( R ∗∗ B μ 2 n ) , where the differentials are determined by the attaching maps of the cellular decomposition of B μ 2 n B\mu_{2^n} B μ 2 n h n h^n h n d 1 d_1 d 1
d 1 ( x [ 2 i ] ) = h n ⋅ x [ 2 i − 1 ] , i ≥ 1 , d_1(x[2i]) = h^n \cdot x[2i-1], \qquad i \ge1, d 1 ( x [ 2 i ]) = h n ⋅ x [ 2 i − 1 ] , i ≥ 1 , where x [ i ] x[i] x [ i ] i i i x x x R ∗ ∗ R_{**} R ∗∗ d 2 d_2 d 2 η \eta η i ≡ 0 , 1 ( m o d 4 ) i \equiv0,1 \pmod4 i ≡ 0 , 1 ( mod 4 ) i ≡ 2 , 3 ( m o d 4 ) i \equiv2,3 \pmod4 i ≡ 2 , 3 ( mod 4 )
d 2 ( x [ i ] ) = η ⋅ x [ i − 2 ] , i ≡ 0 , 1 ( m o d 4 ) , i ≥ 4. d_2(x[i]) = \eta\cdot x[i-2], \qquad i \equiv0,1 \pmod4,\quad i \ge4. d 2 ( x [ i ]) = η ⋅ x [ i − 2 ] , i ≡ 0 , 1 ( mod 4 ) , i ≥ 4. The d 3 d_3 d 3
d 3 ( x [ i ] ) ∈ { ⟨ x , η , h n ⟩ [ i − 3 ] , i ≡ 0 ( m o d 4 ) , i > 0 , ⟨ x , h n , η ⟩ [ i − 3 ] , i ≡ 2 ( m o d 4 ) , i > 2. d_3(x[i]) \in
\begin{cases}
\langle x,\eta,h^n\rangle[i-3], & i \equiv0 \pmod4,\quad i>0,\\
\langle x,h^n,\eta\rangle[i-3], & i \equiv2 \pmod4,\quad i>2.
\end{cases} d 3 ( x [ i ]) ∈ { ⟨ x , η , h n ⟩ [ i − 3 ] , ⟨ x , h n , η ⟩ [ i − 3 ] , i ≡ 0 ( mod 4 ) , i > 0 , i ≡ 2 ( mod 4 ) , i > 2. These Toda brackets are defined because the relevant consecutive composites vanish. When i ≡ 0 ( m o d 4 ) i \equiv0 \pmod4 i ≡ 0 ( mod 4 ) x [ i ] x[i] x [ i ] E 3 E_3 E 3 d 2 d_2 d 2 η ⋅ x = 0 \eta\cdot x = 0 η ⋅ x = 0 i ≡ 2 ( m o d 4 ) i \equiv2 \pmod4 i ≡ 2 ( mod 4 ) d 1 d_1 d 1 h n ⋅ x = 0 h^n \cdot x = 0 h n ⋅ x = 0 h η = η h = 0 h\eta= \eta h = 0 h η = η h = 0 h 0 h 1 = 0 h_0h_1 = 0 h 0 h 1 = 0 [19 Theorem 1.4] ; see also [18 § 5.2, Table 2] .
In most cases, the spectral sequence collapses at the E 4 E_4 E 4 R ∗ ∗ ( B μ 2 n ) R_{**}(B\mu_{2^n}) R ∗∗ ( B μ 2 n ) M 2 F \mathbb{M}_2^F M 2 F
Remark 3.1. In all Atiyah–Hirzebruch spectral sequence charts, for example Figure 10 appearing in this paper, the axes are indexed by (stem, topological degree of cell). The motivic weight w w w
Figure 10. The E 1 E_1 E 1 E ∞ E_\infty E ∞ H Z ∗ ∗ R B μ 2 n HZ^{\mathbb R}_{**}B\mu_{2^n} H Z ∗∗ R B μ 2 n □ \square □ Z [ τ 2 ] \mathbb Z[\tau^2] Z [ τ 2 ] ∙ \bullet ∙ F 2 [ τ 2 ] \mathbb F_2[\tau^2] F 2 [ τ 2 ] ⋄ \diamond ⋄ Z / 2 n [ τ 2 ] \mathbb Z/2^n[\tau^2] Z / 2 n [ τ 2 ] h n h^n h n
Second Step: The Adams Spectral Sequence
The motivic Adams spectral sequence provides a primary computational tool for determining the completed motivic stable homotopy groups of a motivic spectrum.
For an F F F X X X H F 2 HF_2 H F 2
E 2 s , ( t , w ) = Ext A F s , ( t , w ) ( H ∗ ∗ ( X ) , M 2 F ) ⇒ π t − s , w ( X ) ( 2 , η ) ∧ . E_2^{s,(t,w)} = \operatorname{Ext}_{\mathcal{A}_F}^{s,(t,w)}\left(H^{**}(X),\mathbb{M}_2^F\right) \Rightarrow\pi_{t-s,w}(X)^{\wedge}_{(2,\eta)}. E 2 s , ( t , w ) = Ext A F s , ( t , w ) ( H ∗∗ ( X ) , M 2 F ) ⇒ π t − s , w ( X ) ( 2 , η ) ∧ . with differentials
d r : E r s , ( t , w ) ⟼ E r s + r , ( t + r − 1 , w ) . d_r\colon E_r^{s,(t,w)} \longmapsto E_r^{s+r,(t+r-1,w)}. d r : E r s , ( t , w ) ⟼ E r s + r , ( t + r − 1 , w ) . For a motivic cell spectrum X X X ( 2 , η ) (2,\eta) ( 2 , η ) X X X [24 Thm. 1] . Moreover, if cd 2 ( F ( i ) ) < ∞ \operatorname{cd}_{2}(F(i)) < \infty cd 2 ( F ( i )) < ∞ i = − 1 i = \sqrt{-1} i = − 1 ( 2 , η ) (2,\eta) ( 2 , η ) 2 2 2 C \mathbb{C} C R \mathbb{R} R 2 2 2
Here, the groups Ext A F s , ( t , w ) ( X , M 2 F ) \operatorname{Ext}_{\mathcal{A}_{F}}^{s,(t,w)}(X,\mathbb{M}_{2}^{F}) Ext A F s , ( t , w ) ( X , M 2 F )
t − s t-s t − s stem ;
s s s Adams filtration (the filtration degree in the cobar complex);
w w w motivic weight ;
( t , w ) (t,w) ( t , w ) internal degree (corresponding to the grading in A F \mathcal{A}^{F} A F
t − s − w t-s-w t − s − w Milnor–Witt degree (also referred to as the coweight).
Our goal is to compute the 2 2 2 R ∗ ∗ ( B μ 2 n ) R_{**}(B\mu_{2^{n}}) R ∗∗ ( B μ 2 n ) R = H Z R = HZ R = H Z k g l kgl k g l k q kq k q X = B μ 2 n ∧ R X = B\mu_{2^{n}} \wedge R X = B μ 2 n ∧ R
π ∗ ∗ ( B μ 2 n ∧ R ) ≅ R ∗ ∗ ( B μ 2 n ) , \pi_{**}(B\mu_{2^{n}} \wedge R) \cong R_{**}(B\mu_{2^{n}}), π ∗∗ ( B μ 2 n ∧ R ) ≅ R ∗∗ ( B μ 2 n ) , the motivic Adams spectral sequence takes the form
E 2 ∗ ∗ = Ext A F ∗ ∗ ∗ ( H ∗ ∗ ( B μ 2 n ∧ R ) ) → K u ¨ n n e t h ≅ Ext A F ∗ ∗ ∗ ( H ∗ ∗ ( B μ 2 n ) ⊗ H ∗ ∗ ( R ) ) ⇒ R ∗ ∗ ( B μ 2 n ) . E_{2}^{**} = \operatorname{Ext}_{\mathcal{A}_{F}}^{***}\left(H_{**}(B\mu_{2^{n}} \wedge R)\right) \xrightarrow[\mathrm{K\ddot{u}nneth}]{\cong} \operatorname{Ext}_{\mathcal{A}_{F}}^{***}\left(H_{**}(B\mu_{2^{n}}) \otimes H_{**}(R)\right) \Rightarrow R_{**}(B\mu_{2^{n}}). E 2 ∗∗ = Ext A F ∗∗∗ ( H ∗∗ ( B μ 2 n ∧ R ) ) ≅ K u ¨ nneth Ext A F ∗∗∗ ( H ∗∗ ( B μ 2 n ) ⊗ H ∗∗ ( R ) ) ⇒ R ∗∗ ( B μ 2 n ) . The isomorphism labeled “Künneth” follows from the Künneth theorem [17 Thm. 8.7] . In our case, the higher Tor terms vanish since H ∗ ∗ ( B μ 2 n ) H_{**}(B\mu_{2^{n}}) H ∗∗ ( B μ 2 n ) H ∗ ∗ H_{**} H ∗∗
When they occur, the differentials in this spectral sequence can often be understood by comparison with the Atiyah–Hirzebruch spectral sequence, which determines the ranks of the homology groups and detects certain necessary differentials, see for example in Proposition 4.6 . The module structure over Ext A F ∗ ∗ ∗ ( M 2 F ) \operatorname{Ext}_{\mathcal{A}_{F}}^{***}(\mathbb{M}_{2}^{F}) Ext A F ∗∗∗ ( M 2 F ) Theorem 6.2 .
Classical Computations
The connective K K K 2 n 2^{n} 2 n [11 , 12 ] . In this section, we use their results to illustrate the strategy described in the previous section. Note that techniques used in [11 , 12 ] to compute these homology groups differ from the ones used here.
Integral Homology
Proposition 4.1. The 2 2 2 B C 2 n BC_{2^{n}} B C 2 n
H Z k B C 2 n ≅ { Z / 2 n , k > 0 odd , 0 , k > 0 even . HZ_{k}BC_{2^{n}} \cong\begin{cases}
\mathbb{Z}/2^{n}, & k>0\text{ odd},\\
0, & k>0\text{ even}.
\end{cases} H Z k B C 2 n ≅ { Z / 2 n , 0 , k > 0 odd , k > 0 even . Proof. Consider the Atiyah–Hirzebruch spectral sequence (Subsection 3.1):
E 1 = ⨁ each cell i in B C 2 n Σ i H Z ∗ = ⨁ i ≥ 1 Z { x [ i ] } ⟹ g r A H ∗ H Z ∗ B C 2 n . E_1=\bigoplus_{\text{each cell }i\text{ in }BC_{2^n}}\Sigma^i HZ_*=\bigoplus_{i\geq1}\mathbb{Z}\{x[i]\}\Longrightarrow gr^*_{AH}HZ_*BC_{2^n}. E 1 = each cell i in B C 2 n ⨁ Σ i H Z ∗ = i ≥ 1 ⨁ Z { x [ i ]} ⟹ g r A H ∗ H Z ∗ B C 2 n . where H Z ∗ = Z HZ_*=\mathbb{Z} H Z ∗ = Z x [ i ] x[i] x [ i ] x ∈ H Z ∗ x\in HZ_* x ∈ H Z ∗ i i i d 1 d_1 d 1 Figure 1 are
Figure 1. The E 1 E_1 E 1 E ∞ E_\infty E ∞ g r ∗ H Z ∗ B C 2 n gr^*HZ_*BC_{2^n} g r ∗ H Z ∗ B C 2 n □ \square □ Z \mathbb{Z} Z ∙ \bullet ∙ Z / 2 n \mathbb{Z}/2^n Z / 2 n d 1 d_1 d 1
d 1 ( x [ 2 i ] ) = 2 n x [ 2 i − 1 ] , i ≥ 1. d_1(x[2i])=2^n x[2i-1],\qquad i\geq1. d 1 ( x [ 2 i ]) = 2 n x [ 2 i − 1 ] , i ≥ 1. The spectral sequence collapses at E 2 E_2 E 2 E 2 E_2 E 2 H Z ∗ B C 2 n HZ_*BC_{2^n} H Z ∗ B C 2 n
g r A H ∗ H Z k B C 2 n ≅ { Z / 2 n , k > 0 odd , 0 , k > 0 even . gr^*_{AH}HZ_kBC_{2^n}\cong
\begin{cases}
\mathbb{Z}/2^n, & k>0\text{ odd},\\
0, & k>0\text{ even}.
\end{cases} g r A H ∗ H Z k B C 2 n ≅ { Z / 2 n , 0 , k > 0 odd , k > 0 even . There are no possible hidden extensions, so this completely determines the homology groups.
For reference, we compute the same result using the Adams spectral sequence (Subsection 3.2):
E 2 ∗ ∗ ≅ Ext A ∗ ∗ ( H ∗ ( B C 2 n ) ⊗ H ∗ ( H Z ) ) ⟹ H Z ∗ ( B C 2 n ) . E_2^{**}\cong\operatorname{Ext}_{\mathcal{A}}^{**}\left(H^*(BC_{2^n})\otimes H^*(HZ)\right)\Longrightarrow HZ_*(BC_{2^n}). E 2 ∗∗ ≅ Ext A ∗∗ ( H ∗ ( B C 2 n ) ⊗ H ∗ ( H Z ) ) ⟹ H Z ∗ ( B C 2 n ) . Using the isomorphism H ∗ ( H Z ) ≅ A / / A ( 0 ) H^*(HZ)\cong\mathcal{A}//\mathcal{A}(0) H ∗ ( H Z ) ≅ A // A ( 0 ) Proposition 2.5 and applying a change-of-rings isomorphism [29 ] , the E 2 E_2 E 2
E 2 ∗ ∗ ≅ Ext A ( 0 ) ∗ ∗ ( H ∗ ( B C 2 n ) ) . E_2^{**}\cong\operatorname{Ext}_{\mathcal{A}(0)}^{**}\left(H^*(BC_{2^n})\right). E 2 ∗∗ ≅ Ext A ( 0 ) ∗∗ ( H ∗ ( B C 2 n ) ) . From the Sq 1 \operatorname{Sq}^1 Sq 1 H ∗ ( B C 2 n ) H^*(BC_{2^n}) H ∗ ( B C 2 n ) A ( 0 ) \mathcal{A}(0) A ( 0 )
H ∗ ( B C 2 n ) ≅ ⨁ i ≥ 1 Σ i F 2 , H^*(BC_{2^n})\cong\bigoplus_{i\geq1}\Sigma^i\mathbb{F}_2, H ∗ ( B C 2 n ) ≅ i ≥ 1 ⨁ Σ i F 2 , where F 2 \mathbb{F}_2 F 2 A ( 0 ) \mathcal{A}(0) A ( 0 )
Ext A ( 0 ) ∗ ∗ ( F 2 , F 2 ) ≅ F 2 [ h 0 ] , \operatorname{Ext}_{\mathcal{A}(0)}^{**}(\mathbb{F}_2,\mathbb{F}_2)\cong\mathbb{F}_2[h_0], Ext A ( 0 ) ∗∗ ( F 2 , F 2 ) ≅ F 2 [ h 0 ] , where ∣ h 0 ∣ = ( 0 , 1 ) \lvert h_0\rvert=(0,1) ∣ h 0 ∣ = ( 0 , 1 ) ( t − s , s ) (t-s,s) ( t − s , s ) [8 Remark 5.4.4] ).
The E 2 E_2 E 2
E 2 ∗ ∗ ≅ ⨁ k ≥ 1 Ext A ( 0 ) ∗ ∗ ( Σ k F 2 ) ≅ ⨁ k ≥ 1 Σ k Ext A ( 0 ) ∗ ∗ ( F 2 ) ≅ ⨁ k ≥ 1 F 2 [ h 0 ] { x k } . E_2^{**} \cong\bigoplus_{k\geq1}\operatorname{Ext}_{\mathcal{A}(0)}^{**}(\Sigma^k\mathbb{F}_2) \cong\bigoplus_{k\geq1}\Sigma^k\operatorname{Ext}_{\mathcal{A}(0)}^{**}(\mathbb{F}_2) \cong\bigoplus_{k\geq1}\mathbb{F}_2[h_0]\{x_k\}. E 2 ∗∗ ≅ k ≥ 1 ⨁ Ext A ( 0 ) ∗∗ ( Σ k F 2 ) ≅ k ≥ 1 ⨁ Σ k Ext A ( 0 ) ∗∗ ( F 2 ) ≅ k ≥ 1 ⨁ F 2 [ h 0 ] { x k } . Here, x k x_k x k Σ k F 2 \Sigma^k\mathbb{F}_2 Σ k F 2
Comparing with the ranks obtained from the Atiyah–Hirzebruch spectral sequence forces
d n ( x 2 k ) = h 0 n x 2 k − 1 . d_n(x_{2k})=h_0^n x_{2k-1}. d n ( x 2 k ) = h 0 n x 2 k − 1 . All remaining differentials are then determined by h 0 h_0 h 0 E n + 1 E_{n+1} E n + 1
H Z k B C 2 n ≅ { Z / 2 n , k > 0 odd , 0 , otherwise . HZ_kBC_{2^n}\cong
\begin{cases}
\mathbb{Z}/2^n, & k>0\text{ odd},\\
0, & \text{otherwise}.
\end{cases} H Z k B C 2 n ≅ { Z / 2 n , 0 , k > 0 odd , otherwise . Connective real K-theory
4.2. Connective real K-theory. To compute the k o ko k o B C 2 n BC_{2^n} B C 2 n k o ∗ ko_* k o ∗
Remark 4.2. The groups k o ∗ ko_* k o ∗
k o ∗ ≅ Z [ η , α , β ] ( η 3 , 2 η , η α , α 2 − 4 β ) . ko_* \cong\frac{\mathbb{Z}[\eta,\alpha,\beta]}{(\eta^3,2\eta,\eta\alpha,\alpha^2-4\beta)}. k o ∗ ≅ ( η 3 , 2 η , η α , α 2 − 4 β ) Z [ η , α , β ] . Proposition 4.3. The associated graded of k o ∗ B C 2 n ko_*BC_{2^n} k o ∗ B C 2 n
gr A H ∗ k o ∗ ( B C 2 n ) ≅ { Z / 2 n ⊕ ( Z / 2 n − 1 ) ⊕ 2 m , k = 8 m + 1 > 0 , Z / 2 , k = 8 m + 2 > 0 , ( Z / 2 ) ⊕ 2 m + 2 ⊕ ( Z / 2 n ) ⊕ 2 m + 1 , k = 8 m + 3 > 0 , ( Z / 2 n − 1 ) ⊕ 2 m + 2 , k = 8 m + 5 > 0 , ( Z / 2 n ⊕ F 2 ) ⊕ 2 m + 2 , k = 8 m + 7 > 0 , 0 , otherwise . \operatorname{gr}^{*}_{AH}ko_*(BC_{2^n})\cong
\begin{cases}
\mathbb{Z}/2^n\oplus(\mathbb{Z}/2^{n-1})^{\oplus2m}, & k=8m+1>0,\\
\mathbb{Z}/2, & k=8m+2>0,\\
(\mathbb{Z}/2)^{\oplus2m+2}\oplus(\mathbb{Z}/2^n)^{\oplus2m+1}, & k=8m+3>0,\\
(\mathbb{Z}/2^{n-1})^{\oplus2m+2}, & k=8m+5>0,\\
(\mathbb{Z}/2^n\oplus\mathbb{F}_2)^{\oplus2m+2}, & k=8m+7>0,\\
0, & \text{otherwise}.
\end{cases} gr A H ∗ k o ∗ ( B C 2 n ) ≅ ⎩ ⎨ ⎧ Z / 2 n ⊕ ( Z / 2 n − 1 ) ⊕ 2 m , Z /2 , ( Z /2 ) ⊕ 2 m + 2 ⊕ ( Z / 2 n ) ⊕ 2 m + 1 , ( Z / 2 n − 1 ) ⊕ 2 m + 2 , ( Z / 2 n ⊕ F 2 ) ⊕ 2 m + 2 , 0 , k = 8 m + 1 > 0 , k = 8 m + 2 > 0 , k = 8 m + 3 > 0 , k = 8 m + 5 > 0 , k = 8 m + 7 > 0 , otherwise . Proof. Consider the Atiyah–Hirzebruch spectral sequence Subsection 3.1
E 1 = ⨁ each cell i in B C 2 n Σ i k o ∗ ⇒ g r A H ∗ k o ∗ ( B C 2 n ) . E_1=\bigoplus_{\substack{\text{each cell }i\\ \text{in }BC_{2^n}}}\Sigma^i ko_* \Rightarrow gr^*_{AH}ko_*(BC_{2^n}). E 1 = each cell i in B C 2 n ⨁ Σ i k o ∗ ⇒ g r A H ∗ k o ∗ ( B C 2 n ) . We have
d 1 ( x [ 2 i ] ) = 2 n ⋅ x [ 2 i − 1 ] , i > 0. d_1(x[2i])=2^n\cdot x[2i-1],\qquad i>0. d 1 ( x [ 2 i ]) = 2 n ⋅ x [ 2 i − 1 ] , i > 0. Moreover, a cell in degree i ≡ 0 , 1 ( m o d 4 ) i\equiv0,1\pmod{4} i ≡ 0 , 1 ( mod 4 ) i ≡ 2 , 3 ( m o d 4 ) i\equiv2,3\pmod{4} i ≡ 2 , 3 ( mod 4 )
d 2 ( x [ i ] ) = η ⋅ x [ i − 2 ] , i ≡ 0 , 1 ( m o d 4 ) , i ≥ 4. d_2(x[i])=\eta\cdot x[i-2],\qquad i\equiv0,1\pmod{4},\qquad i\geq4. d 2 ( x [ i ]) = η ⋅ x [ i − 2 ] , i ≡ 0 , 1 ( mod 4 ) , i ≥ 4. Finally, using the relation 2 n − 1 α ∈ ⟨ η 2 , η , 2 n ⟩ 2^{n-1}\alpha\in\langle\eta^2,\eta,2^n\rangle 2 n − 1 α ∈ ⟨ η 2 , η , 2 n ⟩ k o ∗ ko_* k o ∗
d 3 ( η 2 [ 4 k ] ) = 2 n − 1 α [ 4 k − 3 ] , k ≥ 1. d_3(\eta^2[4k])=2^{n-1}\alpha[4k-3],\qquad k\geq1. d 3 ( η 2 [ 4 k ]) = 2 n − 1 α [ 4 k − 3 ] , k ≥ 1. Figure 3. The E 1 E_1 E 1 E 2 E_2 E 2 E 3 E_3 E 3 g r A H ∗ k o ∗ ( B C 2 n ) gr^*_{AH}ko_*(BC_{2^n}) g r A H ∗ k o ∗ ( B C 2 n ) Z \mathbb{Z} Z Z / 2 n \mathbb{Z}/2^n Z / 2 n Z / 2 n − 1 \mathbb{Z}/2^{n-1} Z / 2 n − 1 F 2 \mathbb{F}_2 F 2 η \eta η d 1 d_1 d 1 d 2 d_2 d 2 d 3 d_3 d 3
Observe that all the target classes of higher differentials lie in even degrees and are of the form η ⋅ β k [ 1 ] \eta\cdot\beta^k[1] η ⋅ β k [ 1 ] η \eta η η \eta η η \eta η k o ko k o Figure 4 .
Figure 4. The chart is represents g r A H k o ∗ ∗ ( B C 2 n ) \mathrm{gr}^{*}_{AHko_*}(BC_{2^n}) gr A H k o ∗ ∗ ( B C 2 n ) □ \square □ Z \mathbb{Z} Z ⋄ \diamond ⋄ Z / 2 n − 1 \mathbb{Z}/2^{n-1} Z / 2 n − 1 ▹ \triangleright ▹ Z / 2 n − 1 \mathbb{Z}/2^{n-1} Z / 2 n − 1 ∙ \bullet ∙ F 2 \mathbb{F}_2 F 2 η \eta η
To resolve the hidden extension problems, we now consider the Adams spectral sequence. By Proposition 2.5, we have the isomorphism H ∗ ∗ ( k o ) ≅ A / / A ( 1 ) H^{**}(ko) \cong\mathcal{A}//\mathcal{A}(1) H ∗∗ ( k o ) ≅ A // A ( 1 ) E 2 E_2 E 2
E 2 ∗ ∗ ≅ Ext A ( 1 ) ∗ ∗ ( H ∗ ( B C 2 n ) , F 2 ) . E_2^{**} \cong\operatorname{Ext}_{\mathcal{A}(1)}^{**}(H^*(BC_{2^n}),\mathbb{F}_2). E 2 ∗∗ ≅ Ext A ( 1 ) ∗∗ ( H ∗ ( B C 2 n ) , F 2 ) . From Corollary 2.10, we have the decomposition
H ∗ ( B C 2 n ) ≅ ⋁ k ≥ 0 ( Σ 4 k + 2 A ( 1 ) / / E ( 1 ) ⊕ Σ 4 k + 3 A ( 1 ) / / E ( 1 ) ) . H^*(BC_{2^n}) \cong\bigvee_{k\geq0}\left(\Sigma^{4k+2}\mathcal{A}(1)//\mathcal{E}(1)\oplus\Sigma^{4k+3}\mathcal{A}(1)//\mathcal{E}(1)\right). H ∗ ( B C 2 n ) ≅ k ≥ 0 ⋁ ( Σ 4 k + 2 A ( 1 ) // E ( 1 ) ⊕ Σ 4 k + 3 A ( 1 ) // E ( 1 ) ) . Applying Ext A ( 1 ) ∗ ∗ ( − , F 2 ) \operatorname{Ext}_{\mathcal{A}(1)}^{**}(-,\mathbb{F}_2) Ext A ( 1 ) ∗∗ ( − , F 2 )
E 2 ∗ ∗ ≅ Σ Ext A ( 1 ) ∗ ∗ ( F 2 ) ⊕ ⋁ k ≥ 0 ( Σ 4 k + 2 Ext A ( 1 ) ∗ ∗ ( A ( 1 ) / / E ( 1 ) ) ⊕ Σ 4 k + 3 Ext A ( 1 ) ∗ ∗ ( A ( 1 ) / / E ( 1 ) ) ) E_2^{**}\cong\Sigma\operatorname{Ext}_{\mathcal{A}(1)}^{**}(\mathbb{F}_2)\oplus\bigvee_{k\geq0}\left(\Sigma^{4k+2}\operatorname{Ext}_{\mathcal{A}(1)}^{**}(\mathcal{A}(1)//\mathcal{E}(1))\oplus\Sigma^{4k+3}\operatorname{Ext}_{\mathcal{A}(1)}^{**}(\mathcal{A}(1)//\mathcal{E}(1))\right) E 2 ∗∗ ≅ Σ Ext A ( 1 ) ∗∗ ( F 2 ) ⊕ k ≥ 0 ⋁ ( Σ 4 k + 2 Ext A ( 1 ) ∗∗ ( A ( 1 ) // E ( 1 )) ⊕ Σ 4 k + 3 Ext A ( 1 ) ∗∗ ( A ( 1 ) // E ( 1 )) ) ≅ Σ Ext A ( 1 ) ∗ ∗ ( F 2 ) ⊕ ⋁ k ≥ 0 ( Σ 4 k + 2 Ext E ( 1 ) ∗ ∗ ( F 2 ) ⊕ Σ 4 k + 3 Ext E ( 1 ) ∗ ∗ ( F 2 ) ) \cong\Sigma\operatorname{Ext}_{\mathcal{A}(1)}^{**}(\mathbb{F}_2)\oplus\bigvee_{k\geq0}\left(\Sigma^{4k+2}\operatorname{Ext}_{\mathcal{E}(1)}^{**}(\mathbb{F}_2)\oplus\Sigma^{4k+3}\operatorname{Ext}_{\mathcal{E}(1)}^{**}(\mathbb{F}_2)\right) ≅ Σ Ext A ( 1 ) ∗∗ ( F 2 ) ⊕ k ≥ 0 ⋁ ( Σ 4 k + 2 Ext E ( 1 ) ∗∗ ( F 2 ) ⊕ Σ 4 k + 3 Ext E ( 1 ) ∗∗ ( F 2 ) ) ≅ Ext A ( 1 ) ∗ ∗ ( F 2 ) { x 1 } ⊕ ⋁ k ≥ 0 ( Ext E ( 1 ) ∗ ∗ ( F 2 ) { x 4 k + 2 } ⊕ Ext E ( 1 ) ∗ ∗ ( F 2 ) { x 4 k + 3 } ) . \cong\operatorname{Ext}_{\mathcal{A}(1)}^{**}(\mathbb{F}_2)\{x_1\}\oplus\bigvee_{k\geq0}\left(\operatorname{Ext}_{\mathcal{E}(1)}^{**}(\mathbb{F}_2)\{x_{4k+2}\}\oplus\operatorname{Ext}_{\mathcal{E}(1)}^{**}(\mathbb{F}_2)\{x_{4k+3}\}\right). ≅ Ext A ( 1 ) ∗∗ ( F 2 ) { x 1 } ⊕ k ≥ 0 ⋁ ( Ext E ( 1 ) ∗∗ ( F 2 ) { x 4 k + 2 } ⊕ Ext E ( 1 ) ∗∗ ( F 2 ) { x 4 k + 3 } ) . Recall from, for example, [8 Section 4.5 and 4.6] ,
Ext E ( 1 ) ∗ ∗ ( F 2 ) ≅ F 2 [ h 0 , v 1 ] , and Ext A ( 1 ) ∗ ∗ ( F 2 ) ≅ F 2 [ h 0 , h 1 , a , b ] ( h 1 3 , h 0 h 1 , h 1 a , a 2 − h 0 2 b ) \operatorname{Ext}_{\mathcal{E}(1)}^{**}(\mathbb{F}_2)\cong\mathbb{F}_2[h_0,v_1],\qquad\text{and}\qquad\operatorname{Ext}_{\mathcal{A}(1)}^{**}(\mathbb{F}_2)\cong\frac{\mathbb{F}_2[h_0,h_1,a,b]}{(h_1^3,h_0h_1,h_1a,a^2-h_0^2b)} Ext E ( 1 ) ∗∗ ( F 2 ) ≅ F 2 [ h 0 , v 1 ] , and Ext A ( 1 ) ∗∗ ( F 2 ) ≅ ( h 1 3 , h 0 h 1 , h 1 a , a 2 − h 0 2 b ) F 2 [ h 0 , h 1 , a , b ] where ∣ h 0 ∣ = ( 0 , 1 ) |h_0|=(0,1) ∣ h 0 ∣ = ( 0 , 1 ) ∣ h 1 ∣ = ( 1 , 1 ) |h_1|=(1,1) ∣ h 1 ∣ = ( 1 , 1 ) ∣ a ∣ = ( 4 , 3 ) |a|=(4,3) ∣ a ∣ = ( 4 , 3 ) ∣ b ∣ = ( 8 , 4 ) |b|=(8,4) ∣ b ∣ = ( 8 , 4 ) ∣ v 1 ∣ = ( 2 , 1 ) |v_1|=(2,1) ∣ v 1 ∣ = ( 2 , 1 ) E 2 E_2 E 2 Figure 5 .
Figure 5. The E n E_n E n d n d_n d n k o k B C 2 n ko_kBC_2^n k o k B C 2 n F 2 \mathbb{F}_2 F 2 h 0 h_0 h 0 v 1 v_1 v 1 E n E_n E n H ∗ ( B C 2 n ) H^*(BC_2^n) H ∗ ( B C 2 n ) E ( 1 ) E(1) E ( 1 )
Before calculating the differentials, we describe the multiplicative structure on the E 2 E_2 E 2 k o ∗ ( B C 2 n ) ko_*(BC_{2^n}) k o ∗ ( B C 2 n ) [10 Theorem 13.36] . It is induced by the inclusion A ( 0 ) ↪ A ( 1 ) \mathcal{A}(0)\hookrightarrow\mathcal{A}(1) A ( 0 ) ↪ A ( 1 ) E 2 E_2 E 2 H Z ∗ ( B C 2 n ) HZ_*(BC_{2^n}) H Z ∗ ( B C 2 n )
Proposition 4.6. The nontrivial differentials in the Adams spectral sequence converging to k o ∗ ( B C 2 n ) ko_*(BC_{2^n}) k o ∗ ( B C 2 n ) n ≥ 2 n\geq2 n ≥ 2
( 1 ) d n ( x 2 ) = h 0 n x 1 , ( 2 ) d n ( v 1 x 2 ) = h 0 n + 1 x 3 , ( 3 ) d n ( x 6 ) = h 0 n − 1 v 1 x 3 , ( 4 ) d n ( v 1 x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 , ( 5 ) d n ( x 10 ) = h 0 n − 1 v 1 x 7 , ( 6 ) d n ( v 1 x 10 ) = h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 , ( 7 ) d n ( x 14 ) = h 0 n − 1 v 1 x 11 . \begin{aligned}
(1)\quad d_n(x_2)&=h_0^n x_1, &\qquad(2)\quad d_n(v_1x_2)&=h_0^{n+1}x_3,\\
(3)\quad d_n(x_6)&=h_0^{n-1}v_1x_3, &\qquad(4)\quad d_n(v_1x_6)&=h_0^{n+1}x_7+h_0^{n-1}v_1^2x_3,\\
(5)\quad d_n(x_{10})&=h_0^{n-1}v_1x_7, &\qquad(6)\quad d_n(v_1x_{10})&=h_0^{n+1}x_{11}+h_0^{n-1}v_1^2x_7,\\
(7)\quad d_n(x_{14})&=h_0^{n-1}v_1x_{11}.
\end{aligned} ( 1 ) d n ( x 2 ) ( 3 ) d n ( x 6 ) ( 5 ) d n ( x 10 ) ( 7 ) d n ( x 14 ) = h 0 n x 1 , = h 0 n − 1 v 1 x 3 , = h 0 n − 1 v 1 x 7 , = h 0 n − 1 v 1 x 11 . ( 2 ) d n ( v 1 x 2 ) ( 4 ) d n ( v 1 x 6 ) ( 6 ) d n ( v 1 x 10 ) = h 0 n + 1 x 3 , = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 , = h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 , In the case n = 2 n=2 n = 2 k ≥ 4 k\geq4 k ≥ 4
( 8 ) d 2 ( x 4 k + 2 ) = h 0 v 1 x 4 k − 1 , ( 9 ) d 2 ( v 1 2 x 4 k + 2 ) = h 0 v 1 3 x 4 k − 1 , ( 10 ) d 2 ( v 1 x 4 k + 2 ) = h 0 3 x 4 k + 3 + h 0 v 1 2 x 4 k − 1 . \begin{aligned}
(8)\quad d_2(x_{4k+2})&=h_0v_1x_{4k-1}, &\qquad(9)\quad d_2(v_1^2x_{4k+2})&=h_0v_1^3x_{4k-1},\\
(10)\quad d_2(v_1x_{4k+2})&=h_0^3x_{4k+3}+h_0v_1^2x_{4k-1}.
\end{aligned} ( 8 ) d 2 ( x 4 k + 2 ) ( 10 ) d 2 ( v 1 x 4 k + 2 ) = h 0 v 1 x 4 k − 1 , = h 0 3 x 4 k + 3 + h 0 v 1 2 x 4 k − 1 . ( 9 ) d 2 ( v 1 2 x 4 k + 2 ) = h 0 v 1 3 x 4 k − 1 , These differentials are v 1 2 v_1^2 v 1 2
Proof. (1) and (2). The first two differentials are forced by comparison with the Atiyah–Hirzebruch spectral sequence for stems 1–4.
(3). Using the multiplicative relations from Remark 4.5 and the Leibniz rule, we obtain
d n ( h 0 x 6 ) = d n ( x 2 ⋅ v 1 x 2 ) = h 0 n x 1 ⋅ v 1 x 2 + x 2 ⋅ h 0 n + 1 x 3 = h 0 n ⋅ v 1 x 3 . d_n(h_0x_6)=d_n(x_2\cdot v_1x_2)=h_0^n x_1\cdot v_1x_2+x_2\cdot h_0^{n+1}x_3=h_0^n\cdot v_1x_3. d n ( h 0 x 6 ) = d n ( x 2 ⋅ v 1 x 2 ) = h 0 n x 1 ⋅ v 1 x 2 + x 2 ⋅ h 0 n + 1 x 3 = h 0 n ⋅ v 1 x 3 . Since the left-hand side is h 0 d n ( x 6 ) h_0d_n(x_6) h 0 d n ( x 6 ) h 0 h_0 h 0
d n ( x 6 ) = h 0 n − 1 ⋅ v 1 x 3 . d_n(x_6) = h_0^{n-1}\cdot v_1x_3. d n ( x 6 ) = h 0 n − 1 ⋅ v 1 x 3 . (4). Comparison with the Atiyah–Hirzebruch spectral sequence shows that the differential on v 1 x 6 v_1x_6 v 1 x 6 x 7 x_7 x 7 k o 7 ( B C 2 n ) ko_7(BC_{2^n}) k o 7 ( B C 2 n ) 2 n + 3 2^{n+3} 2 n + 3 2 n + 3 2^{n+3} 2 n + 3 d n ( v 1 x 6 ) d_n(v_1x_6) d n ( v 1 x 6 ) h 0 n − 1 v 1 2 x 3 h_0^{n-1}v_1^2x_3 h 0 n − 1 v 1 2 x 3 2 n + 3 2^{n+3} 2 n + 3
d n ( v 1 x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 . d_n(v_1x_6) = h_0^{n+1}x_7+h_0^{n-1}v_1^2x_3. d n ( v 1 x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 . (5). Similarly, using the relation h 0 x 10 = x 2 ⋅ v 1 x 6 h_0x_{10}=x_2\cdot v_1x_6 h 0 x 10 = x 2 ⋅ v 1 x 6 x 2 ⋅ x 3 = 0 x_2\cdot x_3=0 x 2 ⋅ x 3 = 0 x 2 ⋅ x 7 = 0 x_2\cdot x_7=0 x 2 ⋅ x 7 = 0
d n ( h 0 x 10 ) = d n ( x 2 ⋅ v 1 x 6 ) = h 0 n x 1 ⋅ v 1 x 6 + x 2 ⋅ ( h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 ) = h 0 n ⋅ v 1 x 7 . d_n(h_0x_{10})=d_n(x_2\cdot v_1x_6)=h_0^n x_1\cdot v_1x_6+x_2\cdot\left(h_0^{n+1}x_7+h_0^{n-1}v_1^2x_3\right)=h_0^n\cdot v_1x_7. d n ( h 0 x 10 ) = d n ( x 2 ⋅ v 1 x 6 ) = h 0 n x 1 ⋅ v 1 x 6 + x 2 ⋅ ( h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 ) = h 0 n ⋅ v 1 x 7 . Therefore,
d n ( x 10 ) = h 0 n − 1 v 1 x 7 . d_n(x_{10})=h_0^{n-1}v_1x_7. d n ( x 10 ) = h 0 n − 1 v 1 x 7 . (6). For v 1 x 10 v_1x_{10} v 1 x 10 h 0 v 1 x 10 = v 1 x 2 ⋅ v 1 x 6 h_0v_1x_{10}=v_1x_2\cdot v_1x_6 h 0 v 1 x 10 = v 1 x 2 ⋅ v 1 x 6
d n ( h 0 v 1 x 10 ) = d n ( v 1 x 2 ⋅ v 1 x 6 ) = h 0 n + 2 x 11 + h 0 n v 1 2 x 7 . d_n(h_0v_1x_{10})=d_n(v_1x_2\cdot v_1x_6)=h_0^{n+2}x_{11}+h_0^nv_1^2x_7. d n ( h 0 v 1 x 10 ) = d n ( v 1 x 2 ⋅ v 1 x 6 ) = h 0 n + 2 x 11 + h 0 n v 1 2 x 7 . Again, multiplication by h 0 h_0 h 0
d n ( v 1 x 10 ) = h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 . d_n(v_1x_{10})=h_0^{n+1}x_{11}+h_0^{n-1}v_1^2x_7. d n ( v 1 x 10 ) = h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 . (7). Finally, using h 0 x 14 = x 2 ⋅ v 1 x 10 h_0x_{14}=x_2\cdot v_1x_{10} h 0 x 14 = x 2 ⋅ v 1 x 10
d n ( h 0 x 14 ) = d n ( x 2 ⋅ v 1 x 10 ) = h 0 n x 1 ⋅ v 1 x 10 + x 2 ⋅ ( h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 ) = h 0 n v 1 x 11 , d_n(h_0x_{14})=d_n(x_2\cdot v_1x_{10})=h_0^n x_1\cdot v_1x_{10}+x_2\cdot\left(h_0^{n+1}x_{11}+h_0^{n-1}v_1^2x_7\right)=h_0^nv_1x_{11}, d n ( h 0 x 14 ) = d n ( x 2 ⋅ v 1 x 10 ) = h 0 n x 1 ⋅ v 1 x 10 + x 2 ⋅ ( h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 ) = h 0 n v 1 x 11 , and hence
d n ( x 14 ) = h 0 n − 1 v 1 x 11 . d_n(x_{14})=h_0^{n-1}v_1x_{11}. d n ( x 14 ) = h 0 n − 1 v 1 x 11 . (8), (9), and (10). For n = 2 n=2 n = 2 k o ∗ ( B C 4 ) ko_*(BC_4) k o ∗ ( B C 4 )
Bruner and Greenlees compute the groups k o ∗ ( B C 4 ) ko_*(BC_4) k o ∗ ( B C 4 ) [11 Example 7.3.3] , obtaining
k o k ( B C 4 ) ≅ { Z / 2 2 m + 1 ⊕ Z / 2 , k = 8 m + 1 , m ≥ 0 , Z / 2 , k = 8 m + 2 , m ≥ 0 , Z / 2 4 m + 3 ⊕ Z / 2 2 m + 1 , k = 8 m + 3 , m ≥ 0 , Z / 2 2 m + 2 , k = 8 m + 5 , m ≥ 0 , Z / 2 4 m + 5 ⊕ Z / 2 2 m + 1 , k = 8 m + 7 , m ≥ 0 , 0 , otherwise . ko_k(BC_4)\cong
\begin{cases}
\mathbb{Z}/2^{2m+1}\oplus\mathbb{Z}/2, & k=8m+1,\quad m\ge0,\\
\mathbb{Z}/2, & k=8m+2,\quad m\ge0,\\
\mathbb{Z}/2^{4m+3}\oplus\mathbb{Z}/2^{2m+1}, & k=8m+3,\quad m\ge0,\\
\mathbb{Z}/2^{2m+2}, & k=8m+5,\quad m\ge0,\\
\mathbb{Z}/2^{4m+5}\oplus\mathbb{Z}/2^{2m+1}, & k=8m+7,\quad m\ge0,\\
0, & \text{otherwise}.
\end{cases} k o k ( B C 4 ) ≅ ⎩ ⎨ ⎧ Z / 2 2 m + 1 ⊕ Z /2 , Z /2 , Z / 2 4 m + 3 ⊕ Z / 2 2 m + 1 , Z / 2 2 m + 2 , Z / 2 4 m + 5 ⊕ Z / 2 2 m + 1 , 0 , k = 8 m + 1 , m ≥ 0 , k = 8 m + 2 , m ≥ 0 , k = 8 m + 3 , m ≥ 0 , k = 8 m + 5 , m ≥ 0 , k = 8 m + 7 , m ≥ 0 , otherwise . Comparing these groups with the E 2 E_2 E 2 Figure 4 , the above d 2 d_2 d 2 E ∞ E_\infty E ∞ k o ko k o d 2 ( v 1 x 10 ) d_2(v_1x_{10}) d 2 ( v 1 x 10 ) h 0 3 x 11 h_0^3x_{11} h 0 3 x 11 Z / 2 7 \mathbb{Z}/2^7 Z / 2 7 k o 11 ( B C 4 ) ko_{11}(BC_4) k o 11 ( B C 4 ) E 2 E_2 E 2 E ∞ E_\infty E ∞
In particular, the differentials are linear over h 0 h_0 h 0 a a a a x = h 0 v 1 2 x ax=h_0v_1^2x a x = h 0 v 1 2 x
h 0 d n ( v 1 2 x ) = d n ( h 0 v 1 2 x ) = d n ( a x ) = a d n ( x ) = h 0 v 1 2 d n ( x ) . h_0d_n(v_1^2x)=d_n(h_0v_1^2x)=d_n(ax)=a\,d_n(x)=h_0v_1^2d_n(x). h 0 d n ( v 1 2 x ) = d n ( h 0 v 1 2 x ) = d n ( a x ) = a d n ( x ) = h 0 v 1 2 d n ( x ) . By h 0 h_0 h 0
d n ( v 1 2 x ) = v 1 2 d n ( x ) . d_n(v_1^2x)=v_1^2d_n(x). d n ( v 1 2 x ) = v 1 2 d n ( x ) . Therefore, all the displayed differentials are v 1 2 v_1^2 v 1 2
There are no further nontrivial differentials through stem 14 for arbitrary n ≥ 2 n \ge2 n ≥ 2 n = 2 n=2 n = 2
Remark 4.8. Observe that it is enough to determine the differentials on the indecomposable classes v 1 x 2 k − 2 v_1x_{2k-2} v 1 x 2 k − 2 k > 1 k>1 k > 1 E 2 E_2 E 2 k < 4 k < 4 k < 4 v 1 x 2 k − 2 v_1x_{2k-2} v 1 x 2 k − 2 k > 3 k > 3 k > 3
Figure 6. The E ∞ E_\infty E ∞ k o ∗ B C 2 n ko_*BC_{2^n} k o ∗ B C 2 n F 2 \mathbb{F}_2 F 2 h 0 h_0 h 0
Connective complex K-theory
Finally, we compute the k u ku k u B C 2 n BC_{2^n} B C 2 n k o ko k o k o → k u ko \to ku k o → k u k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n ) k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n )
Proposition 4.9. The associated graded of k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n )
gr A H ∗ k u ∗ ( B C 2 n ) ≅ { ( Z / 2 n ) ⊕ ( j + 1 ) , k = 2 j + 1 , j ≥ 0 , 0 , otherwise . \operatorname{gr}_{AH}^{*} ku_*(BC_{2^n}) \cong
\begin{cases}
(\mathbb{Z}/2^n)^{\oplus(j+1)}, & k = 2j+1,\quad j \ge0,\\
0, & \text{otherwise}.
\end{cases} gr A H ∗ k u ∗ ( B C 2 n ) ≅ { ( Z / 2 n ) ⊕ ( j + 1 ) , 0 , k = 2 j + 1 , j ≥ 0 , otherwise . Proof. Consider the Atiyah–Hirzebruch spectral sequence from Subsection 3.1:
E 1 = ⨁ cells of dimension i in B C 2 n Σ i k u ∗ ⟹ gr A H ∗ k u ∗ ( B C 2 n ) , E_1 = \bigoplus_{\substack{\text{cells of dimension }i\\\text{in }BC_{2^n}}} \Sigma^i ku_* \Longrightarrow\operatorname{gr}_{AH}^{*} ku_*(BC_{2^n}), E 1 = cells of dimension i in B C 2 n ⨁ Σ i k u ∗ ⟹ gr A H ∗ k u ∗ ( B C 2 n ) , @page 19/46 starts_mid=1 ends_mid=1 k < 4 k < 4 k < 4 v 1 x 2 k − 2 v_1x_{2k-2} v 1 x 2 k − 2 k > 3 k > 3 k > 3
Figure 6 (PDF p. 19). The E ∞ E_\infty E ∞ k o ∗ B C 2 n ko_*BC_{2^n} k o ∗ B C 2 n F 2 \mathbb{F}_2 F 2 h 0 h_0 h 0
Connective complex K-theory
Finally, we compute the k u ku k u B C 2 n BC_{2^n} B C 2 n k o ko k o k o → k u ko \to ku k o → k u k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n ) k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n )
Proposition 4.9. The associated graded of k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n )
gr A H ∗ k u ∗ ( B C 2 n ) ≅ { ( Z / 2 n ) ⊕ ( j + 1 ) , k = 2 j + 1 , j ≥ 0 , 0 , otherwise . \operatorname{gr}_{AH}^{*} ku_*(BC_{2^n}) \cong
\begin{cases}
(\mathbb{Z}/2^n)^{\oplus(j+1)}, & k = 2j+1,\quad j \ge0,\\
0, & \text{otherwise}.
\end{cases} gr A H ∗ k u ∗ ( B C 2 n ) ≅ { ( Z / 2 n ) ⊕ ( j + 1 ) , 0 , k = 2 j + 1 , j ≥ 0 , otherwise . Proof. Consider the Atiyah–Hirzebruch spectral sequence from Subsection 3.1:
E 1 = ⨁ cells of dimension i in B C 2 n Σ i k u ∗ ⟹ gr A H ∗ k u ∗ ( B C 2 n ) , E_1 = \bigoplus_{\substack{\text{cells of dimension }i\\\text{in }BC_{2^n}}} \Sigma^i ku_* \Longrightarrow\operatorname{gr}_{AH}^{*} ku_*(BC_{2^n}), E 1 = cells of dimension i in B C 2 n ⨁ Σ i k u ∗ ⟹ gr A H ∗ k u ∗ ( B C 2 n ) , where k u 2 j = Z ku_{2j}=\mathbb{Z} k u 2 j = Z j ≥ 0 j\geq0 j ≥ 0 0 0 0 d 1 d_1 d 1
d 1 ( x [ 2 i ] ) = 2 n x [ 2 i − 1 ] , i ≥ 1 , d_1(x[2i])=2^n x[2i-1],\qquad i\geq1, d 1 ( x [ 2 i ]) = 2 n x [ 2 i − 1 ] , i ≥ 1 , where x [ i ] x[i] x [ i ] x ∈ H Z ∗ x\in H\mathbb{Z}_* x ∈ H Z ∗ i i i
Figure 7. The E 1 E_1 E 1 E ∞ E_\infty E ∞ gr A H ∗ k u ∗ ( B C 2 n ) \operatorname{gr}^{*}_{AH}ku_*(BC_{2^n}) gr A H ∗ k u ∗ ( B C 2 n ) □ \square □ Z \mathbb{Z} Z ∙ \bullet ∙ Z / 2 n \mathbb{Z}/2^n Z / 2 n
The spectral sequence collapses at E 2 E_2 E 2 □ \square □
To determine the k u ∗ ku_* k u ∗
E 2 = Ext A ∗ ∗ ( H ∗ ( B C 2 n ∧ k u ) ) ≅ K u ¨ nneth Ext A ∗ ∗ ( H ∗ ( B C 2 n ) ⊗ H ∗ ( k u ) ) ⇒ k u ∗ ( B C 2 n ) . E_2=\operatorname{Ext}^{**}_{\mathcal{A}}\left(H^*(BC_{2^n}\wedge ku)\right)\underset{\text{Künneth}}{\cong}\operatorname{Ext}^{**}_{\mathcal{A}}\left(H^*(BC_{2^n})\otimes H^*(ku)\right)\Rightarrow ku_*(BC_{2^n}). E 2 = Ext A ∗∗ ( H ∗ ( B C 2 n ∧ k u ) ) K u ¨ nneth ≅ Ext A ∗∗ ( H ∗ ( B C 2 n ) ⊗ H ∗ ( k u ) ) ⇒ k u ∗ ( B C 2 n ) . Recall from Proposition 2.5 that, as A \mathcal{A} A
H ∗ ( k u ) ≅ A / / E ( 1 ) . H^*(ku)\cong\mathcal{A}//E(1). H ∗ ( k u ) ≅ A // E ( 1 ) . Applying the change-of-rings isomorphism, we may rewrite the E 2 E_2 E 2
E 2 ≅ Ext E ( 1 ) ∗ ∗ ( H ∗ ( B C 2 n ) ) . E_2\cong\operatorname{Ext}^{**}_{E(1)}\left(H^*(BC_{2^n})\right). E 2 ≅ Ext E ( 1 ) ∗∗ ( H ∗ ( B C 2 n ) ) . Since both Sq 1 \operatorname{Sq}^1 Sq 1 Q 1 Q_1 Q 1 H ∗ ( B C 2 n ) H^*(BC_{2^n}) H ∗ ( B C 2 n )
E 2 ≅ ⨁ k ≥ 1 Ext E ( 1 ) ∗ ∗ ( F 2 ) { x ~ k } ≅ ⨁ k ≥ 1 F 2 [ h 0 , v 1 ] { x ~ k } . E_2\cong\bigoplus_{k\geq1}\operatorname{Ext}^{**}_{E(1)}(\mathbb{F}_2)\{\widetilde{x}_k\}\cong\bigoplus_{k\geq1}\mathbb{F}_2[h_0,v_1]\{\widetilde{x}_k\}. E 2 ≅ k ≥ 1 ⨁ Ext E ( 1 ) ∗∗ ( F 2 ) { x k } ≅ k ≥ 1 ⨁ F 2 [ h 0 , v 1 ] { x k } . The differentials in this spectral sequence are determined by those in the Adams spectral sequence for k o ∗ ( B C 2 n ) ko_*(BC_{2^n}) k o ∗ ( B C 2 n ) E ( 1 ) → j A ( 1 ) E(1)\xrightarrow{j}\mathcal{A}(1) E ( 1 ) j A ( 1 )
Proposition 4.10. On the E 2 E_2 E 2 E ( 1 ) → j A ( 1 ) E(1)\xrightarrow{j}\mathcal{A}(1) E ( 1 ) j A ( 1 )
Ext A ( 1 ) ∗ ∗ ( H ∗ ( B C 2 n ) ) → j ∗ Ext E ( 1 ) ∗ ∗ ( H ∗ ( B C 2 n ) ) , \operatorname{Ext}^{**}_{\mathcal{A}(1)}\left(H^*(BC_{2^n})\right)\xrightarrow{j^*}\operatorname{Ext}^{**}_{E(1)}\left(H^*(BC_{2^n})\right), Ext A ( 1 ) ∗∗ ( H ∗ ( B C 2 n ) ) j ∗ Ext E ( 1 ) ∗∗ ( H ∗ ( B C 2 n ) ) , sends the indicated classes for k ≥ 0 k\geq0 k ≥ 0 i = 1 , 4 k + 2 i=1,4k+2 i = 1 , 4 k + 2 4 k + 3 4k+3 4 k + 3
More generally, for k , j ≥ 0 k,j \ge0 k , j ≥ 0 ε ∈ { 0 , 1 } \varepsilon\in\{0,1\} ε ∈ { 0 , 1 }
v 1 2 j + ε x 4 k + 2 ⟼ v 1 2 j + ε x ~ 4 k + 2 + ε h 0 v 1 2 j x ~ 4 k + 4 v 1 2 j + ε x 4 k + 3 ⟼ v 1 2 j + ε x ~ 4 k + 3 + ε h 0 v 1 2 j x ~ 4 k + 5 . \begin{aligned}
v_1^{2j+\varepsilon}x_{4k+2}&\longmapsto v_1^{2j+\varepsilon}\widetilde{x}_{4k+2}+\varepsilon h_0v_1^{2j}\widetilde{x}_{4k+4}\\
v_1^{2j+\varepsilon}x_{4k+3}&\longmapsto v_1^{2j+\varepsilon}\widetilde{x}_{4k+3}+\varepsilon h_0v_1^{2j}\widetilde{x}_{4k+5}.
\end{aligned} v 1 2 j + ε x 4 k + 2 v 1 2 j + ε x 4 k + 3 ⟼ v 1 2 j + ε x 4 k + 2 + ε h 0 v 1 2 j x 4 k + 4 ⟼ v 1 2 j + ε x 4 k + 3 + ε h 0 v 1 2 j x 4 k + 5 . Proof. Consider the two chains of inclusions
A ( 0 ) ↪ i E ( 1 ) ↪ j A ( 1 ) and E ( Q 1 ) ↪ k E ( 1 ) ↪ j A ( 1 ) , \mathcal{A}(0)\mathbin{\overset{i}{\hookrightarrow}}\mathcal{E}(1)\mathbin{\overset{j}{\hookrightarrow}}\mathcal{A}(1)
\qquad\text{and}\qquad
\mathbb{E}(Q_1)\mathbin{\overset{k}{\hookrightarrow}}\mathcal{E}(1)\mathbin{\overset{j}{\hookrightarrow}}\mathcal{A}(1), A ( 0 ) ↪ i E ( 1 ) ↪ j A ( 1 ) and E ( Q 1 ) ↪ k E ( 1 ) ↪ j A ( 1 ) , where E ( Q 1 ) \mathbb{E}(Q_1) E ( Q 1 ) Q 1 Q_1 Q 1 E ( 1 ) \mathcal{E}(1) E ( 1 ) A ( 0 ) \mathcal{A}(0) A ( 0 ) E ( Q 1 ) \mathbb{E}(Q_1) E ( Q 1 ) x i A ( 0 ) x_i^{\mathcal{A}(0)} x i A ( 0 ) x i E ( 1 ) x_i^{\mathcal{E}(1)} x i E ( 1 ) x i E ( Q 1 ) x_i^{\mathbb{E}(Q_1)} x i E ( Q 1 ) A ( 0 ) \mathcal{A}(0) A ( 0 ) E ( 1 ) \mathcal{E}(1) E ( 1 ) E ( Q 1 ) \mathbb{E}(Q_1) E ( Q 1 )
The images of the map induced by the inclusion A ( 0 ) ↪ j ∘ i A ( 1 ) \mathcal{A}(0)\mathbin{\overset{j\circ i}{\hookrightarrow}}\mathcal{A}(1) A ( 0 ) ↪ j ∘ i A ( 1 ) [10 Theorem 13.30] . The induced map i ∗ i^* i ∗ E ( 1 ) \mathcal{E}(1) E ( 1 ) A ( 0 ) \mathcal{A}(0) A ( 0 )
h 0 E ( 1 ) ⟼ h 0 A ( 0 ) and v 1 E ( 1 ) ⟼ 0. h_0^{\mathcal{E}(1)}\longmapsto h_0^{\mathcal{A}(0)}
\qquad\text{and}\qquad
v_1^{\mathcal{E}(1)}\longmapsto0. h 0 E ( 1 ) ⟼ h 0 A ( 0 ) and v 1 E ( 1 ) ⟼ 0. The map k ∗ k^* k ∗ E ( 1 ) \mathcal{E}(1) E ( 1 ) E ( Q 1 ) \mathbb{E}(Q_1) E ( Q 1 )
h 0 E ( 1 ) ⟼ 0 and v 1 E ( 1 ) ⟼ v 1 E ( Q 1 ) , h_0^{\mathcal{E}(1)}\longmapsto0
\qquad\text{and}\qquad
v_1^{\mathcal{E}(1)}\longmapsto v_1^{\mathbb{E}(Q_1)}, h 0 E ( 1 ) ⟼ 0 and v 1 E ( 1 ) ⟼ v 1 E ( Q 1 ) , while the map ( k ∗ ∘ j ∗ ) (k^* \circ j^*) ( k ∗ ∘ j ∗ ) A ( 1 ) \mathcal{A}(1) A ( 1 ) E ( Q 1 ) \mathbb{E}(Q_1) E ( Q 1 )
h 0 A ( 1 ) ⟼ 0 and v 1 x i A ( 1 ) ⟼ v 1 E ( Q 1 ) x i E ( Q 1 ) , for i = 1 , 4 k + 2 , 4 k + 3. h_0^{\mathcal{A}(1)}\longmapsto0
\qquad\text{and}\qquad
v_1x_i^{\mathcal{A}(1)}\longmapsto v_1^{\mathbb{E}(Q_1)}x_i^{\mathbb{E}(Q_1)},
\qquad\text{for }i=1,4k+2,4k+3. h 0 A ( 1 ) ⟼ 0 and v 1 x i A ( 1 ) ⟼ v 1 E ( Q 1 ) x i E ( Q 1 ) , for i = 1 , 4 k + 2 , 4 k + 3. These give the relevant images summarized in the following table, where M M M H ∗ ( B C 2 n ) H^*(BC_{2^n}) H ∗ ( B C 2 n ) i = 1 , 4 k + 2 , 4 k + 3 i=1,4k+2,4k+3 i = 1 , 4 k + 2 , 4 k + 3 r = 4 k + 2 , 4 k + 3 r=4k+2,4k+3 r = 4 k + 2 , 4 k + 3
It remains to determine p , q ∈ F 2 p,q \in\mathbb{F}_2 p , q ∈ F 2 v 1 x r v_1x_r v 1 x r E ( 1 ) \mathcal{E}(1) E ( 1 ) A ( 0 ) \mathcal{A}(0) A ( 0 ) p h 0 x r + 2 A ( 0 ) p h_0x_{r+2}^{\mathcal{A}(0)} p h 0 x r + 2 A ( 0 ) i ∗ ∘ j ∗ ( v 1 x r ) i^* \circ j^*(v_1x_r) i ∗ ∘ j ∗ ( v 1 x r ) h 0 x r + 2 A ( 0 ) h_0x_{r+2}^{\mathcal{A}(0)} h 0 x r + 2 A ( 0 ) p = 1 p=1 p = 1 E ( Q 1 ) \mathrm{E}(\mathrm{Q}_{1}) E ( Q 1 ) q v 1 x r E ( Q 1 ) qv_{1}x_{r}^{\mathrm{E}(\mathrm{Q}_{1})} q v 1 x r E ( Q 1 ) k ∗ ∘ j ∗ ( v 1 x r ) k^{*}\circ j^{*}(v_{1}x_{r}) k ∗ ∘ j ∗ ( v 1 x r ) v 1 x r E ( Q 1 ) v_{1}x_{r}^{\mathrm{E}(\mathrm{Q}_{1})} v 1 x r E ( Q 1 ) q = 1 q=1 q = 1
v 1 x r ⟼ h 0 x r + 2 E ( 1 ) + v 1 x r E ( 1 ) . v_{1}x_{r}\longmapsto h_{0}x_{r+2}^{\mathcal{E}(1)}+v_{1}x_{r}^{\mathcal{E}(1)}. v 1 x r ⟼ h 0 x r + 2 E ( 1 ) + v 1 x r E ( 1 ) . Finally, the map is a a a a a a h 0 v 1 2 h_{0}v_{1}^{2} h 0 v 1 2 a a a h 0 h_{0} h 0 h 0 d ( v 1 2 x ) = h 0 v 1 2 d ( x ) h_{0}d(v_{1}^{2}x)=h_{0}v_{1}^{2}d(x) h 0 d ( v 1 2 x ) = h 0 v 1 2 d ( x ) h 0 h_{0} h 0 v 1 2 v_{1}^{2} v 1 2
v 1 2 j x r ⟼ v 1 2 j x r E ( 1 ) , v 1 2 j + 1 x r ⟼ v 1 2 j + 1 x r E ( 1 ) + h 0 v 1 2 j x r + 2 E ( 1 ) . v_{1}^{2j}x_{r}\longmapsto v_{1}^{2j}x_{r}^{\mathcal{E}(1)},\qquad v_{1}^{2j+1}x_{r}\longmapsto v_{1}^{2j+1}x_{r}^{\mathcal{E}(1)}+h_{0}v_{1}^{2j}x_{r+2}^{\mathcal{E}(1)}. v 1 2 j x r ⟼ v 1 2 j x r E ( 1 ) , v 1 2 j + 1 x r ⟼ v 1 2 j + 1 x r E ( 1 ) + h 0 v 1 2 j x r + 2 E ( 1 ) . Taking r = 4 k + 2 r=4k+2 r = 4 k + 2 r = 4 k + 3 r=4k+3 r = 4 k + 3 x i E ( 1 ) x_{i}^{\mathcal{E}(1)} x i E ( 1 ) x ~ i \widetilde{x}_{i} x i □ \square □
Remark 4.11. Another important point is the multiplicative structure on the E 2 E_{2} E 2 k u ∗ ( B C 2 n ) \mathrm{ku}_{*}(BC_{2^{n}}) ku ∗ ( B C 2 n ) E 2 E_{2} E 2 H Z ∗ ( B C 2 n ) \mathrm{HZ}_{*}(BC_{2^{n}}) HZ ∗ ( B C 2 n ) [10 Proposition 13.18] . The relations are
x ~ i 2 = 0 , x ~ i ⋅ x ~ j = ( i + j i ) x ~ i + j . \widetilde{x}_{i}^{2}=0,\qquad\widetilde{x}_{i}\cdot\widetilde{x}_{j}=\binom{i+j}{i}\widetilde{x}_{i+j}. x i 2 = 0 , x i ⋅ x j = ( i i + j ) x i + j . Proposition 4.12. The nontrivial differentials in the Adams spectral sequence converging to k u ∗ ( B C 2 n ) \mathrm{ku}_{*}(BC_{2^{n}}) ku ∗ ( B C 2 n )
For n = 2 n=2 n = 2
d 2 ( x ~ 2 ) = h 0 2 x ~ 1 and for k > 1 , d 2 ( x ~ 2 k ) = h 0 2 x ~ 2 k − 1 + h 0 v 1 x ~ 2 k − 3 . d_{2}(\widetilde{x}_{2})=h_{0}^{2}\widetilde{x}_{1}\qquad\text{and}\qquad\text{for }k>1,\ d_{2}(\widetilde{x}_{2k})=h_{0}^{2}\widetilde{x}_{2k-1}+h_{0}v_{1}\widetilde{x}_{2k-3}. d 2 ( x 2 ) = h 0 2 x 1 and for k > 1 , d 2 ( x 2 k ) = h 0 2 x 2 k − 1 + h 0 v 1 x 2 k − 3 . For n > 2 n>2 n > 2
d n ( x ~ 2 ) = h 0 n x ~ 1 , and for 2 ≤ k ≤ 7 , d n ( x ~ 2 k ) = h 0 n x ~ 2 k − 1 + h 0 n − 1 v 1 x ~ 2 k − 3 . d_{n}(\widetilde{x}_{2})=h_{0}^{n}\widetilde{x}_{1},\qquad\text{and}\qquad\text{for }2\leq k\leq7,\ d_{n}(\widetilde{x}_{2k})=h_{0}^{n}\widetilde{x}_{2k-1}+h_{0}^{n-1}v_{1}\widetilde{x}_{2k-3}. d n ( x 2 ) = h 0 n x 1 , and for 2 ≤ k ≤ 7 , d n ( x 2 k ) = h 0 n x 2 k − 1 + h 0 n − 1 v 1 x 2 k − 3 . These differentials are h 0 h_{0} h 0 v 1 v_{1} v 1
Proof. First, for all n n n
k u 1 ( B C 2 n ) ≅ Z / 2 n . \mathrm{ku}_{1}(BC_{2^{n}})\cong\mathbb{Z}/2^{n}. ku 1 ( B C 2 n ) ≅ Z / 2 n . To recover this group, the Adams spectral sequence must support the differential
d n ( x ~ 2 ) = h 0 n x ~ 1 . d_{n}(\widetilde{x}_{2})=h_{0}^{n}\widetilde{x}_{1}. d n ( x 2 ) = h 0 n x 1 . Since v 1 v_{1} v 1 v 1 v_{1} v 1
d n ( v 1 x ~ 2 ) = h 0 n v 1 x ~ 1 . d_{n}(v_{1}\widetilde{x}_{2})=h_{0}^{n}v_{1}\widetilde{x}_{1}. d n ( v 1 x 2 ) = h 0 n v 1 x 1 . Under the map of Adams spectral sequences induced by the inclusion E ( 1 ) ↪ A ( 1 ) \mathcal{E}(1)\hookrightarrow\mathcal{A}(1) E ( 1 ) ↪ A ( 1 ) Proposition 4.10 , we have
v 1 x 2 ⟼ h 0 x ~ 4 + v 1 x ~ 2 . v_{1}x_{2}\longmapsto h_{0}\widetilde{x}_{4}+v_{1}\widetilde{x}_{2}. v 1 x 2 ⟼ h 0 x 4 + v 1 x 2 . By Proposition 4.6 , d n k u ( v 1 x 2 ) = h 0 n + 1 x 3 d_{n}^{\mathrm{ku}}(v_{1}x_{2})=h_{0}^{n+1}x_{3} d n ku ( v 1 x 2 ) = h 0 n + 1 x 3
h 0 d n k u ( x ~ 4 ) + v 1 d n k u ( x ~ 2 ) = h 0 n + 1 x ~ 3 . h_{0}d_{n}^{\mathrm{ku}}(\widetilde{x}_{4})+v_{1}d_{n}^{\mathrm{ku}}(\widetilde{x}_{2})=h_{0}^{n+1}\widetilde{x}_{3}. h 0 d n ku ( x 4 ) + v 1 d n ku ( x 2 ) = h 0 n + 1 x 3 . Since multiplication by h 0 h_{0} h 0
d n k u ( x ~ 4 ) = h 0 n x ~ 3 + h 0 n − 1 v 1 x ~ 1 . d_{n}^{\mathrm{ku}}(\widetilde{x}_{4})=h_{0}^{n}\widetilde{x}_{3}+h_{0}^{n-1}v_{1}\widetilde{x}_{1}. d n ku ( x 4 ) = h 0 n x 3 + h 0 n − 1 v 1 x 1 . Next, using x ~ 6 = x ~ 2 x ~ 4 \widetilde{x}_{6}=\widetilde{x}_{2}\widetilde{x}_{4} x 6 = x 2 x 4
d n k u ( x ~ 6 ) = h 0 n x ~ 1 x ~ 4 + x ~ 2 ( h 0 n x ~ 3 + h 0 n − 1 v 1 x ~ 1 ) = h 0 n x ~ 5 + h 0 n − 1 v 1 x ~ 3 . d_{n}^{\mathrm{ku}}(\widetilde{x}_{6})=h_{0}^{n}\widetilde{x}_{1}\widetilde{x}_{4}+\widetilde{x}_{2}\left(h_{0}^{n}\widetilde{x}_{3}+h_{0}^{n-1}v_{1}\widetilde{x}_{1}\right)=h_{0}^{n}\widetilde{x}_{5}+h_{0}^{n-1}v_{1}\widetilde{x}_{3}. d n ku ( x 6 ) = h 0 n x 1 x 4 + x 2 ( h 0 n x 3 + h 0 n − 1 v 1 x 1 ) = h 0 n x 5 + h 0 n − 1 v 1 x 3 . Similarly, v 1 x 6 ⟼ h 0 x ~ 8 + v 1 x ~ 6 v_1x_6 \longmapsto h_0\widetilde{x}_8+v_1\widetilde{x}_6 v 1 x 6 ⟼ h 0 x 8 + v 1 x 6 Proposition 4.6 ,
d n k o ( v 1 x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 . d_n^{ko}(v_1x_6)=h_0^{n+1}x_7+h_0^{n-1}v_1^2x_3. d n k o ( v 1 x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 . Naturality gives h 0 d n k u ( x ~ 8 ) + v 1 d n k u ( x ~ 6 ) = h 0 n + 1 x ~ 7 + h 0 n − 1 v 1 2 x 3 h_0d_n^{ku}(\widetilde{x}_8)+v_1d_n^{ku}(\widetilde{x}_6)=h_0^{n+1}\widetilde{x}_7+h_0^{n-1}v_1^2x_3 h 0 d n k u ( x 8 ) + v 1 d n k u ( x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 x ~ 6 \widetilde{x}_6 x 6 h 0 h_0 h 0
d n k u ( x ~ 8 ) = h 0 n x ~ 7 + h 0 n − 1 v 1 x ~ 5 . d_n^{ku}(\widetilde{x}_8)=h_0^n\widetilde{x}_7+h_0^{n-1}v_1\widetilde{x}_5. d n k u ( x 8 ) = h 0 n x 7 + h 0 n − 1 v 1 x 5 . Repeating these two arguments determines the remaining differentials in the stated range. □
Figure 8. The E n E_n E n d n d_n d n k u ∗ B C 2 n ku_*BC_{2^n} k u ∗ B C 2 n F 2 \mathbb{F}_2 F 2 h 0 h_0 h 0 κ i \kappa_i κ i h 0 κ i + v 1 κ i − 1 h_0\kappa_i+v_1\kappa_{i-1} h 0 κ i + v 1 κ i − 1 x 3 ↦ v 1 x 1 + h 0 x 3 x_3\mapsto v_1x_1+h_0x_3 x 3 ↦ v 1 x 1 + h 0 x 3 v 1 v_1 v 1
Remark 4.13. Again, observe that it is enough to determine the differentials on the indecomposable classes x ~ 2 k \widetilde{x}_{2^k} x 2 k E 2 E_2 E 2 k o ko k o k > 3 k>3 k > 3
These differentials propagate by h 0 h_0 h 0 v 1 v_1 v 1 E n + 1 E_{n+1} E n + 1 E n + 1 E_{n+1} E n + 1
There are higher differentials, so the module structure can be read directly from the Adams E ∞ E_\infty E ∞ Figure 9 .
Figure 9. The E ∞ E_\infty E ∞ k u ∗ B C 2 n ku_*BC_{2^n} k u ∗ B C 2 n F 2 \mathbb{F}_2 F 2 h 0 h_0 h 0 v 1 v_1 v 1
Integral Motivic Homology
We begin with computing the integral homology H Z ∗ ∗ ( B μ 2 n ) HZ_{**}(B\mu_{2^n}) H Z ∗∗ ( B μ 2 n ) H ∗ ∗ ( B μ 2 n ) H^{**}(B\mu_{2^n}) H ∗∗ ( B μ 2 n ) A F ( 0 ) \mathcal{A}^{F}(0) A F ( 0 )
Integral motivic homology over algebraically closed fields
Note that it is enough to compute the integral motivic homology over C \mathbb{C} C Proposition 2.6 that, analogously, we have
H ∗ ∗ ( H Z C ) ≅ A C / / A C ( 0 ) , H^{**}(HZ^{\mathbb{C}}) \cong\mathcal{A}^{\mathbb{C}} // \mathcal{A}^{\mathbb{C}}(0), H ∗∗ ( H Z C ) ≅ A C // A C ( 0 ) , and from Corollary 2.12 we have
H ∗ ∗ B μ 2 n ≅ ⨁ k ≥ 1 Σ k , ⌊ k + 1 2 ⌋ M 2 C . H^{**}B\mu_{2^n} \cong\bigoplus_{k\geq1}\Sigma^{k,\left\lfloor\frac{k+1}{2}\right\rfloor}\mathbb{M}^{\mathbb{C}}_2. H ∗∗ B μ 2 n ≅ k ≥ 1 ⨁ Σ k , ⌊ 2 k + 1 ⌋ M 2 C . Using these to simplify the E 2 E_2 E 2 H Z ∗ ∗ C B μ 2 n HZ^{\mathbb{C}}_{**}B\mu_{2^n} H Z ∗∗ C B μ 2 n
E 2 ≅ Ext A C ( 0 ) ∗ ∗ ∗ ( H ∗ ∗ B μ 2 n ) ≅ ⨁ k ≥ 1 M 2 C [ h 0 ] { x k } ⟹ H Z ∗ ∗ C B μ 2 n , E_2 \cong\operatorname{Ext}_{\mathcal{A}^{\mathbb{C}}(0)}^{***}\left(H^{**}B\mu_{2^n}\right) \cong\bigoplus_{k\geq1}\mathbb{M}^{\mathbb{C}}_2[h_0]\{x_k\} \Longrightarrow HZ^{\mathbb{C}}_{**}B\mu_{2^n}, E 2 ≅ Ext A C ( 0 ) ∗∗∗ ( H ∗∗ B μ 2 n ) ≅ k ≥ 1 ⨁ M 2 C [ h 0 ] { x k } ⟹ H Z ∗∗ C B μ 2 n , where ∣ x k ∣ = ( k , ⌊ k + 1 2 ⌋ ) \lvert x_k\rvert=\left(k,\left\lfloor\frac{k+1}{2}\right\rfloor\right) ∣ x k ∣ = ( k , ⌊ 2 k + 1 ⌋ )
The Betti realization functor from motivic spectra over C \mathbb{C} C
d n ( x 2 k ) = h 0 n x 2 k − 1 , d_n(x_{2k})=h_0^n x_{2k-1}, d n ( x 2 k ) = h 0 n x 2 k − 1 , and are h 0 h_0 h 0
Theorem 5.1. The integral C \mathbb{C} C B μ 2 n B\mu_{2^n} B μ 2 n
H Z ∗ ∗ C ( B μ 2 n ) ≅ ⨁ k ≥ 1 Σ 2 k + 1 , k + 1 H Z ∗ ∗ C / 2 [ τ ] . HZ^{\mathbb{C}}_{**}(B\mu_{2^n}) \cong\bigoplus_{k\geq1} \Sigma^{2k+1,k+1}HZ^{\mathbb{C}}_{**}/2[\tau]. H Z ∗∗ C ( B μ 2 n ) ≅ k ≥ 1 ⨁ Σ 2 k + 1 , k + 1 H Z ∗∗ C /2 [ τ ] . The theorem shows that the Adams chart for H Z ∗ ∗ C ( B μ 2 n ) HZ^{\mathbb{C}}_{**}(B\mu_{2^n}) H Z ∗∗ C ( B μ 2 n ) H Z ∗ ( B C 2 n ) HZ_{*}(BC_{2^n}) H Z ∗ ( B C 2 n ) Figure 2 , where each ∙ \bullet ∙ F 2 [ τ ] \mathbb{F}_2[\tau] F 2 [ τ ] F 2 \mathbb{F}_2 F 2
Figure 2. The E n E_n E n E n + 1 = E ∞ E_{n+1}=E_\infty E n + 1 = E ∞ H Z ∗ B C 2 n HZ_*BC_{2^n} H Z ∗ B C 2 n F 2 \mathbb{F}_2 F 2 d n d_n d n
Integral motivic homology over real closed fields
Over R \mathbb{R} R ρ \rho ρ R \mathbb{R} R R \mathbb{R} R
We now use this as input for the Atiyah–Hirzebruch spectral sequence.
Proposition 5.3. The associated graded of H Z ∗ ∗ R ( B μ 2 n ) HZ^{\mathbb{R}}_{**}(B\mu_{2^n}) H Z ∗∗ R ( B μ 2 n )
gr A H ∗ H Z ∗ , ∗ R B μ 2 n ≅ { Z / 2 n [ τ 2 ] ⊕ ⨁ j ≥ k F 2 [ τ 2 ] k > 0 even , ⨁ j ≥ k F 2 [ τ 2 ] k > 1 odd , ⨁ j ≥ 0 F 2 [ τ 2 ] k ≤ 1. \operatorname{gr}^{*}_{AH}HZ^{\mathbb{R}}_{*,*}B\mu_{2^n}\cong
\begin{cases}
\mathbb{Z}/2^n[\tau^2]\oplus\bigoplus_{j\geq k}\mathbb{F}_2[\tau^2] & k>0\text{ even},\\
\bigoplus_{j\geq k}\mathbb{F}_2[\tau^2] & k>1\text{ odd},\\
\bigoplus_{j\geq0}\mathbb{F}_2[\tau^2] & k\leq1.
\end{cases} gr A H ∗ H Z ∗ , ∗ R B μ 2 n ≅ ⎩ ⎨ ⎧ Z / 2 n [ τ 2 ] ⊕ ⨁ j ≥ k F 2 [ τ 2 ] ⨁ j ≥ k F 2 [ τ 2 ] ⨁ j ≥ 0 F 2 [ τ 2 ] k > 0 even , k > 1 odd , k ≤ 1. Proof. Consider the Atiyah–Hirzebruch spectral sequence (described in Subsection 3.1):
E 1 = ⨁ each cell i in B μ 2 n Σ i H Z ∗ ∗ R = ⨁ i ≥ 1 Σ i H Z ∗ ∗ R ⇒ H Z ∗ ∗ R B μ 2 n , E_1=\bigoplus_{\text{each cell }i\text{ in }B\mu_{2^n}}\Sigma^i HZ^{\mathbb{R}}_{**}=\bigoplus_{i\geq1}\Sigma^i HZ^{\mathbb{R}}_{**}\Rightarrow HZ^{\mathbb{R}}_{**}B\mu_{2^n}, E 1 = each cell i in B μ 2 n ⨁ Σ i H Z ∗∗ R = i ≥ 1 ⨁ Σ i H Z ∗∗ R ⇒ H Z ∗∗ R B μ 2 n , with nontrivial differentials d 1 ( x [ 2 i ] ) = h n x [ 2 i − 1 ] d_1(x[2i])=h^n x[2i-1] d 1 ( x [ 2 i ]) = h n x [ 2 i − 1 ] i ≥ 1 i\geq1 i ≥ 1 E 2 E_2 E 2 ρ \rho ρ
To resolve extension problems, we consider the motivic Adams spectral sequence (Subsection 3.2). Again,
H ∗ ∗ ( H Z R ) ≅ A R / A R ( 0 ) and H ∗ ∗ B μ 2 n ≅ ⨁ k ≥ 1 Σ [ k , ⌊ k + 1 2 ⌋ ] M 2 R H^{**}(HZ^{\mathbb R}) \cong\mathcal A^{\mathbb R}/\mathcal A^{\mathbb R(0)} \qquad\text{and} \qquad H^{**}B\mu_{2^n} \cong\bigoplus_{k\geq1}\Sigma^{[k,\lfloor\frac{k+1}{2}\rfloor]}\mathbb M^{\mathbb R}_2 H ∗∗ ( H Z R ) ≅ A R / A R ( 0 ) and H ∗∗ B μ 2 n ≅ k ≥ 1 ⨁ Σ [ k , ⌊ 2 k + 1 ⌋] M 2 R gives
E 2 ≅ Ext A R ( 0 ) ∗ ∗ ∗ ( H ∗ ∗ B μ 2 n ) ≅ ⨁ k ≥ 1 Ext A R ( 0 ) ∗ ∗ ∗ ( Σ [ k , ⌊ k + 1 2 ⌋ ] M 2 R ) ≅ ⨁ k ≥ 1 Σ [ k , ⌊ k + 1 2 ⌋ ] F 2 [ h 0 , τ 2 , ρ ] ρ h 0 { x k } , E_2 \cong\operatorname{Ext}_{\mathcal A^{\mathbb R(0)}}^{**\!*}(H^{**}B\mu_{2^n}) \cong\bigoplus_{k\geq1}\operatorname{Ext}_{\mathcal A^{\mathbb R(0)}}^{**\!*}\left(\Sigma^{[k,\lfloor\frac{k+1}{2}\rfloor]}\mathbb M^{\mathbb R}_2\right) \cong\bigoplus_{k\geq1}\Sigma^{[k,\lfloor\frac{k+1}{2}\rfloor]}\frac{\mathbb F_2[h_0,\tau^2,\rho]}{\rho h_0}\{x_k\}, E 2 ≅ Ext A R ( 0 ) ∗∗ ∗ ( H ∗∗ B μ 2 n ) ≅ k ≥ 1 ⨁ Ext A R ( 0 ) ∗∗ ∗ ( Σ [ k , ⌊ 2 k + 1 ⌋] M 2 R ) ≅ k ≥ 1 ⨁ Σ [ k , ⌊ 2 k + 1 ⌋] ρ h 0 F 2 [ h 0 , τ 2 , ρ ] { x k } , where ∣ x k ∣ = ( k , ⌊ k + 1 2 ⌋ ) |x_k|=(k,\lfloor\frac{k+1}{2}\rfloor) ∣ x k ∣ = ( k , ⌊ 2 k + 1 ⌋)
d n ( x 2 k ) = h 0 n x 2 k − 1 . d_n(x_{2k})=h_0^n x_{2k-1}. d n ( x 2 k ) = h 0 n x 2 k − 1 . Also, the differentials are τ 2 \tau^2 τ 2 τ 2 \tau^2 τ 2 H Z ∗ ∗ R HZ^{\mathbb R}_{**} H Z ∗∗ R Figure 11 . This proves the following Theorem 5.4.
Figure 11. The E n E_n E n E n + 1 = E ∞ E_{n+1}=E_\infty E n + 1 = E ∞ H Z ∗ R B C 2 n HZ^{\mathbb R}_{*}BC_{2^n} H Z ∗ R B C 2 n ∙ = F 2 [ τ 2 ] \bullet=\mathbb F_2[\tau^2] ∙ = F 2 [ τ 2 ] d n d_n d n
Theorem 5.4. The 2-completed integral homology of B μ 2 n B\mu_{2^n} B μ 2 n
H Z ∗ ∗ R ( B μ 2 n ) ≅ ⨁ k ≥ 1 Σ 2 k − 1 , k H Z ∗ ∗ R / h n ⊕ ⨁ k ≥ 1 Σ 2 k , k ρ Z / 2 [ ρ , τ 2 ] . H\mathbb{Z}^{\mathbb{R}}_{\ast\ast}(B\mu_{2^n}) \cong\bigoplus_{k\geq1}\Sigma^{2k-1,k}H\mathbb{Z}^{\mathbb{R}}_{\ast\ast}/h^n \oplus\bigoplus_{k\geq1}\Sigma^{2k,k}\rho\,\mathbb{Z}/2[\rho,\tau^2]. H Z ∗∗ R ( B μ 2 n ) ≅ k ≥ 1 ⨁ Σ 2 k − 1 , k H Z ∗∗ R / h n ⊕ k ≥ 1 ⨁ Σ 2 k , k ρ Z /2 [ ρ , τ 2 ] . Effective algebraic K-theory
As discussed in Subsection 2.3, in motivic homotopy theory, we have the notion of “connective” cover of the algebraic K-theory spectrum K G L KGL K G L K U KU K U
We now analyze the effective algebraic K-theory of B μ 2 n B\mu_{2^n} B μ 2 n
H F ∗ ∗ ( k g l F ) ≅ A F / / E F ( 1 ) . HF^{\ast\ast}(kgl^{F}) \cong\mathcal{A}^{F}//\mathcal{E}^{F}(1). H F ∗∗ ( k g l F ) ≅ A F // E F ( 1 ) . Again, calculations are done only over C \mathbb{C} C R \mathbb{R} R
Effective algebraic K-theory over algebraically closed fields
Proposition 6.1. The associated graded of k g l ∗ ∗ C ( B μ 2 n ) kgl^{\mathbb{C}}_{\ast\ast}(B\mu_{2^n}) k g l ∗∗ C ( B μ 2 n )
gr A H ∗ k g l k , w C ( B μ 2 n ) ≅ { ( Z / 2 n [ τ ] ) ⊕ ( i + 1 ) if k = 2 i + 1 > 0 , 0 otherwise. \operatorname{gr}^{\ast}_{AH}kgl^{\mathbb{C}}_{k,w}(B\mu_{2^n}) \cong\begin{cases} \left(\mathbb{Z}/2^n[\tau]\right)^{\oplus(i+1)} & \text{if } k=2i+1>0, \\ 0 & \text{otherwise.} \end{cases} gr A H ∗ k g l k , w C ( B μ 2 n ) ≅ { ( Z / 2 n [ τ ] ) ⊕ ( i + 1 ) 0 if k = 2 i + 1 > 0 , otherwise. Proof. Consider the Atiyah–Hirzebruch spectral sequence (Subsection 3.1):
E 1 = ⨁ each cell i in B μ 2 n Σ i , ⌊ i + 1 2 ⌋ k g l ∗ ∗ C ⟹ gr A H ∗ k g l ∗ ∗ C ( B μ 2 n ) , E_1=\bigoplus_{\text{each cell }i\text{ in }B\mu_{2^n}}\Sigma^{i,\lfloor\frac{i+1}{2}\rfloor}kgl^{\mathbb{C}}_{\ast\ast}\Longrightarrow\operatorname{gr}^{\ast}_{AH}kgl^{\mathbb{C}}_{\ast\ast}(B\mu_{2^n}), E 1 = each cell i in B μ 2 n ⨁ Σ i , ⌊ 2 i + 1 ⌋ k g l ∗∗ C ⟹ gr A H ∗ k g l ∗∗ C ( B μ 2 n ) , where k g l n , ∗ C ≅ Z 2 [ τ ] kgl^{\mathbb{C}}_{n,\ast}\cong\mathbb{Z}_2[\tau] k g l n , ∗ C ≅ Z 2 [ τ ] n = 2 k ≥ 0 n=2k\geq0 n = 2 k ≥ 0 ∣ h 0 ∣ = ( 0 , 1 , 0 ) |h_0|=(0,1,0) ∣ h 0 ∣ = ( 0 , 1 , 0 ) ∣ v 1 ∣ = ( 2 , 1 , 1 ) |v_1|=(2,1,1) ∣ v 1 ∣ = ( 2 , 1 , 1 ) d 1 d_1 d 1
d 1 ( x [ 2 i ] ) = 2 n ⋅ x [ 2 i − 1 ] , i ≥ 1. d_1(x[2i])=2^n\cdot x[2i-1],\qquad i\geq1. d 1 ( x [ 2 i ]) = 2 n ⋅ x [ 2 i − 1 ] , i ≥ 1. For degree reasons, this spectral sequence collapses at E 2 E_2 E 2
Theorem 6.2. The E ∞ E_\infty E ∞ k g l ∗ ∗ C B μ 2 n kgl^{\mathbb{C}}_{\ast\ast}B\mu_{2^n} k g l ∗∗ C B μ 2 n E ∞ E_\infty E ∞ k g l ∗ ∗ C kgl^{\mathbb{C}}_{\ast\ast} k g l ∗∗ C
For arbitrary n n n
E ∞ ( k g l k , ∗ C B μ 2 n ) ≅ { Z / 2 n + i [ τ ] ⊕ ( Z / 2 n − 1 [ τ ] ) ⊕ i if 0 < k = 2 i + 1 ≤ 14 , 0 otherwise. E_\infty\left(kgl^{\mathbb{C}}_{k,\ast}B\mu_{2^n}\right)\cong\begin{cases}\mathbb{Z}/2^{n+i}[\tau]\oplus\left(\mathbb{Z}/2^{n-1}[\tau]\right)^{\oplus i} & \text{if }0<k=2i+1\leq14, \\ 0 & \text{otherwise.}\end{cases} E ∞ ( k g l k , ∗ C B μ 2 n ) ≅ { Z / 2 n + i [ τ ] ⊕ ( Z / 2 n − 1 [ τ ] ) ⊕ i 0 if 0 < k = 2 i + 1 ≤ 14 , otherwise. For n = 2 n=2 n = 2
E ∞ ( k g l k , ∗ C B μ 4 ) ≅ { Z / 2 2 + i [ τ ] ⊕ ( Z / 2 [ τ ] ) ⊕ i if 0 < k = 2 i + 1 , 0 otherwise. E_\infty\left(kgl^{\mathbb{C}}_{k,\ast}B\mu_4\right)\cong\begin{cases}\mathbb{Z}/2^{2+i}[\tau]\oplus\left(\mathbb{Z}/2[\tau]\right)^{\oplus i} & \text{if }0<k=2i+1, \\ 0 & \text{otherwise.}\end{cases} E ∞ ( k g l k , ∗ C B μ 4 ) ≅ { Z / 2 2 + i [ τ ] ⊕ ( Z /2 [ τ ] ) ⊕ i 0 if 0 < k = 2 i + 1 , otherwise. Proof. Recall that, as in the classical case, Proposition 2.6 gives an isomorphism
H ∗ ∗ ( k g l C ) ≅ A C / E C ( 1 ) , H^{**}(kgl^{\mathbb C}) \cong\mathcal{A}^{\mathbb C}/\mathcal{E}^{\mathbb C}(1), H ∗∗ ( k g l C ) ≅ A C / E C ( 1 ) , and from Corollary 2.12 we have
H ∗ ∗ B μ 2 n ≅ ⨁ k ≥ 0 Σ k , ⌊ k + 1 2 ⌋ M 2 C . H^{**}B\mu_{2^n} \cong\bigoplus_{k\geq0}\Sigma^{k,\left\lfloor\frac{k+1}{2}\right\rfloor}\mathbb{M}^{\mathbb C}_2. H ∗∗ B μ 2 n ≅ k ≥ 0 ⨁ Σ k , ⌊ 2 k + 1 ⌋ M 2 C . Using these together with the change-of-rings isomorphism to simplify the E 2 E_2 E 2
E 2 ∗ ∗ ≅ ⨁ k ≥ 1 Σ k Ext C ( 1 ) ∗ ∗ ∗ ( M 2 ) ≅ ⨁ k ≥ 1 M 2 C [ h 0 , v 1 ] { x ~ k } ⇒ k g l ∗ ∗ C ( B μ 2 n ) . E_2^{**} \cong\bigoplus_{k\geq1}\Sigma^k\operatorname{Ext}_{\mathcal{C}(1)}^{***}(\mathbb{M}_2) \cong\bigoplus_{k\geq1}\mathbb{M}^{\mathbb C}_2[h_0,v_1]\{\widetilde{x}_k\}\Rightarrow kgl^{\mathbb C}_{**}(B\mu_{2^n}). E 2 ∗∗ ≅ k ≥ 1 ⨁ Σ k Ext C ( 1 ) ∗∗∗ ( M 2 ) ≅ k ≥ 1 ⨁ M 2 C [ h 0 , v 1 ] { x k } ⇒ k g l ∗∗ C ( B μ 2 n ) . with ∣ x ~ k ∣ = ( k , ⌊ k + 1 2 ⌋ ) \lvert\widetilde{x}_k\rvert=(k,\lfloor\frac{k+1}{2}\rfloor) ∣ x k ∣ = ( k , ⌊ 2 k + 1 ⌋) ∣ h 0 ∣ = ( 1 , 1 , 1 ) \lvert h_0\rvert=(1,1,1) ∣ h 0 ∣ = ( 1 , 1 , 1 ) ∣ v 1 ∣ = ( 3 , 1 , 1 ) \lvert v_1\rvert=(3,1,1) ∣ v 1 ∣ = ( 3 , 1 , 1 ) τ \tau τ
For arbitrary n n n
d n ( x ~ 2 ) = h 0 n x ~ 1 , and for 2 ≤ k ≤ 7 , d n ( x ~ 2 k ) = h 0 n x ~ 2 k − 1 + h 0 n − 1 v 1 x ~ 2 k − 3 . d_n(\widetilde{x}_2)=h_0^n\widetilde{x}_1,\qquad\text{and}\qquad\text{for }2\leq k\leq7,\ d_n(\widetilde{x}_{2k})=h_0^n\widetilde{x}_{2k-1}+h_0^{n-1}v_1\widetilde{x}_{2k-3}. d n ( x 2 ) = h 0 n x 1 , and for 2 ≤ k ≤ 7 , d n ( x 2 k ) = h 0 n x 2 k − 1 + h 0 n − 1 v 1 x 2 k − 3 . Propagation by h 0 h_0 h 0 v 1 v_1 v 1 E n + 1 E_{n+1} E n + 1
These differentials continue beyond stem 14 for n = 2 n=2 n = 2
d 2 ( x ~ 2 ) = h 0 2 x ~ 1 and for k > 1 , d 2 ( x ~ 2 k ) = h 0 2 x ~ 2 k − 1 + h 0 v 1 x ~ 2 k − 3 . d_2(\widetilde{x}_2)=h_0^2\widetilde{x}_1\qquad\text{and}\qquad\text{for }k>1,\ d_2(\widetilde{x}_{2k})=h_0^2\widetilde{x}_{2k-1}+h_0v_1\widetilde{x}_{2k-3}. d 2 ( x 2 ) = h 0 2 x 1 and for k > 1 , d 2 ( x 2 k ) = h 0 2 x 2 k − 1 + h 0 v 1 x 2 k − 3 . The Adams chart for C-motivic k g l ∗ ∗ C ( B μ 2 n ) kgl^{\mathbb C}_{**}(B\mu_{2^n}) k g l ∗∗ C ( B μ 2 n ) k u ∗ ( B C 2 n ) ku_*(BC_{2^n}) k u ∗ ( B C 2 n ) Figure 9 , where each ∙ \bullet ∙ F 2 [ τ ] \mathbb{F}_2[\tau] F 2 [ τ ] F 2 \mathbb{F}_2 F 2
Effective algebraic K-theory over real closed fields
Next, we compute k g l ∗ ∗ R B μ 2 n kgl^{\mathbb R}_{**}B\mu_{2^n} k g l ∗∗ R B μ 2 n [23 ] , we have
Ext C ( 1 ) ∗ ∗ ∗ ( M 2 R ) ≅ F 2 [ τ 4 , ρ , h 0 , v 1 , τ 2 h 0 ] ( ρ h 0 , ρ 3 v 1 , ρ ⋅ [ τ 2 h 0 ] ) , (1) \operatorname{Ext}_{\mathcal{C}(1)}^{***}(\mathbb{M}^{\mathbb R}_2)\cong\frac{\mathbb{F}_2[\tau^4,\rho,h_0,v_1,\tau^2h_0]}{(\rho h_0,\rho^3v_1,\rho\cdot[\tau^2h_0])}, \tag*{(1)} Ext C ( 1 ) ∗∗∗ ( M 2 R ) ≅ ( ρ h 0 , ρ 3 v 1 , ρ ⋅ [ τ 2 h 0 ]) F 2 [ τ 4 , ρ , h 0 , v 1 , τ 2 h 0 ] , (1) as a ring.
The R \mathbb{R} R
Figure 12. The E 2 = E ∞ E_2=E_\infty E 2 = E ∞ k g l ∗ ∗ R kgl^{\mathbb{R}}_{**} k g l ∗∗ R F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ] ρ 3 \rho^3 ρ 3 v 1 v_1 v 1
Consider the Atiyah–Hirzebruch spectral sequence
E 1 = ⨁ each cell i in B μ 2 n Σ i , ⌊ i + 1 2 ⌋ k g l ∗ ∗ R ⇒ k g l ∗ ∗ R B μ 2 n . E_1=\bigoplus_{\text{each cell }i\text{ in }B\mu_{2^n}}\Sigma^{i,\left\lfloor\frac{i+1}{2}\right\rfloor}kgl^{\mathbb{R}}_{**}\Rightarrow kgl^{\mathbb{R}}_{**}B\mu_{2^n}. E 1 = each cell i in B μ 2 n ⨁ Σ i , ⌊ 2 i + 1 ⌋ k g l ∗∗ R ⇒ k g l ∗∗ R B μ 2 n . where x [ i ] x[i] x [ i ] x ∈ k g l ∗ ∗ R x\in kgl^{\mathbb{R}}_{**} x ∈ k g l ∗∗ R
Proposition 6.3. The differentials are given by:
d 1 ( x [ 2 i ] ) = h n ⋅ x [ 2 i − 1 ] d_1(x[2i])=h^n\cdot x[2i-1] d 1 ( x [ 2 i ]) = h n ⋅ x [ 2 i − 1 ] i ≥ 1 i\geq1 i ≥ 1
no non-trivial d 2 d_2 d 2 d 3 d_3 d 3
Proof. (1) Each even-dimensional cell of B μ 2 n B\mu_{2^n} B μ 2 n h n h^n h n d 1 d_1 d 1
(2) The d 2 d_2 d 2 k g l kgl k g l d 2 d_2 d 2
For the d 3 d_3 d 3
⟨ x [ i ] , η , h n ⟩ (for i ≡ 0 ( m o d 4 ) , i > 0 ) \langle x[i],\eta,h^n\rangle\quad\text{(for }i\equiv0\pmod4,\ i>0\text{)} ⟨ x [ i ] , η , h n ⟩ (for i ≡ 0 ( mod 4 ) , i > 0 ) and
⟨ x [ i ] , h n , η ⟩ (for i ≡ 2 ( m o d 4 ) , i > 2 ) . \langle x[i],h^n,\eta\rangle\quad\text{(for }i\equiv2\pmod4,\ i>2\text{)}. ⟨ x [ i ] , h n , η ⟩ (for i ≡ 2 ( mod 4 ) , i > 2 ) . Using the relation h n − 1 v 1 z ∈ ⟨ z , η , h n ⟩ h^{n-1}v_1z\in\langle z,\eta,h^n\rangle h n − 1 v 1 z ∈ ⟨ z , η , h n ⟩ h n − 1 v 1 z ∈ ⟨ z , h n , η ⟩ h^{n-1}v_1z\in\langle z,h^n,\eta\rangle h n − 1 v 1 z ∈ ⟨ z , h n , η ⟩
d 3 ( z [ 4 k ] ) = h n − 1 v 1 ⋅ z [ 4 k − 3 ] d_3(z[4k])=h^{n-1}v_1\cdot z[4k-3] d 3 ( z [ 4 k ]) = h n − 1 v 1 ⋅ z [ 4 k − 3 ] and
d 3 ( z [ 4 k − 2 ] ) = h n − 1 v 1 ⋅ z [ 4 k − 1 ] ∀ k ≥ 1. d_3(z[4k-2])=h^{n-1}v_1\cdot z[4k-1]\quad\forall k\geq1. d 3 ( z [ 4 k − 2 ]) = h n − 1 v 1 ⋅ z [ 4 k − 1 ] ∀ k ≥ 1. However, there are no d 3 d_3 d 3 h n − 1 h^{n-1} h n − 1
:::
Remark 6.4. In this paper, our primary technique is to compute the Atiyah–Hirzebruch differentials using the cellular filtration. However, beyond the E 3 E_3 E 3 Figure 13 .
Figure 13. The chart represents the E 1 E_1 E 1 E 4 E_4 E 4 k g l ∗ ∗ R B μ 2 n kgl^{\mathbb{R}}_{**}B\mu_{2^n} k g l ∗∗ R B μ 2 n ∙ = Z 2 [ τ − 4 ] \bullet= \mathbb{Z}_2[\tau^{-4}] ∙ = Z 2 [ τ − 4 ] ⋄ = Z / 2 n [ τ − 4 ] \diamond= \mathbb{Z}/2^n[\tau^{-4}] ⋄ = Z / 2 n [ τ − 4 ] ρ \rho ρ v 1 v_1 v 1
To determine whether these differentials actually occur (see Figure 14 ) and to resolve the hidden extension problems, we appeal to the Adams spectral sequence (Subsection 3.2).
Figure 14. The E n E_n E n d n d_n d n E ∞ E_\infty E ∞ k g l k , ∗ R B μ 2 n kgl^{\mathbb{R}}_{k,*}B\mu_{2^n} k g l k , ∗ R B μ 2 n ∙ \bullet ∙ F 2 [ τ 2 ] \mathbb{F}_2[\tau^2] F 2 [ τ 2 ] ⋄ \diamond ⋄ F 2 [ τ 2 , ρ ] / ρ 3 \mathbb{F}_2[\tau^2,\rho]/\rho^3 F 2 [ τ 2 , ρ ] / ρ 3 h 0 h_0 h 0 κ i \kappa_i κ i h 0 κ i + v 1 κ i − 2 h_0\kappa_i+v_1\kappa_{i-2} h 0 κ i + v 1 κ i − 2 v 1 v_1 v 1 ρ \rho ρ
Theorem 6.5. As a module over the E ∞ E_\infty E ∞ k g l ∗ ∗ R kgl^{\mathbb{R}}_{**} k g l ∗∗ R E ∞ E_\infty E ∞ k g l ∗ ∗ R B μ 2 n kgl^{\mathbb{R}}_{**}B\mu_{2^n} k g l ∗∗ R B μ 2 n
(1) For arbitrary n n n
E ∞ ( k g l k , ∗ R B μ 2 n ) ≅ { ( Z / 2 n + i [ τ 2 ] ) ⊕ 2 ⊕ ( Z / 2 n − 1 [ τ 2 ] ) ⊕ 2 i ⊕ F 2 [ τ 2 ] ⊕ ( i + 1 ) ⊕ F 2 [ τ 2 , ρ ] , 0 < k = 2 i + 1 ≤ 14 , F 2 [ τ 2 ] ⊕ i ⊕ F 2 [ τ 2 , ρ ] , 0 < k = 2 i ≤ 14 , F 2 [ τ 2 , ρ ] , otherwise. E_\infty\left(kgl^{\mathbb{R}}_{k,*}B\mu_{2^n}\right) \cong\begin{cases}
\left(\mathbb{Z}/2^{n+i}[\tau^2]\right)^{\oplus2} \oplus\left(\mathbb{Z}/2^{n-1}[\tau^2]\right)^{\oplus2i} \oplus\mathbb{F}_2[\tau^2]^{\oplus(i+1)} \oplus\mathbb{F}_2[\tau^2,\rho], & 0 < k=2i+1 \le14, \\
\mathbb{F}_2[\tau^2]^{\oplus i} \oplus\mathbb{F}_2[\tau^2,\rho], & 0 < k=2i \le14, \\
\mathbb{F}_2[\tau^2,\rho], & \text{otherwise.}
\end{cases} E ∞ ( k g l k , ∗ R B μ 2 n ) ≅ ⎩ ⎨ ⎧ ( Z / 2 n + i [ τ 2 ] ) ⊕ 2 ⊕ ( Z / 2 n − 1 [ τ 2 ] ) ⊕ 2 i ⊕ F 2 [ τ 2 ] ⊕ ( i + 1 ) ⊕ F 2 [ τ 2 , ρ ] , F 2 [ τ 2 ] ⊕ i ⊕ F 2 [ τ 2 , ρ ] , F 2 [ τ 2 , ρ ] , 0 < k = 2 i + 1 ≤ 14 , 0 < k = 2 i ≤ 14 , otherwise. (2) For n = 2 n=2 n = 2
E ∞ ( k g l k , ∗ R B μ 4 ) ≅ { ( Z / 2 i + 2 [ τ 2 ] ) ⊕ 2 ⊕ ( Z / 2 [ τ 2 ] ) ⊕ 2 i ⊕ F 2 [ τ 2 ] ⊕ ( i + 1 ) ⊕ F 2 [ τ 2 , ρ ] , 0 < k = 2 i + 1 , F 2 [ τ 2 ] ⊕ i ⊕ F 2 [ τ 2 , ρ ] , 0 < k = 2 i , F 2 [ τ 2 , ρ ] , otherwise. E_\infty\left(kgl^{\mathbb{R}}_{k,*}B\mu_4\right) \cong\begin{cases}
\left(\mathbb{Z}/2^{i+2}[\tau^2]\right)^{\oplus2} \oplus\left(\mathbb{Z}/2[\tau^2]\right)^{\oplus2i} \oplus\mathbb{F}_2[\tau^2]^{\oplus(i+1)} \oplus\mathbb{F}_2[\tau^2,\rho], & 0 < k=2i+1, \\
\mathbb{F}_2[\tau^2]^{\oplus i} \oplus\mathbb{F}_2[\tau^2,\rho], & 0 < k=2i, \\
\mathbb{F}_2[\tau^2,\rho], & \text{otherwise.}
\end{cases} E ∞ ( k g l k , ∗ R B μ 4 ) ≅ ⎩ ⎨ ⎧ ( Z / 2 i + 2 [ τ 2 ] ) ⊕ 2 ⊕ ( Z /2 [ τ 2 ] ) ⊕ 2 i ⊕ F 2 [ τ 2 ] ⊕ ( i + 1 ) ⊕ F 2 [ τ 2 , ρ ] , F 2 [ τ 2 ] ⊕ i ⊕ F 2 [ τ 2 , ρ ] , F 2 [ τ 2 , ρ ] , 0 < k = 2 i + 1 , 0 < k = 2 i , otherwise. Proof. Recall from Proposition 2.6 ,
H ∗ ∗ ( k g l R ) ≅ A R / / E R ( 1 ) . H^{**}(kgl^{\mathbb{R}}) \cong A^{\mathbb{R}} // \mathcal{E}^{\mathbb{R}}(1). H ∗∗ ( k g l R ) ≅ A R // E R ( 1 ) . and from Corollary 2.12 we have
H ∗ ∗ B μ 2 n ≅ ⨁ k ≥ 1 Σ k , ⌊ k + 1 2 ⌋ M 2 R . (2) H^{**}B\mu_{2^n} \cong\bigoplus_{k\ge1}\Sigma^{k,\left\lfloor\frac{k+1}{2}\right\rfloor}\mathbb{M}^{\mathbb{R}}_2. \tag*{(2)} H ∗∗ B μ 2 n ≅ k ≥ 1 ⨁ Σ k , ⌊ 2 k + 1 ⌋ M 2 R . (2) Using these and change of rings isomorphism to simplify the E 2 E_2 E 2
E 2 ≅ ⨁ k ≥ 1 Ext E ( 1 ) R ∗ ∗ ∗ ( M 2 R ) { x ~ k } . E_2 \cong\bigoplus_{k\geq1}\operatorname{Ext}^{***}_{\mathcal{E}(1)_{\mathbb{R}}}\left(M^{\mathbb{R}}_2\right)\{\widetilde{x}_k\}. E 2 ≅ k ≥ 1 ⨁ Ext E ( 1 ) R ∗∗∗ ( M 2 R ) { x k } . The extension of scalars from the R \mathbb{R} R C \mathbb{C} C C \mathbb{C} C R \mathbb{R} R ρ \rho ρ [20 ] for details. In the C \mathbb{C} C Theorem 6.2 . Since the extension-of-scalars map preserves the generators h 0 h_0 h 0 v 1 v_1 v 1 x ~ j \widetilde{x}_j x j ρ \rho ρ R \mathbb{R} R
Therefore, for arbitrary n n n
d n ( x ~ 2 ) = h 0 n x ~ 1 , and for 2 ≤ k ≤ 7 , d n ( x ~ 2 k ) = h 0 n x ~ 2 k − 1 + h 0 n − 1 v 1 x ~ 2 k − 3 . d_n(\widetilde{x}_2)=h_0^n\widetilde{x}_1,\qquad\text{and}\qquad\text{for }2\leq k\leq7,\ d_n(\widetilde{x}_{2k})=h_0^n\widetilde{x}_{2k-1}+h_0^{n-1}v_1\widetilde{x}_{2k-3}. d n ( x 2 ) = h 0 n x 1 , and for 2 ≤ k ≤ 7 , d n ( x 2 k ) = h 0 n x 2 k − 1 + h 0 n − 1 v 1 x 2 k − 3 . These differentials continue beyond stem 14 for n = 2 n=2 n = 2
d 2 ( x ~ 2 ) = h 0 2 x ~ 1 and for k > 1 , d 2 ( x ~ 2 k ) = h 0 2 x ~ 2 k − 1 + h 0 v 1 x ~ 2 k − 3 . d_2(\widetilde{x}_2)=h_0^2\widetilde{x}_1\qquad\text{and}\qquad\text{for }k>1,\ d_2(\widetilde{x}_{2k})=h_0^2\widetilde{x}_{2k-1}+h_0v_1\widetilde{x}_{2k-3}. d 2 ( x 2 ) = h 0 2 x 1 and for k > 1 , d 2 ( x 2 k ) = h 0 2 x 2 k − 1 + h 0 v 1 x 2 k − 3 . Moreover, these propagate by h 0 h_0 h 0 v 1 v_1 v 1 τ 2 h 0 \tau^2h_0 τ 2 h 0 x ~ 2 i \widetilde{x}_{2i} x 2 i E n + 1 E_{n+1} E n + 1 Figure 14 , which rules out the possibility of any further nontrivial differentials through stem 14 for arbitrary n n n n = 2 n=2 n = 2
Remark 6.6. After this calculation, we can definitely say that the higher differentials in the Atiyah–Hirzebruch spectral sequence do not occur, and the spectral sequence therefore yields the complete associated graded object.
We emphasize that this reverse approach of deducing Atiyah–Hirzebruch differentials via the Adams spectral sequence is efficient and conceptually powerful, allowing spectral sequences to be used in-combination to ultimately leading to the desired result.
Very Effective Hermitian K-Theory
As discussed in Subsection 2.3, in motivic homotopy theory, we have the notion of “connective” cover of the Hermitian K K K K Q KQ K Q K K K K O KO K O
In this section, our goal is to compute the very effective Hermitian K K K B μ 2 n B\mu_{2^n} B μ 2 n k g l kgl k g l
Again, using the Lefschetz principle described in Subsection 2.5, we can extend the computation over C \mathbb{C} C R \mathbb{R} R
Very Effective Hermitian K-theory over algebraically closed fields
In this section we compute k q ∗ ∗ C ( B μ 2 n ) kq^{\mathbb{C}}_{**}(B\mu_{2^n}) k q ∗∗ C ( B μ 2 n ) k q ∗ ∗ C kq^{\mathbb{C}}_{**} k q ∗∗ C
Remark 7.1. As shown in [26 Theorem 4.10] with chart [IS11, Figure A.1], the bigraded homotopy ring of very effective Hermitian K K K
k q ∗ ∗ C : = π ∗ ∗ ( k q ) ≅ Z [ τ , η , α , β ] ( 2 η , τ η 3 , η α , α 2 − 4 β ) . kq^{\mathbb{C}}_{**}:=\pi_{**}(kq)\cong\frac{\mathbb{Z}[\tau,\eta,\alpha,\beta]}{(2\eta,\tau\eta^{3},\eta\alpha,\alpha^{2}-4\beta)}. k q ∗∗ C := π ∗∗ ( k q ) ≅ ( 2 η , τ η 3 , η α , α 2 − 4 β ) Z [ τ , η , α , β ] . So, let’s start with the Atiyah–Hirzebruch spectral sequence, which gives
Proposition 7.2. The associated graded groups are given by
gr A H ∗ k q ∗ , ∗ C ( B μ 2 n ) ≅ { Z / 2 n [ τ ] ⊕ ( Z / 2 n − 1 ) ⊕ 2 m [ τ ] , k = 8 m + 1 > 0 , F 2 [ τ ] , k = 8 m + 2 > 0 , ( F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( Z / 2 n [ τ ] ) ⊕ 2 m + 1 , k = 8 m + 3 > 0 , ( F 2 ) ⊕ m + 1 , k = 8 m + 4 , 8 m + 6 , 8 m > 0 , ( Z / 2 n − 1 ) ⊕ 2 m + 2 [ τ ] ⊕ ( F 2 ) ⊕ m + 1 , k = 8 m + 5 > 0 , ( Z / 2 n [ τ ] ⊕ F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( F 2 ) ⊕ m + 1 , k = 8 m + 7 > 0. \operatorname{gr}^{*}_{AH}kq^{\mathbb{C}}_{*,*}(B\mu_{2^n})\cong
\begin{cases}
\mathbb{Z}/2^n[\tau]\oplus\left(\mathbb{Z}/2^{n-1}\right)^{\oplus2m}[\tau], & k=8m+1>0,\\
\mathbb{F}_2[\tau], & k=8m+2>0,\\
\left(\mathbb{F}_2[\tau]\right)^{\oplus2m+2}\oplus\left(\mathbb{Z}/2^n[\tau]\right)^{\oplus2m+1}, & k=8m+3>0,\\
\left(\mathbb{F}_2\right)^{\oplus m+1}, & k=8m+4,8m+6,8m>0,\\
\left(\mathbb{Z}/2^{n-1}\right)^{\oplus2m+2}[\tau]\oplus\left(\mathbb{F}_2\right)^{\oplus m+1}, & k=8m+5>0,\\
\left(\mathbb{Z}/2^n[\tau]\oplus\mathbb{F}_2[\tau]\right)^{\oplus2m+2}\oplus\left(\mathbb{F}_2\right)^{\oplus m+1}, & k=8m+7>0.
\end{cases} gr A H ∗ k q ∗ , ∗ C ( B μ 2 n ) ≅ ⎩ ⎨ ⎧ Z / 2 n [ τ ] ⊕ ( Z / 2 n − 1 ) ⊕ 2 m [ τ ] , F 2 [ τ ] , ( F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( Z / 2 n [ τ ] ) ⊕ 2 m + 1 , ( F 2 ) ⊕ m + 1 , ( Z / 2 n − 1 ) ⊕ 2 m + 2 [ τ ] ⊕ ( F 2 ) ⊕ m + 1 , ( Z / 2 n [ τ ] ⊕ F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( F 2 ) ⊕ m + 1 , k = 8 m + 1 > 0 , k = 8 m + 2 > 0 , k = 8 m + 3 > 0 , k = 8 m + 4 , 8 m + 6 , 8 m > 0 , k = 8 m + 5 > 0 , k = 8 m + 7 > 0. Proof. By Subsection 3.1, the E 1 E_1 E 1
E 1 = ⨁ each cell i in B μ 2 n Σ i , ⌊ i + 1 2 ⌋ k q ∗ ∗ C ⟹ gr A H ∗ ∗ k q ∗ ∗ C ( B μ 2 n ) , E_1=\bigoplus_{\text{each cell }i\text{ in }B\mu_{2^n}}\Sigma^{i,\left\lfloor\frac{i+1}{2}\right\rfloor}kq^{\mathbb{C}}_{**}\Longrightarrow\operatorname{gr}^{**}_{AH}kq^{\mathbb{C}}_{**}(B\mu_{2^n}), E 1 = each cell i in B μ 2 n ⨁ Σ i , ⌊ 2 i + 1 ⌋ k q ∗∗ C ⟹ gr A H ∗∗ k q ∗∗ C ( B μ 2 n ) , @page 32/46 starts_mid=1 ends_mid=1 which rules out the possibility of any further nontrivial differentials through stem 14 for arbitrary n n n n = 2 n=2 n = 2
Remark 6.6. After this calculation, we can definitely say that the higher differentials in the Atiyah–Hirzebruch spectral sequence do not occur, and the spectral sequence therefore yields the complete associated graded object.
We emphasize that this reverse approach of deducing Atiyah–Hirzebruch differentials via the Adams spectral sequence is efficient and conceptually powerful, allowing spectral sequences to be used in-combination to ultimately leading to the desired result.
Very Effective Hermitian K-Theory
As discussed in Subsection 2.3, in motivic homotopy theory, we have the notion of “connective” cover of the Hermitian K K K K Q KQ K Q K K K K O KO K O
In this section, our goal is to compute the very effective Hermitian K K K B μ 2 n B\mu_{2^n} B μ 2 n k g l kgl k g l
Again, using the Lefschetz principle described in Subsection 2.5, we can extend the computation over C \mathbb{C} C R \mathbb{R} R
Very Effective Hermitian K-theory over algebraically closed fields
In this section we compute k q ∗ ∗ C ( B μ 2 n ) kq^{\mathbb{C}}_{**}(B\mu_{2^n}) k q ∗∗ C ( B μ 2 n ) k q ∗ ∗ C kq^{\mathbb{C}}_{**} k q ∗∗ C
Remark 7.1. As shown in [26 Theorem 4.10] with chart [IS11, Figure A.1], the bigraded homotopy ring of very effective Hermitian K K K
k q ∗ ∗ C : = π ∗ ∗ ( k q ) ≅ Z [ τ , η , α , β ] ( 2 η , τ η 3 , η α , α 2 − 4 β ) . kq^{\mathbb{C}}_{**}:=\pi_{**}(kq)\cong\frac{\mathbb{Z}[\tau,\eta,\alpha,\beta]}{(2\eta,\tau\eta^{3},\eta\alpha,\alpha^{2}-4\beta)}. k q ∗∗ C := π ∗∗ ( k q ) ≅ ( 2 η , τ η 3 , η α , α 2 − 4 β ) Z [ τ , η , α , β ] . So, let’s start with the Atiyah–Hirzebruch spectral sequence, which gives
Proposition 7.2. The associated graded groups are given by
gr A H ∗ k q ∗ , ∗ C ( B μ 2 n ) ≅ { Z / 2 n [ τ ] ⊕ ( Z / 2 n − 1 ) ⊕ 2 m [ τ ] , k = 8 m + 1 > 0 , F 2 [ τ ] , k = 8 m + 2 > 0 , ( F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( Z / 2 n [ τ ] ) ⊕ 2 m + 1 , k = 8 m + 3 > 0 , ( F 2 ) ⊕ m + 1 , k = 8 m + 4 , 8 m + 6 , 8 m > 0 , ( Z / 2 n − 1 ) ⊕ 2 m + 2 [ τ ] ⊕ ( F 2 ) ⊕ m + 1 , k = 8 m + 5 > 0 , ( Z / 2 n [ τ ] ⊕ F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( F 2 ) ⊕ m + 1 , k = 8 m + 7 > 0. \operatorname{gr}^{*}_{AH}kq^{\mathbb{C}}_{*,*}(B\mu_{2^n})\cong
\begin{cases}
\mathbb{Z}/2^n[\tau]\oplus\left(\mathbb{Z}/2^{n-1}\right)^{\oplus2m}[\tau], & k=8m+1>0,\\
\mathbb{F}_2[\tau], & k=8m+2>0,\\
\left(\mathbb{F}_2[\tau]\right)^{\oplus2m+2}\oplus\left(\mathbb{Z}/2^n[\tau]\right)^{\oplus2m+1}, & k=8m+3>0,\\
\left(\mathbb{F}_2\right)^{\oplus m+1}, & k=8m+4,8m+6,8m>0,\\
\left(\mathbb{Z}/2^{n-1}\right)^{\oplus2m+2}[\tau]\oplus\left(\mathbb{F}_2\right)^{\oplus m+1}, & k=8m+5>0,\\
\left(\mathbb{Z}/2^n[\tau]\oplus\mathbb{F}_2[\tau]\right)^{\oplus2m+2}\oplus\left(\mathbb{F}_2\right)^{\oplus m+1}, & k=8m+7>0.
\end{cases} gr A H ∗ k q ∗ , ∗ C ( B μ 2 n ) ≅ ⎩ ⎨ ⎧ Z / 2 n [ τ ] ⊕ ( Z / 2 n − 1 ) ⊕ 2 m [ τ ] , F 2 [ τ ] , ( F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( Z / 2 n [ τ ] ) ⊕ 2 m + 1 , ( F 2 ) ⊕ m + 1 , ( Z / 2 n − 1 ) ⊕ 2 m + 2 [ τ ] ⊕ ( F 2 ) ⊕ m + 1 , ( Z / 2 n [ τ ] ⊕ F 2 [ τ ] ) ⊕ 2 m + 2 ⊕ ( F 2 ) ⊕ m + 1 , k = 8 m + 1 > 0 , k = 8 m + 2 > 0 , k = 8 m + 3 > 0 , k = 8 m + 4 , 8 m + 6 , 8 m > 0 , k = 8 m + 5 > 0 , k = 8 m + 7 > 0. Proof. By Subsection 3.1, the E 1 E_1 E 1
E 1 = ⨁ each cell i in B μ 2 n Σ i , ⌊ i + 1 2 ⌋ k q ∗ ∗ C ⟹ gr A H ∗ ∗ k q ∗ ∗ C ( B μ 2 n ) , E_1=\bigoplus_{\text{each cell }i\text{ in }B\mu_{2^n}}\Sigma^{i,\left\lfloor\frac{i+1}{2}\right\rfloor}kq^{\mathbb{C}}_{**}\Longrightarrow\operatorname{gr}^{**}_{AH}kq^{\mathbb{C}}_{**}(B\mu_{2^n}), E 1 = each cell i in B μ 2 n ⨁ Σ i , ⌊ 2 i + 1 ⌋ k q ∗∗ C ⟹ gr A H ∗∗ k q ∗∗ C ( B μ 2 n ) , with nontrivial differentials computed similar to classical ones
d 1 ( x [ 2 i ] ) = 2 n x [ 2 i − 1 ] , i ≥ 1 , d_1(x[2i]) = 2^n x[2i-1], \qquad i \ge1, d 1 ( x [ 2 i ]) = 2 n x [ 2 i − 1 ] , i ≥ 1 , d 2 ( x [ i ] ) = n x [ i − 2 ] , i = 4 k , 4 k + 1 ≥ 4 , d_2(x[i]) = n x[i-2], \qquad i = 4k, 4k+1 \ge4, d 2 ( x [ i ]) = n x [ i − 2 ] , i = 4 k , 4 k + 1 ≥ 4 , d 3 ( τ η 2 [ 4 i ] ) = 2 n − 1 α [ 4 i − 3 ] as 2 n − 1 α ∈ ⟨ τ η 2 , η , 2 n ⟩ shown in Fig. 15 . d_3(\tau\eta^2[4i]) = 2^{n-1}\alpha[4i-3] \qquad\text{as } 2^{n-1}\alpha\in\langle\tau\eta^2,\eta,2^n\rangle\text{ shown in } \text{Fig. 15}. d 3 ( τ η 2 [ 4 i ]) = 2 n − 1 α [ 4 i − 3 ] as 2 n − 1 α ∈ ⟨ τ η 2 , η , 2 n ⟩ shown in Fig. 15 . The spectral sequence converges at E 4 E_4 E 4 η \eta η α \alpha α β \beta β
Figure 15. The E 3 E_3 E 3 E ∞ E_\infty E ∞ gr A H ∗ k q ∗ ∗ ( B μ 2 n ) \operatorname{gr}^{*}_{AH}kq^{*}_{*}(B\mu_{2^n}) gr A H ∗ k q ∗ ∗ ( B μ 2 n ) Z / 2 n [ τ ] \mathbb{Z}/2^n[\tau] Z / 2 n [ τ ] Z / 2 n − 1 [ τ ] \mathbb{Z}/2^{n-1}[\tau] Z / 2 n − 1 [ τ ] F 2 [ τ ] \mathbb{F}_2[\tau] F 2 [ τ ] F 2 \mathbb{F}_2 F 2 η \eta η d 3 d_3 d 3
Again extension problems are resolved using the motivic Adams spectral sequence in Subsection 3.2. So, recall from Proposition 2.6 that, analogously, we have
H ∗ ∗ ( k q C ) ≅ A C / / A C ( 1 ) . H^{**}(kq^{\mathbb{C}}) \cong\mathcal{A}^{\mathbb{C}}//\mathcal{A}^{\mathbb{C}}(1). H ∗∗ ( k q C ) ≅ A C // A C ( 1 ) . Using this and a change of rings isomorphism to simplify the E 2 E_2 E 2
E 2 = Ext A C ( 1 ) ∗ ∗ ∗ ( H ∗ ∗ ( B μ 2 n ) ) ⇒ k q ∗ ∗ C ( B μ 2 n ) . E_2 = \operatorname{Ext}^{***}_{\mathcal{A}^{\mathbb{C}}(1)}\left(H^{**}(B\mu_{2^n})\right) \Rightarrow kq^{\mathbb{C}}_{**}(B\mu_{2^n}). E 2 = Ext A C ( 1 ) ∗∗∗ ( H ∗∗ ( B μ 2 n ) ) ⇒ k q ∗∗ C ( B μ 2 n ) . From Corollary 2.13 we have
H ∗ ∗ ( B μ 2 n ) = Σ 1 , 1 M 2 C ⨁ ⨁ k ≥ 0 ( Σ 4 k + 2 , 2 k + 1 A C ( 1 ) / / E C ( 1 ) ⊕ Σ 4 k + 3 , 2 k + 2 A C ( 1 ) / / E C ( 1 ) ) . H^{**}(B\mu_{2^n}) = \Sigma^{1,1}M^{\mathbb{C}}_2 \bigoplus\bigoplus_{k\ge0}\left(\Sigma^{4k+2,2k+1}\mathcal{A}^{\mathbb{C}}(1)//\mathcal{E}^{\mathbb{C}}(1) \oplus\Sigma^{4k+3,2k+2}\mathcal{A}^{\mathbb{C}}(1)//\mathcal{E}^{\mathbb{C}}(1)\right). H ∗∗ ( B μ 2 n ) = Σ 1 , 1 M 2 C ⨁ k ≥ 0 ⨁ ( Σ 4 k + 2 , 2 k + 1 A C ( 1 ) // E C ( 1 ) ⊕ Σ 4 k + 3 , 2 k + 2 A C ( 1 ) // E C ( 1 ) ) . Consequently, we get
E 2 ∗ ∗ ∗ = Ext A C ( 1 ) ∗ ∗ ∗ ( Σ M 2 C ) ⊕ ⨁ k ≥ 0 ( Ext E C ( 1 ) ∗ ∗ ∗ ( Σ 4 k + 2 , 2 k + 1 M 2 C ) ⊕ Ext E C ( 1 ) ∗ ∗ ∗ ( Σ 4 k + 3 , 2 k + 2 M 2 C ) ) , E_2^{***} = \operatorname{Ext}^{***}_{\mathcal{A}^{\mathbb{C}}(1)}( \Sigma M^{\mathbb{C}}_2) \oplus\bigoplus_{k\ge0}\left(\operatorname{Ext}^{***}_{\mathcal{E}^{\mathbb{C}}(1)}(\Sigma^{4k+2,2k+1}M^{\mathbb{C}}_2) \oplus\operatorname{Ext}^{***}_{\mathcal{E}^{\mathbb{C}}(1)}(\Sigma^{4k+3,2k+2}M^{\mathbb{C}}_2)\right), E 2 ∗∗∗ = Ext A C ( 1 ) ∗∗∗ ( Σ M 2 C ) ⊕ k ≥ 0 ⨁ ( Ext E C ( 1 ) ∗∗∗ ( Σ 4 k + 2 , 2 k + 1 M 2 C ) ⊕ Ext E C ( 1 ) ∗∗∗ ( Σ 4 k + 3 , 2 k + 2 M 2 C ) ) , with
Ext E C ( 1 ) ∗ ∗ ∗ ( M 2 C ) ≅ M 2 C [ h 0 , v 1 ] \operatorname{Ext}^{***}_{\mathcal{E}^{\mathbb{C}}(1)}(M^{\mathbb{C}}_2) \cong M^{\mathbb{C}}_2[h_0,v_1] Ext E C ( 1 ) ∗∗∗ ( M 2 C ) ≅ M 2 C [ h 0 , v 1 ] and
Ext A ( 1 ) ∗ ∗ ∗ ( M 2 C ) ≅ M 2 C [ h 0 , h 1 , a , b ] ( h 0 h 1 , τ h 1 3 , h 1 a , a 2 = 4 b ) \operatorname{Ext}_{\mathcal{A}(1)}^{***}\left(M_{2}^{\mathbb{C}}\right)\cong\frac{M_{2}^{\mathbb{C}}[h_{0},h_{1},a,b]}{(h_{0}h_{1},\tau h_{1}^{3},h_{1}a,a^{2}=4b)} Ext A ( 1 ) ∗∗∗ ( M 2 C ) ≅ ( h 0 h 1 , τ h 1 3 , h 1 a , a 2 = 4 b ) M 2 C [ h 0 , h 1 , a , b ] (by Remark 7.1).
Now, we can proceed to analyze the differentials giving us the following theorem.
Theorem 7.3. The E ∞ E_{\infty} E ∞ k q ∗ ∗ C B μ 2 n kq^{\mathbb{C}}_{**}B\mu_{2^{n}} k q ∗∗ C B μ 2 n E ∞ E_{\infty} E ∞ k q ∗ ∗ C kq^{\mathbb{C}}_{**} k q ∗∗ C
(1) For arbitrary n > 1 n>1 n > 1
E ∞ ( k q ∗ ∗ C B μ 2 n ) ≅ E ∞ ( k q ∗ ∗ C ) { x 1 } ( h 0 n , a h 0 n − 1 ) ⊕ ⨁ i = 1 3 M 2 C [ h 0 , v 1 2 ] h 0 n + 2 i − 1 { x 4 i − 1 } ⊕ M 2 C [ h 0 , v 1 2 ] h 0 n − 1 { v 1 x 4 i − 1 } / ( a + h 0 v 1 2 , v 1 2 h 0 n − 1 x 3 + h 0 n + 1 x 7 , v 1 2 h 0 n − 1 x 7 + h 0 n + 1 x 11 ) . E_{\infty}\left(kq^{\mathbb{C}}_{**}B\mu_{2^{n}}\right)\cong\frac{E_{\infty}\left(kq^{\mathbb{C}}_{**}\right)\{x_{1}\}}{(h_{0}^{n},ah_{0}^{n-1})}\oplus\bigoplus_{i=1}^{3}\frac{M_{2}^{\mathbb{C}}[h_{0},v_{1}^{2}]}{h_{0}^{n+2i-1}}\{x_{4i-1}\}\oplus\frac{M_{2}^{\mathbb{C}}[h_{0},v_{1}^{2}]}{h_{0}^{n-1}}\{v_{1}x_{4i-1}\}
\Big/\left(a+h_{0}v_{1}^{2},v_{1}^{2}h_{0}^{n-1}x_{3}+h_{0}^{n+1}x_{7},v_{1}^{2}h_{0}^{n-1}x_{7}+h_{0}^{n+1}x_{11}\right). E ∞ ( k q ∗∗ C B μ 2 n ) ≅ ( h 0 n , a h 0 n − 1 ) E ∞ ( k q ∗∗ C ) { x 1 } ⊕ i = 1 ⨁ 3 h 0 n + 2 i − 1 M 2 C [ h 0 , v 1 2 ] { x 4 i − 1 } ⊕ h 0 n − 1 M 2 C [ h 0 , v 1 2 ] { v 1 x 4 i − 1 } / ( a + h 0 v 1 2 , v 1 2 h 0 n − 1 x 3 + h 0 n + 1 x 7 , v 1 2 h 0 n − 1 x 7 + h 0 n + 1 x 11 ) . (2) For n = 2 n=2 n = 2
k q ∗ , ∗ C B μ 4 ≅ { Z / 2 2 m + 1 [ τ ] ⊕ Z / 2 [ τ ] , k = 8 m + 1 , m ≥ 0 , Z / 2 [ τ ] , k = 8 m + 2 , m ≥ 0 , Z / 2 4 m + 3 [ τ ] ⊕ Z / 2 2 m + 1 [ τ ] , k = 8 m + 3 , m ≥ 0 , Z / 2 2 m + 2 [ τ ] , k = 8 m + 5 , m ≥ 0 , Z / 2 4 m + 5 [ τ ] ⊕ Z / 2 2 m + 1 [ τ ] , k = 8 m + 7 , m ≥ 0 , 0 , otherwise . kq^{\mathbb{C}}_{*,*}B\mu_{4}\cong
\begin{cases}
\mathbb{Z}/2^{2m+1}[\tau]\oplus\mathbb{Z}/2[\tau], & k=8m+1,\quad m\geq0,\\
\mathbb{Z}/2[\tau], & k=8m+2,\quad m\geq0,\\
\mathbb{Z}/2^{4m+3}[\tau]\oplus\mathbb{Z}/2^{2m+1}[\tau], & k=8m+3,\quad m\geq0,\\
\mathbb{Z}/2^{2m+2}[\tau], & k=8m+5,\quad m\geq0,\\
\mathbb{Z}/2^{4m+5}[\tau]\oplus\mathbb{Z}/2^{2m+1}[\tau], & k=8m+7,\quad m\geq0,\\
0, & \text{otherwise}.
\end{cases} k q ∗ , ∗ C B μ 4 ≅ ⎩ ⎨ ⎧ Z / 2 2 m + 1 [ τ ] ⊕ Z /2 [ τ ] , Z /2 [ τ ] , Z / 2 4 m + 3 [ τ ] ⊕ Z / 2 2 m + 1 [ τ ] , Z / 2 2 m + 2 [ τ ] , Z / 2 4 m + 5 [ τ ] ⊕ Z / 2 2 m + 1 [ τ ] , 0 , k = 8 m + 1 , m ≥ 0 , k = 8 m + 2 , m ≥ 0 , k = 8 m + 3 , m ≥ 0 , k = 8 m + 5 , m ≥ 0 , k = 8 m + 7 , m ≥ 0 , otherwise . Proof. Continuing from the previous discussion, the E 2 E_{2} E 2 k q ∗ ∗ C ( B μ 2 n ) kq^{\mathbb{C}}_{**}(B\mu_{2^{n}}) k q ∗∗ C ( B μ 2 n ) Figure 16 . The nontrivial differentials are as follows. For arbitrary n n n
Figure 16. The E n E_n E n d n d_n d n E n + 1 = E ∞ E_{n+1}=E_\infty E n + 1 = E ∞ k q ∗ ∗ B μ 2 n kq_{**}B\mu_{2^n} k q ∗∗ B μ 2 n F 2 [ τ ] \mathbb{F}_2[\tau] F 2 [ τ ] ∘ \circ ∘ F 2 \mathbb{F}_2 F 2 h 0 h_0 h 0 v 1 v_1 v 1
d n ( x 2 ) = h 0 n x 1 , d n ( v 1 x 2 ) = h 0 n + 1 x 3 , d n ( v 1 2 x 2 ) = a h 0 n − 1 x 1 , d n ( x 6 ) = h 0 n − 1 v 1 x 3 , d n ( v 1 x 6 ) = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 , d n ( v 1 2 x 6 ) = h 0 n − 1 v 1 3 x 3 , d n ( x 10 ) = h 0 n − 1 v 1 x 7 , d n ( v 1 x 10 ) = h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 , d n ( v 1 2 x 10 ) = h 0 n − 1 v 1 3 x 7 , d n ( x 14 ) = h 0 n − 1 v 1 x 11 . \begin{aligned}
d_{n}(x_{2})&=h_{0}^{n}x_{1}, & d_{n}(v_{1}x_{2})&=h_{0}^{n+1}x_{3},\\
d_{n}(v_{1}^{2}x_{2})&=ah_{0}^{n-1}x_{1}, & d_{n}(x_{6})&=h_{0}^{n-1}v_{1}x_{3},\\
d_{n}(v_{1}x_{6})&=h_{0}^{n+1}x_{7}+h_{0}^{n-1}v_{1}^{2}x_{3}, & d_{n}(v_{1}^{2}x_{6})&=h_{0}^{n-1}v_{1}^{3}x_{3},\\
d_{n}(x_{10})&=h_{0}^{n-1}v_{1}x_{7}, & d_{n}(v_{1}x_{10})&=h_{0}^{n+1}x_{11}+h_{0}^{n-1}v_{1}^{2}x_{7},\\
d_{n}(v_{1}^{2}x_{10})&=h_{0}^{n-1}v_{1}^{3}x_{7}, & d_{n}(x_{14})&=h_{0}^{n-1}v_{1}x_{11}.
\end{aligned} d n ( x 2 ) d n ( v 1 2 x 2 ) d n ( v 1 x 6 ) d n ( x 10 ) d n ( v 1 2 x 10 ) = h 0 n x 1 , = a h 0 n − 1 x 1 , = h 0 n + 1 x 7 + h 0 n − 1 v 1 2 x 3 , = h 0 n − 1 v 1 x 7 , = h 0 n − 1 v 1 3 x 7 , d n ( v 1 x 2 ) d n ( x 6 ) d n ( v 1 2 x 6 ) d n ( v 1 x 10 ) d n ( x 14 ) = h 0 n + 1 x 3 , = h 0 n − 1 v 1 x 3 , = h 0 n − 1 v 1 3 x 3 , = h 0 n + 1 x 11 + h 0 n − 1 v 1 2 x 7 , = h 0 n − 1 v 1 x 11 . In the case n = 2 n=2 n = 2 k > 3 k>3 k > 3
d 2 ( x 4 k + 2 ) = h 0 v 1 x 4 k − 1 , d_{2}(x_{4k+2})=h_{0}v_{1}x_{4k-1}, d 2 ( x 4 k + 2 ) = h 0 v 1 x 4 k − 1 , d 2 ( v 1 2 x 4 k + 2 ) = h 0 v 1 3 x 4 k − 1 , d_{2}(v_{1}^{2}x_{4k+2})=h_{0}v_{1}^{3}x_{4k-1}, d 2 ( v 1 2 x 4 k + 2 ) = h 0 v 1 3 x 4 k − 1 , d 2 ( v 1 x 4 k + 2 ) = h 0 3 x 4 k + 3 + h 0 v 1 2 x 4 k − 1 . d_{2}(v_{1}x_{4k+2})=h_{0}^{3}x_{4k+3}+h_{0}v_{1}^{2}x_{4k-1}. d 2 ( v 1 x 4 k + 2 ) = h 0 3 x 4 k + 3 + h 0 v 1 2 x 4 k − 1 . These differentials are τ \tau τ v 1 2 v_{1}^{2} v 1 2 E n + 1 E_{n+1} E n + 1 h 0 h_{0} h 0 v 1 2 v_{1}^{2} v 1 2 E n + 1 E_{n+1} E n + 1 E n + 1 E_{n+1} E n + 1 E ∞ E_{\infty} E ∞ Figure 16 . □
Very Effective Hermitian K-theory over real closed fields
Finally, we compute k q ∗ ∗ R B μ 2 n kq^{\mathbb{R}}_{**}B\mu_{2^n} k q ∗∗ R B μ 2 n ρ \rho ρ B μ 2 n B\mu_{2^n} B μ 2 n stem − - − weight pairs.
To begin the Atiyah–Hirzebruch spectral sequence, we first need to understand its input, k q ∗ ∗ R kq^{\mathbb{R}}_{**} k q ∗∗ R [23 ] uses the ρ \rho ρ
E 1 = Ext A ( 1 ) C ∗ ∗ ∗ ( M 2 C ) [ ρ ] ⇒ Ext A ( 1 ) R ∗ ∗ ∗ ( M 2 R ) . E_1=\operatorname{Ext}^{***}_{A(1)_{\mathbb{C}}}(M^{\mathbb{C}}_2)[\rho]\Rightarrow\operatorname{Ext}^{***}_{A(1)_{\mathbb{R}}}(M^{\mathbb{R}}_2). E 1 = Ext A ( 1 ) C ∗∗∗ ( M 2 C ) [ ρ ] ⇒ Ext A ( 1 ) R ∗∗∗ ( M 2 R ) . to compute the E 2 E_2 E 2 k q ∗ ∗ R kq^{\mathbb{R}}_{**} k q ∗∗ R
E 2 = Ext A ( 1 ) R ∗ ∗ ∗ ( M 2 R ) ⇒ k q ∗ ∗ R . E_2=\operatorname{Ext}^{***}_{A(1)_{\mathbb{R}}}(M^{\mathbb{R}}_2)\Rightarrow kq^{\mathbb{R}}_{**}. E 2 = Ext A ( 1 ) R ∗∗∗ ( M 2 R ) ⇒ k q ∗∗ R . The Adams spectral sequence collapses, giving a complete computation of k q ∗ ∗ R kq^{\mathbb{R}}_{**} k q ∗∗ R coweight = = = stem − - − weight , as shown in Figure 17 . See [21 ] for detailed calculations.
Figure 17. Ext A R ( 1 ) ∗ ∗ ∗ ( M 2 R ) \operatorname{Ext}^{***}_{A_{\mathbb{R}}(1)}(M^{\mathbb{R}}_2) Ext A R ( 1 ) ∗∗∗ ( M 2 R ) k q ∗ ∗ R kq^{\mathbb{R}}_{**} k q ∗∗ R ∙ \bullet ∙ F 2 [ τ − 4 ] \mathbb{F}_2[\tau^{-4}] F 2 [ τ − 4 ] [21 Section 12.1] , which can be extracted from [23 ] .
We now analyze the Atiyah–Hirzebruch spectral sequence
E 1 = ⨁ each cell i in B μ 2 n Σ i , ⌊ i + 1 2 ⌋ k q ∗ ∗ R ⇒ k q ∗ ∗ R B μ 2 n . E_1=\bigoplus_{\substack{\text{each cell }i\\\text{in }B\mu_{2n}}}\Sigma^{i,\left\lfloor\frac{i+1}{2}\right\rfloor}kq^{\mathbb{R}}_{**}\Rightarrow kq^{\mathbb{R}}_{**}B\mu_{2n}. E 1 = each cell i in B μ 2 n ⨁ Σ i , ⌊ 2 i + 1 ⌋ k q ∗∗ R ⇒ k q ∗∗ R B μ 2 n . Since B μ 2 n B\mu_{2n} B μ 2 n E 1 E_1 E 1 k q ∗ ∗ R kq^{\mathbb{R}}_{**} k q ∗∗ R [21 Section 6] . The E 1 E_1 E 1 h n h^n h n d 1 d_1 d 1 h n h^n h n
Thus, the E 2 E_2 E 2
E 2 i , ∗ ∗ ≅ { k q ∗ ∗ / h n if i = 2 k + 1 > 0 , ker ( h n : k q ∗ ∗ → k q ∗ ∗ ) if i = 2 k > 0. E_2^{i,**}\cong
\begin{cases}
kq_{**}/h^n & \text{if }i=2k+1>0,\\
\ker\left(h^n:kq_{**}\to kq_{**}\right) & \text{if }i=2k>0.
\end{cases} E 2 i , ∗∗ ≅ { k q ∗∗ / h n ker ( h n : k q ∗∗ → k q ∗∗ ) if i = 2 k + 1 > 0 , if i = 2 k > 0. From the chart perspective, for odd filtrations (i = 2 k + 1 i=2k+1 i = 2 k + 1 E 2 E_2 E 2 h n h^n h n h n h^n h n i = 2 k + 2 i=2k+2 i = 2 k + 2 h n h^n h n h n h^n h n Figure 18 for even filtrations and Figure 19 for odd filtrations.
Figure 18. Charts representing the even filtrations of the E 2 E_2 E 2 k q ∗ ∗ R B μ 2 n kq^{\mathbb{R}}_{**}B\mu_{2n} k q ∗∗ R B μ 2 n F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Figure 19. Charts representing the odd filtrations of the E 2 E_2 E 2 k q ∗ ∗ R B μ 2 n kq^{\mathbb{R}}_{**}B\mu_{2n} k q ∗∗ R B μ 2 n F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Next, the d 2 d_2 d 2 4 i 4i 4 i 4 i + 1 4i+1 4 i + 1 4 i − 2 4i-2 4 i − 2 4 i − 1 4i-1 4 i − 1
d 2 ( x [ 4 i ] ) = η ⋅ x [ 4 i − 2 ] , i > 0 , d 2 ( x [ 4 i + 1 ] ) = η ⋅ x [ 4 i − 1 ] , i > 0. d_2(x[4i])=\eta\cdot x[4i-2],\quad i>0,\qquad d_2(x[4i+1])=\eta\cdot x[4i-1],\quad i>0. d 2 ( x [ 4 i ]) = η ⋅ x [ 4 i − 2 ] , i > 0 , d 2 ( x [ 4 i + 1 ]) = η ⋅ x [ 4 i − 1 ] , i > 0. Recall, d 2 d_2 d 2 t − s t-s t − s E 3 E_3 E 3 η \eta η E 2 E_2 E 2
E 3 i , ∗ ∗ ≅ { ker ( η : k q ∗ ∗ / h n → k q ∗ ∗ / h n ) if i = 4 k + 1 > 4 , ker ( h n ) / η if i = 4 k + 2 > 0 , ( k q ∗ ∗ / h n ) / η if i = 4 k + 3 , ker ( η : ker ( h n ) → ker ( h n ) ) if i = 4 k + 4 > 0. E_3^{i,**}\cong
\begin{cases}
\ker(\eta:kq_{**}/h^n\to kq_{**}/h^n) & \text{if }i=4k+1>4,\\
\ker(h^n)/\eta& \text{if }i=4k+2>0,\\
(kq_{**}/h^n)/\eta& \text{if }i=4k+3,\\
\ker(\eta:\ker(h^n)\to\ker(h^n)) & \text{if }i=4k+4>0.
\end{cases} E 3 i , ∗∗ ≅ ⎩ ⎨ ⎧ ker ( η : k q ∗∗ / h n → k q ∗∗ / h n ) ker ( h n ) / η ( k q ∗∗ / h n ) / η ker ( η : ker ( h n ) → ker ( h n )) if i = 4 k + 1 > 4 , if i = 4 k + 2 > 0 , if i = 4 k + 3 , if i = 4 k + 4 > 0. From the chart perspective, each row of the E 3 E_3 E 3 E 2 E_2 E 2 η \eta η η \eta η Figure 20 , Figure 21 , Figure 22 , and Figure 23 for each congruence class of the filtration modulo 4.
Figure 20. Charts represents filtrations congruent to 0 mod 4 of the E 3 E_3 E 3 F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Figure 21. Charts represents filtrations congruent to 1 mod 4 and greater than 1 organized by coweights. Each • represents F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Figure 22. Charts represents filtrations congruent to 2 mod 4 of the E 3 E_3 E 3 F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Figure 23. Charts represents filtrations congruent to 3 mod 4 of the E 3 E_3 E 3 F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Proposition 7.4. Up to Bott periodicity, the only nontrivial d 3 d_3 d 3
d 3 ( h 1 ⋅ τ h 1 ) = h 0 n − 1 a . d_3(h_1\cdot\tau h_1)=h_0^{n-1}a. d 3 ( h 1 ⋅ τ h 1 ) = h 0 n − 1 a . Proof. Recall that the d 3 d_3 d 3 t − s t-s t − s
d 3 ( x [ i ] ) ∈ { ⟨ x [ i ] , η , h ⟩ , i ≡ 0 ( m o d 4 ) , i > 0 , ⟨ x [ i ] , h , η ⟩ , i ≡ 2 ( m o d 4 ) , i > 2. d_3(x[i])\in\begin{cases}
\langle x[i],\eta,h\rangle, & i\equiv0\pmod{4},\ i>0,\\
\langle x[i],h,\eta\rangle, & i\equiv2\pmod{4},\ i>2.
\end{cases} d 3 ( x [ i ]) ∈ { ⟨ x [ i ] , η , h ⟩ , ⟨ x [ i ] , h , η ⟩ , i ≡ 0 ( mod 4 ) , i > 0 , i ≡ 2 ( mod 4 ) , i > 2. In particular,
h 0 n − 1 a ∈ ⟨ τ h 1 ⋅ h 1 , h 1 , h 0 n ⟩ , h_0^{n-1}a\in\langle\tau h_1\cdot h_1,h_1,h_0^n\rangle, h 0 n − 1 a ∈ ⟨ τ h 1 ⋅ h 1 , h 1 , h 0 n ⟩ , which gives a d 3 d_3 d 3 Figure 20 to Figure 21 , from the class h 1 ⋅ τ h 1 h_1\cdot\tau h_1 h 1 ⋅ τ h 1 Figure 20 , corresponding to coweight 1 mod 4) to h 0 n − 1 a h_0^{n-1}a h 0 n − 1 a n + 2 n+2 n + 2 Figure 21 , corresponding to coweight 2 mod 4).
On the other hand,
d 3 ( ρ τ h 1 ⋅ h 1 ) = ρ h 0 n − 1 a = 0 , d_3(\rho\tau h_1\cdot h_1)=\rho h_0^{n-1}a=0, d 3 ( ρ τ h 1 ⋅ h 1 ) = ρ h 0 n − 1 a = 0 , so the class ρ τ h 1 ⋅ h 1 \rho\tau h_1\cdot h_1 ρ τ h 1 ⋅ h 1 Figure 20 and Figure 21 , survives to the next page, since there are no nontrivial targets for the differential.
There are no possible d 3 d_3 d 3 Figure 22 to Figure 23 , because ⟨ ρ , h 0 n , h 1 ⟩ = 0 \langle\rho,h_0^n,h_1\rangle=0 ⟨ ρ , h 0 n , h 1 ⟩ = 0 d 3 d_3 d 3
Remark 7.5. The row in filtration i = 1 i=1 i = 1 E 2 E_2 E 2 d 2 d_2 d 2
By assembling these charts after applying the d 3 d_3 d 3 E 4 E_4 E 4 Figure 24 .
Figure 24. Chart representing the whole E 4 E_4 E 4 gr A H ∗ k q ∗ ∗ R ( B μ 2 n ) \operatorname{gr}^{*}_{AH}kq^{\mathbb{R}}_{**}(B\mu_{2^n}) gr A H ∗ k q ∗∗ R ( B μ 2 n ) □ = Z / 2 n − 1 [ τ 4 ] \square=\mathbb{Z}/2^{n-1}[\tau^4] □ = Z / 2 n − 1 [ τ 4 ] ⋄ = Z / 2 n − 1 [ τ 4 ] \diamond=\mathbb{Z}/2^{n-1}[\tau^4] ⋄ = Z / 2 n − 1 [ τ 4 ] ∙ = F 2 [ τ 4 ] \bullet=\mathbb{F}_2[\tau^4] ∙ = F 2 [ τ 4 ]
Finally, note that the d 4 d_4 d 4 k q ∗ ∗ R B μ 2 n kq^{\mathbb{R}}_{**}B\mu_{2^n} k q ∗∗ R B μ 2 n E 4 E_4 E 4 gr A H ∗ k q ∗ ∗ R B μ 2 n \operatorname{gr}^{*}_{AH}kq^{\mathbb{R}}_{**}B\mu_{2^n} gr A H ∗ k q ∗∗ R B μ 2 n
To solve extension problems we use the motivic Adams spectral sequence, which has E 2 E_2 E 2
E 2 = Ext A R ( 1 ) ∗ ∗ ∗ ( H ∗ ∗ ( B μ 2 n ) ) ⇒ k q ∗ ∗ R ( B μ 2 n ) . E_2=\operatorname{Ext}^{***}_{A_{\mathbb{R}}(1)}\left(H^{**}(B\mu_{2^n})\right)\Rightarrow kq^{\mathbb{R}}_{**}(B\mu_{2^n}). E 2 = Ext A R ( 1 ) ∗∗∗ ( H ∗∗ ( B μ 2 n ) ) ⇒ k q ∗∗ R ( B μ 2 n ) . From Corollary 2.13 we have
H ∗ ∗ ( B μ 2 n ) = Σ 1 , 1 M 2 R ⊕ ⨁ k ≥ 0 ( Σ 4 k + 2 , 2 k + 1 A R ( 1 ) / / E R ( 1 ) ⊕ Σ 4 k + 3 , 2 k + 2 A R ( 1 ) / / E R ( 1 ) ) . H^{**}(B\mu_{2^n})=\Sigma^{1,1}M^{\mathbb{R}}_2\oplus\bigoplus_{k\geq0}\left(\Sigma^{4k+2,2k+1}A^{\mathbb{R}}(1)//E^{\mathbb{R}}(1)\oplus\Sigma^{4k+3,2k+2}A^{\mathbb{R}}(1)//E^{\mathbb{R}}(1)\right). H ∗∗ ( B μ 2 n ) = Σ 1 , 1 M 2 R ⊕ k ≥ 0 ⨁ ( Σ 4 k + 2 , 2 k + 1 A R ( 1 ) // E R ( 1 ) ⊕ Σ 4 k + 3 , 2 k + 2 A R ( 1 ) // E R ( 1 ) ) . Consequently,
E 2 ∗ ∗ ∗ = Σ 1 , 1 Ext A R ( 1 ) ∗ ∗ ∗ ( M 2 R ) ⊕ ⨁ k ≥ 0 ( Σ 4 k + 2 , 2 k + 1 Ext E R ( 1 ) ∗ ∗ ∗ ( M 2 R ) ⊕ Σ 4 k + 3 , 2 k + 2 Ext E R ( 1 ) ∗ ∗ ∗ ( M 2 R ) ) . E_2^{***}=\Sigma^{1,1}\operatorname{Ext}^{***}_{A_{\mathbb{R}}(1)}(M^{\mathbb{R}}_2)\oplus\bigoplus_{k\geq0}\left(\Sigma^{4k+2,2k+1}\operatorname{Ext}^{***}_{E^{\mathbb{R}}(1)}(M^{\mathbb{R}}_2)\oplus\Sigma^{4k+3,2k+2}\operatorname{Ext}^{***}_{E^{\mathbb{R}}(1)}(M^{\mathbb{R}}_2)\right). E 2 ∗∗∗ = Σ 1 , 1 Ext A R ( 1 ) ∗∗∗ ( M 2 R ) ⊕ k ≥ 0 ⨁ ( Σ 4 k + 2 , 2 k + 1 Ext E R ( 1 ) ∗∗∗ ( M 2 R ) ⊕ Σ 4 k + 3 , 2 k + 2 Ext E R ( 1 ) ∗∗∗ ( M 2 R ) ) . These summands are drawn in the charts below, see Figure 25 .
Figure 25. Chart representing the E n E_n E n k q ∗ ∗ R B μ 2 n kq^{\mathbb{R}}_{**} B\mu_{2^n} k q ∗∗ R B μ 2 n ∙ \bullet ∙ F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ] ρ \rho ρ
Theorem 7.6. The E ∞ E_\infty E ∞ k q ∗ ∗ R ( B μ 2 n ) kq^{\mathbb{R}}_{**}(B\mu_{2^n}) k q ∗∗ R ( B μ 2 n ) E ∞ ( k q ∗ ∗ R ) E_\infty(kq^{\mathbb{R}}_{**}) E ∞ ( k q ∗∗ R ) Figure 26 .
Figure 26. The chart represents E ∞ ( k q k , w R ( B μ 2 n ) ) E_\infty(kq^{\mathbb{R}}_{k,w}(B\mu_{2^n})) E ∞ ( k q k , w R ( B μ 2 n )) F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ]
Proof. Since we have already determined the E 2 E_2 E 2 k q ∗ ∗ R / h kq^{\mathbb{R}}_{**}/h k q ∗∗ R / h
For arbitrary n n n
The differentials in stem 1–4 are obtained by comparison with the Atiyah–Hirzebruch spectral sequence where the ρ \rho ρ ρ \rho ρ Z / 2 n [ τ 4 ] \mathbb{Z}/2^n[\tau^4] Z / 2 n [ τ 4 ] F 2 [ τ 4 ] \mathbb{F}_2[\tau^4] F 2 [ τ 4 ] Z / 2 n − 1 [ τ 4 ] \mathbb{Z}/2^{n-1}[\tau^4] Z / 2 n − 1 [ τ 4 ]
d n ( x 2 ) = h 0 n x 1 , d n ( h 0 τ 2 x 2 ) = τ 2 h 0 n + 1 x 1 . d_n(x_2) = h_0^n x_1,\qquad d_n(h_0\tau^2x_2) = \tau^2h_0^{n+1}x_1. d n ( x 2 ) = h 0 n x 1 , d n ( h 0 τ 2 x 2 ) = τ 2 h 0 n + 1 x 1 . since d n d_n d n n n n h 0 τ 2 h_0\tau^2 h 0 τ 2 h 0 τ 2 h_0\tau^2 h 0 τ 2 d n ( h 0 τ 2 x 2 ) = τ 2 h 0 n + 1 x 1 d_n(h_0\tau^2x_2)=\tau^2h_0^{n+1}x_1 d n ( h 0 τ 2 x 2 ) = τ 2 h 0 n + 1 x 1
The differential d n ( v 1 x 2 ) = h 0 n + 1 x 3 d_n(v_1x_2)=h_0^{n+1}x_3 d n ( v 1 x 2 ) = h 0 n + 1 x 3 Figure 24 .
In filtration 3, there remain two copies of Z / 2 \mathbb{Z}/2 Z /2 F 2 \mathbb{F}_2 F 2 ρ \rho ρ F 2 \mathbb{F}_2 F 2 ρ \rho ρ v 1 x 2 v_1x_2 v 1 x 2 x 3 x_3 x 3
The h 0 τ 2 h_0\tau^2 h 0 τ 2
d n ( h 0 τ 2 v 1 x 2 ) = h 0 τ 2 d n ( v 1 x 2 ) = h 0 n + 2 τ 2 x 3 . d_n(h_0\tau^2v_1x_2)=h_0\tau^2d_n(v_1x_2)=h_0^{n+2}\tau^2x_3. d n ( h 0 τ 2 v 1 x 2 ) = h 0 τ 2 d n ( v 1 x 2 ) = h 0 n + 2 τ 2 x 3 . This is also reflected in the Atiyah–Hirzebruch spectral sequence: the entire tower on h 0 v 1 τ 2 x 2 h_0v_1\tau^2x_2 h 0 v 1 τ 2 x 2 τ 2 x 3 \tau^2x_3 τ 2 x 3 h 0 n + 1 τ 2 x 3 h_0^{n+1}\tau^2x_3 h 0 n + 1 τ 2 x 3
The extension-of-scalars functor again induces a morphism of Adams spectral sequences compatible with all differentials. Under this morphism, ρ \rho ρ h 0 h_0 h 0 v 1 v_1 v 1 x j x_j x j C \mathbb{C} C
The corresponding C \mathbb{C} C Theorem 7.3 . Naturality therefore gives
d n ( x 4 k + 2 ) = h 0 n − 1 v 1 x 4 k − 1 and d n ( v 1 x 4 k + 2 ) = h 0 n + 1 x 4 k + 3 + h 0 n − 1 v 1 2 x 4 k − 1 . d_n(x_{4k+2})=h_0^{n-1}v_1x_{4k-1}\quad\text{and}\quad d_n(v_1x_{4k+2})=h_0^{n+1}x_{4k+3}+h_0^{n-1}v_1^2x_{4k-1}. d n ( x 4 k + 2 ) = h 0 n − 1 v 1 x 4 k − 1 and d n ( v 1 x 4 k + 2 ) = h 0 n + 1 x 4 k + 3 + h 0 n − 1 v 1 2 x 4 k − 1 . Thus, in the R \mathbb{R} R ρ \rho ρ ρ \rho ρ C \mathbb{C} C
These differentials exhibit v 1 2 v_1^2 v 1 2
For n = 2 n=2 n = 2 C \mathbb{C} C R \mathbb{R} R R \mathbb{R} R ρ \rho ρ ρ \rho ρ k > 0 k>0 k > 0 n = 2 n=2 n = 2
This completes the computation of the E n E_n E n E n + 1 E_{n+1} E n + 1 Figure 26 . □
Appendix A. An eta-invariant approach to differentials on indecomposable classes
In this appendix, we use the eta invariant to detect an element in k o 7 ( B C 2 n ) ko_7(BC_{2^n}) k o 7 ( B C 2 n ) 2 n + 3 2^{n+3} 2 n + 3 [13 Theorem 4.1] to estimate its order.
Proposition A.1. For n ≥ 2 n \ge2 n ≥ 2 k o 7 ( B C 2 n ) ko_7(BC_{2^n}) k o 7 ( B C 2 n ) 2 n + 3 2^{n+3} 2 n + 3
:::@references
Before proving the proposition, we recall the necessary background on lens spaces and eta invariants.
Let k ≥ 2 k \ge2 k ≥ 2 q 1 , … , q ℓ q_1,\ldots,q_\ell q 1 , … , q ℓ k k k
L ( k ; q 1 , … , q ℓ ) = S 2 ℓ − 1 / C k , L(k;q_1,\ldots,q_\ell)=S^{2\ell-1}/C_k, L ( k ; q 1 , … , q ℓ ) = S 2 ℓ − 1 / C k , is given by a fixed generator acting by the action ( z 1 , … , z ℓ ) ↦ ( e 2 π i q 1 / k z 1 , … , e 2 π i q ℓ / k z ℓ ) (z_1,\ldots,z_\ell)\mapsto\left(e^{2\pi i q_1/k}z_1,\ldots,e^{2\pi i q_\ell/k}z_\ell\right) ( z 1 , … , z ℓ ) ↦ ( e 2 π i q 1 / k z 1 , … , e 2 π i q ℓ / k z ℓ ) c : L ( k ; q 1 , … , q ℓ ) → B C k c:L(k;q_1,\ldots,q_\ell)\to BC_k c : L ( k ; q 1 , … , q ℓ ) → B C k
If ℓ \ell ℓ k k k [13 the discussion preceding Theorem 4.6] . After choosing a spin structure, the pair
( L ( k ; q 1 , … , q ℓ ) , c ) \left(L(k;q_1,\ldots,q_\ell),c\right) ( L ( k ; q 1 , … , q ℓ ) , c ) determines a class
[ L ( k ; q 1 , … , q ℓ ) , c ] ∈ Ω 2 ℓ − 1 S p i n ( B C k ) . \left[L(k;q_1,\ldots,q_\ell),c\right]\in\Omega^{\mathrm{Spin}}_{2\ell-1}(BC_k). [ L ( k ; q 1 , … , q ℓ ) , c ] ∈ Ω 2 ℓ − 1 Spin ( B C k ) . Applying the Atiyah–Bott–Shapiro orientation gives a class in real connective K K K
Ω 2 ℓ − 1 S p i n ( B C k ) ⟶ k o 2 ℓ − 1 ( B C k ) , \Omega^{\mathrm{Spin}}_{2\ell-1}(BC_k)\longrightarrow ko_{2\ell-1}(BC_k), Ω 2 ℓ − 1 Spin ( B C k ) ⟶ k o 2 ℓ − 1 ( B C k ) , which is 7-connected; see also [1 ] for details.
To detect the order of this class, we use the periodicization map
p : k o 2 ℓ − 1 ( B C k ) ⟶ K O 2 ℓ − 1 ( B C k ) , p:ko_{2\ell-1}(BC_k)\longrightarrow KO_{2\ell-1}(BC_k), p : k o 2 ℓ − 1 ( B C k ) ⟶ K O 2 ℓ − 1 ( B C k ) , followed by an eta invariant on periodic K O KO K O [13 Theorem 4.1] , every virtual representation ρ \rho ρ C k C_k C k
η ( ρ ) : K O 2 ℓ − 1 ( B C k ) ⟶ R / Z . \eta(\rho):KO_{2\ell-1}(BC_k)\longrightarrow\mathbb{R}/\mathbb{Z}. η ( ρ ) : K O 2 ℓ − 1 ( B C k ) ⟶ R / Z . Composing with periodicization gives a homomorphism
η ‾ ( ρ ) : = η ( ρ ) ∘ p : k o 2 ℓ − 1 ( B C k ) ⟶ R / Z . \overline{\eta}(\rho):=\eta(\rho)\circ p:ko_{2\ell-1}(BC_k)\longrightarrow\mathbb{R}/\mathbb{Z}. η ( ρ ) := η ( ρ ) ∘ p : k o 2 ℓ − 1 ( B C k ) ⟶ R / Z . The main idea of the proof proceeds as follows:
Since a group homomorphism cannot increase the order of an element, if η ‾ ( ρ ) ( x ) \overline{\eta}(\rho)(x) η ( ρ ) ( x ) m m m x ∈ k o 2 ℓ − 1 ( B C k ) x\in ko_{2\ell-1}(BC_k) x ∈ k o 2 ℓ − 1 ( B C k ) m m m
Proof. Set k = 2 n k=2^n k = 2 n ρ s : C k → U ( 1 ) \rho_s:C_k\to U(1) ρ s : C k → U ( 1 ) C k C_k C k e 2 π i s / k e^{2\pi i s/k} e 2 π i s / k ρ s − ρ s ′ \rho_s-\rho_{s'} ρ s − ρ s ′
η ‾ s , s ′ : = η ‾ ( ρ s − ρ s ′ ) : k o 2 ℓ − 1 ( B C k ) ⟶ R / Z . \overline{\eta}_{s,s'}:=\overline{\eta}(\rho_s-\rho_{s'}):ko_{2\ell-1}(BC_k)\longrightarrow\mathbb{R}/\mathbb{Z}. η s , s ′ := η ( ρ s − ρ s ′ ) : k o 2 ℓ − 1 ( B C k ) ⟶ R / Z . Writing η s D i r a c \eta^{\mathrm{Dirac}}_s η s Dirac ρ s \rho_s ρ s
η ‾ s , s ′ = ( η s D i r a c − η s ′ D i r a c ) ∘ p . \overline{\eta}_{s,s'}=\left(\eta^{\mathrm{Dirac}}_s-\eta^{\mathrm{Dirac}}_{s'}\right)\circ p. η s , s ′ = ( η s Dirac − η s ′ Dirac ) ∘ p . We now specialize to stem 7, so take ℓ = 4 \ell=4 ℓ = 4 L n : = L ( 2 n ; 1 , 1 , 1 , 1 ) L_n:=L(2^n;1,1,1,1) L n := L ( 2 n ; 1 , 1 , 1 , 1 ) c n : L n → B C 2 n c_n:L_n\to BC_{2^n} c n : L n → B C 2 n ℓ = 4 \ell=4 ℓ = 4 L n L_n L n
[ L n , c n ] ∈ Ω 7 S p i n ( B C 2 n ) ↦ x n ∈ k o 7 ( B C 2 n ) . [L_n,c_n]\in\Omega^{\mathrm{Spin}}_7(BC_{2^n})\mapsto x_n\in ko_7(BC_{2^n}). [ L n , c n ] ∈ Ω 7 Spin ( B C 2 n ) ↦ x n ∈ k o 7 ( B C 2 n ) . We detect the order of x n x_n x n
For q = ( q 1 , … , q ℓ ) \mathbf{q}=(q_1,\ldots,q_\ell) q = ( q 1 , … , q ℓ ) [22 Theorem 2.5(c)] gives
η s Dirac ( L ( 2 n ; q ) ) = − 1 2 n A ^ ℓ ( s + 2 n − 1 ; q 1 , … , q ℓ ) ∈ R / Z , \eta_s^{\operatorname{Dirac}}\left(L(2^n;\mathbf{q})\right)=-\frac{1}{2^n}\widehat{A}_{\ell}\left(s+2^{n-1};q_1,\ldots,q_\ell\right)\in\mathbb{R}/\mathbb{Z}, η s Dirac ( L ( 2 n ; q ) ) = − 2 n 1 A ℓ ( s + 2 n − 1 ; q 1 , … , q ℓ ) ∈ R / Z , where
A ^ ℓ ( s ; x ) = ∑ a + 2 b = ℓ s a a ! A ^ b ( x ) . \widehat{A}_{\ell}(s;\mathbf{x})=\sum_{a+2b=\ell}\frac{s^a}{a!}\widehat{A}_b(\mathbf{x}). A ℓ ( s ; x ) = a + 2 b = ℓ ∑ a ! s a A b ( x ) . Here A ^ \widehat{A} A A ^ \widehat{A} A [14 Section 23.1] .
For the calculation in stem 7, we use
A ^ 1 ( x ) = − 1 24 ∑ i x i 2 and A ^ 2 ( x ) = 1 5760 ( − 4 ∑ i < j x i 2 x j 2 + 7 ( ∑ i x i 2 ) 2 ) . \widehat{A}_1(\mathbf{x})=-\frac{1}{24}\sum_i x_i^2
\qquad\text{and}\qquad
\widehat{A}_2(\mathbf{x})=\frac{1}{5760}\left(-4\sum_{i<j}x_i^2x_j^2+7\left(\sum_i x_i^2\right)^2\right). A 1 ( x ) = − 24 1 i ∑ x i 2 and A 2 ( x ) = 5760 1 − 4 i < j ∑ x i 2 x j 2 + 7 ( i ∑ x i 2 ) 2 . For ℓ = 4 \ell=4 ℓ = 4 q = ( 1 , 1 , 1 , 1 ) \mathbf{q}=(1,1,1,1) q = ( 1 , 1 , 1 , 1 )
A ^ 1 ( 1 , 1 , 1 , 1 ) = − 1 6 and A ^ 2 ( 1 , 1 , 1 , 1 ) = 1 5760 ( − 4 ∑ i < j 1 + 7 ( ∑ i = 1 4 1 ) 2 ) = 11 720 . \widehat{A}_1(1,1,1,1)=-\frac{1}{6}
\qquad\text{and}\qquad
\widehat{A}_2(1,1,1,1)=\frac{1}{5760}\left(-4\sum_{i<j}1+7\left(\sum_{i=1}^{4}1\right)^2\right)=\frac{11}{720}. A 1 ( 1 , 1 , 1 , 1 ) = − 6 1 and A 2 ( 1 , 1 , 1 , 1 ) = 5760 1 − 4 i < j ∑ 1 + 7 ( i = 1 ∑ 4 1 ) 2 = 720 11 . It follows that
A ^ 4 ( s ; 1 , 1 , 1 , 1 ) = s 4 4 ! + s 2 2 ! A ^ 1 ( 1 , 1 , 1 , 1 ) + A ^ 2 ( 1 , 1 , 1 , 1 ) = s 4 24 − s 2 12 + 11 720 . \widehat{A}_4(s;1,1,1,1)=\frac{s^4}{4!}+\frac{s^2}{2!}\widehat{A}_1(1,1,1,1)+\widehat{A}_2(1,1,1,1)=\frac{s^4}{24}-\frac{s^2}{12}+\frac{11}{720}. A 4 ( s ; 1 , 1 , 1 , 1 ) = 4 ! s 4 + 2 ! s 2 A 1 ( 1 , 1 , 1 , 1 ) + A 2 ( 1 , 1 , 1 , 1 ) = 24 s 4 − 12 s 2 + 720 11 . Consequently,
A ^ 4 ( s + 2 n − 1 ; 1 , 1 , 1 , 1 ) = ( s + 2 n − 1 ) 4 24 − ( s + 2 n − 1 ) 2 12 + 11 720 , \widehat{A}_4(s+2^{n-1};1,1,1,1)=\frac{\left(s+2^{n-1}\right)^4}{24}-\frac{\left(s+2^{n-1}\right)^2}{12}+\frac{11}{720}, A 4 ( s + 2 n − 1 ; 1 , 1 , 1 , 1 ) = 24 ( s + 2 n − 1 ) 4 − 12 ( s + 2 n − 1 ) 2 + 720 11 , and hence
η s Dirac ( L ( 2 n ; 1 , 1 , 1 , 1 ) ) = − 1 2 n ( ( s + 2 n − 1 ) 4 24 − ( s + 2 n − 1 ) 2 12 + 11 720 ) ∈ R / Z . \eta_s^{\operatorname{Dirac}}\left(L(2^n;1,1,1,1)\right)=-\frac{1}{2^n}\left(\frac{\left(s+2^{n-1}\right)^4}{24}-\frac{\left(s+2^{n-1}\right)^2}{12}+\frac{11}{720}\right)\in\mathbb{R}/\mathbb{Z}. η s Dirac ( L ( 2 n ; 1 , 1 , 1 , 1 ) ) = − 2 n 1 ( 24 ( s + 2 n − 1 ) 4 − 12 ( s + 2 n − 1 ) 2 + 720 11 ) ∈ R / Z . We now evaluate the relative eta invariant η 1 Dirac − η 0 Dirac \eta_1^{\operatorname{Dirac}}-\eta_0^{\operatorname{Dirac}} η 1 Dirac − η 0 Dirac L n L_n L n
( η 1 Dirac − η 0 Dirac ) ( L n ) = − 1 2 n ( ( 2 n − 1 + 1 ) 4 24 − ( 2 n − 1 + 1 ) 2 12 − 2 4 n − 4 24 + 2 2 n − 2 12 ) = − ( 2 n − 1 + 1 ) 4 − 2 ( 2 n − 1 + 1 ) 2 − 2 4 n − 4 + 2 2 n − 1 24 ⋅ 2 n ∈ R / Z . \begin{aligned}
\left(\eta_1^{\operatorname{Dirac}}-\eta_0^{\operatorname{Dirac}}\right)(L_n)=-\frac{1}{2^n}\left(\frac{\left(2^{n-1}+1\right)^4}{24}-\frac{\left(2^{n-1}+1\right)^2}{12}-\frac{2^{4n-4}}{24}+\frac{2^{2n-2}}{12}\right) \\
&=-\frac{\left(2^{n-1}+1\right)^4-2\left(2^{n-1}+1\right)^2-2^{4n-4}+2^{2n-1}}{24\cdot2^n}\in\mathbb{R}/\mathbb{Z}.
\end{aligned} ( η 1 Dirac − η 0 Dirac ) ( L n ) = − 2 n 1 ( 24 ( 2 n − 1 + 1 ) 4 − 12 ( 2 n − 1 + 1 ) 2 − 24 2 4 n − 4 + 12 2 2 n − 2 ) = − 24 ⋅ 2 n ( 2 n − 1 + 1 ) 4 − 2 ( 2 n − 1 + 1 ) 2 − 2 4 n − 4 + 2 2 n − 1 ∈ R / Z . Set
u n = ( 2 n − 1 + 1 ) 4 − 2 ( 2 n − 1 + 1 ) 2 − 2 4 n − 4 + 2 2 n − 1 . u_n=\left(2^{n-1}+1\right)^4-2\left(2^{n-1}+1\right)^2-2^{4n-4}+2^{2n-1}. u n = ( 2 n − 1 + 1 ) 4 − 2 ( 2 n − 1 + 1 ) 2 − 2 4 n − 4 + 2 2 n − 1 . Then
( η 1 Dirac − η 0 Dirac ) ( L n ) = − u n 3 ⋅ 2 n + 3 ∈ R / Z . \left(\eta_1^{\operatorname{Dirac}}-\eta_0^{\operatorname{Dirac}}\right)(L_n)=-\frac{u_n}{3\cdot2^{n+3}}\in\mathbb{R}/\mathbb{Z}. ( η 1 Dirac − η 0 Dirac ) ( L n ) = − 3 ⋅ 2 n + 3 u n ∈ R / Z . For n ≥ 2 n\geq2 n ≥ 2 u n u_n u n
( 2 n − 1 + 1 ) 4 ≡ 1 ( m o d 2 ) . \left(2^{n-1}+1\right)^4\equiv1\pmod{2}. ( 2 n − 1 + 1 ) 4 ≡ 1 ( mod 2 ) . Thus u n u_n u n 2 2 2 η ˉ 1 , 0 ( x n ) ∈ R / Z \bar{\eta}_{1,0}(x_n) \in\mathbb{R}/\mathbb{Z} η ˉ 1 , 0 ( x n ) ∈ R / Z 2 n + 3 2^{n+3} 2 n + 3
The Atiyah–Hirzebruch spectral sequence shows that k o 7 ( B C 2 n ) ko_7(BC_{2^n}) k o 7 ( B C 2 n ) 2 2 2
k o 7 ( B C 2 n ) ≅ ⨁ i = 1 r Z / 2 j i . ko_7(BC_{2^n}) \cong\bigoplus_{i=1}^{r} \mathbb{Z}/2^{j_i}. k o 7 ( B C 2 n ) ≅ i = 1 ⨁ r Z / 2 j i . Suppose, for contradiction, that
j i ≤ n + 2 for every 1 ≤ i ≤ r . j_i \le n+2 \qquad\text{for every } 1 \le i \le r. j i ≤ n + 2 for every 1 ≤ i ≤ r . Then multiplication by 2 n + 2 2^{n+2} 2 n + 2 2 n + 2 x n = 0 2^{n+2}x_n=0 2 n + 2 x n = 0 η ˉ 1 , 0 \bar{\eta}_{1,0} η ˉ 1 , 0 2 n + 2 η ˉ 1 , 0 ( x n ) = 0 2^{n+2}\bar{\eta}_{1,0}(x_n)=0 2 n + 2 η ˉ 1 , 0 ( x n ) = 0 η ˉ 1 , 0 ( x n ) \bar{\eta}_{1,0}(x_n) η ˉ 1 , 0 ( x n ) 2 n + 3 2^{n+3} 2 n + 3 j i ≥ n + 3 j_i \ge n+3 j i ≥ n + 3 i i i 2 n + 3 2^{n+3} 2 n + 3
Remark A.2. The eta-invariant method described above also applies in higher stems. However, by itself it provides limited information about the differentials supported by indecomposable classes.
Remark A.3. Another possible approach uses the Bockstein homology spectral sequence. This approach is used in [7 ] and [6 ] , where this spectral sequence is combined with the May–Milgram theorem to relate Bockstein differentials to differentials between h 0 h_0 h 0
University of Kentucky
Email address: perna.dhankhar@uky.edu
