Introduction
Quandles are algebraic structures that arise from conjugation in groups and colorings of knot diagrams. The rack space R X R_X R X X X X [11 ] . For an augmented rack, Clauwens equipped a covering of the associated rack space with a strictly associative monoid structure and showed that this monoid is equivalent to a loop space whenever it is connected [10 Section 2.5] . Mostovoy established a comparison between this monoid and the geometric realization of an associated simplicial group [22 Theorem 3.1] . Building on these comparisons, we construct an explicit marked delooping from the augmentation data. We apply its local form to integral quandle homology computations and to the study of the multiplication on the covering monoid.
For a connected quandle X X X x 0 ∈ X x_0 \in X x 0 ∈ X G = Inn ( X ) G = \operatorname{Inn}(X) G = Inn ( X ) H = Stab G ( x 0 ) H = \operatorname{Stab}_G(x_0) H = Stab G ( x 0 ) M X = B ( G ; X ) M_X = B(G; X) M X = B ( G ; X ) Y Y Y G G G H H H S x 0 S_{x_0} S x 0 B M X ≃ Y BM_X \simeq Y B M X ≃ Y B G BG B G R X R_X R X j : B G → Y j : BG \to Y j : B G → Y Theorem 2.1 ). This integral model requires neither finiteness nor faithfulness.
In the finite case, for a prime p p p p ∤ ∣ H ∣ p \nmid|H| p ∤ ∣ H ∣ A ∞ A_{\infty} A ∞
( M X ) ( p ) ≃ A ∞ Ω ( ( S 2 ∨ ( B G ) p ∧ ) ( p ) ) (M_X)_{(p)} \simeq_{A_{\infty}} \Omega\left((S^2 \vee(BG)^{\wedge}_p)_{(p)}\right) ( M X ) ( p ) ≃ A ∞ Ω ( ( S 2 ∨ ( B G ) p ∧ ) ( p ) ) (Theorem 3.2 and Proposition 3.3 ). Here the subscript ( p ) (p) ( p ) p p p ∧ p \wedge_p ∧ p p p p p p p ( B G ) p ∧ (BG)^{\wedge}_p ( B G ) p ∧ A ∞ A_{\infty} A ∞ ( M X ) ( p ) (M_X)_{(p)} ( M X ) ( p ) Theorem 3.6 ).
Combining this model with decompositions of the rack and quandle complexes, we compute quandle homology from the homology of an auxiliary loop space with coefficients in Z ( p ) \mathbb{Z}_{(p)} Z ( p ) [24 Corollary 6.2] shows that, in degrees n ≥ 2 n \ge2 n ≥ 2 ∣ X ∣ |X| ∣ X ∣ H n Q ( X ; Z ) H_n^Q(X;\mathbb{Z}) H n Q ( X ; Z ) C m C_m C m m m m
Theorem A (Cyclic Alexander quandles). Let p p p r ≥ 1 r \ge1 r ≥ 1 ω ∈ ( Z / p r Z ) × \omega\in(\mathbb{Z}/p^r\mathbb{Z})^\times ω ∈ ( Z / p r Z ) × d > 1 d > 1 d > 1 p − 1 p - 1 p − 1 X = Z / p r Z X = \mathbb{Z}/p^r\mathbb{Z} X = Z / p r Z x ∗ y = ω x + ( 1 − ω ) y x * y = \omega x + (1 - \omega)y x ∗ y = ω x + ( 1 − ω ) y
H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ ( C p r ) b n ( n ≥ 2 ) H_1^Q(X;\mathbb{Z}) \cong\mathbb{Z}, \qquad H_n^Q(X;\mathbb{Z}) \cong(C_{p^r})^{b_n} \quad(n \ge2) H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ ( C p r ) b n ( n ≥ 2 ) where the nonnegative integers b n b_n b n
∑ n ≥ 2 b n t n = t 2 d − 1 ( 1 + t ) 1 − t 2 d − 2 − t 2 d − 1 − t 2 d . \sum_{n \ge2} b_n t^n = \frac{t^{2d-1}(1+t)}{1-t^{2d-2}-t^{2d-1}-t^{2d}}. n ≥ 2 ∑ b n t n = 1 − t 2 d − 2 − t 2 d − 1 − t 2 d t 2 d − 1 ( 1 + t ) . In the case r = 1 r = 1 r = 1 F p \mathbb{F}_p F p [24 Theorem 1.1, Corollary 5.9] . Ishikawa also computed the quandle homology of connected dihedral quandles using a different spectral sequence [15 ] . Theorem A extends the additive computation in [24 Theorem 1.1] to the above cyclic family of prime-power order. The multiplicities b n b_n b n r r r p r p^r p r Corollary 6.9 gives the corresponding prime-by-prime formula for every finite connected cyclic Alexander quandle ( A , T ) (A,T) ( A , T ) gcd ( ∣ A ∣ , ord ( T ) ) = 1 \gcd(|A|,\operatorname{ord}(T)) = 1 g cd( ∣ A ∣ , ord ( T )) = 1 § 7 . In particular, the result applies to every dihedral quandle of odd order greater than one. At a good prime p p p p p p Theorem 6.8 ).
Theorem B (The tetrahedral quandle). Let F 4 = F 2 [ ω ] / ( ω 2 + ω + 1 ) \mathbb{F}_4 = \mathbb{F}_2[\omega]/(\omega^2 + \omega+ 1) F 4 = F 2 [ ω ] / ( ω 2 + ω + 1 ) X = F 4 X = \mathbb{F}_4 X = F 4 x ∗ y = ω x + ( 1 − ω ) y x * y = \omega x + (1 - \omega)y x ∗ y = ω x + ( 1 − ω ) y
H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ C 2 a n ⊕ C 4 b n ( n ≥ 2 ) H_1^Q(X;\mathbb{Z}) \cong\mathbb{Z}, \qquad H_n^Q(X;\mathbb{Z}) \cong C_2^{a_n} \oplus C_4^{b_n} \quad(n \ge2) H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ C 2 a n ⊕ C 4 b n ( n ≥ 2 ) where the nonnegative integers a n , b n a_n,b_n a n , b n
∑ n ≥ 2 b n t n = t 3 1 − t − t 3 , ∑ n ≥ 2 ( a n + b n ) t n = t 2 ( 1 + t ) 1 − t − t 2 − t 3 . \sum_{n \ge2} b_n t^n = \frac{t^3}{1-t-t^3}, \qquad\sum_{n \ge2}(a_n+b_n)t^n = \frac{t^2(1+t)}{1-t-t^2-t^3}. n ≥ 2 ∑ b n t n = 1 − t − t 3 t 3 , n ≥ 2 ∑ ( a n + b n ) t n = 1 − t − t 2 − t 3 t 2 ( 1 + t ) . Theorem B establishes the recurrence for the multiplicities of the C 4 C_4 C 4 [23 Example 13(ii)] . Their prediction for the entire group, however, requires a correction: in degree 8, the group is H 8 Q ( X ; Z ) ≅ C 2 33 ⊕ C 4 4 H_8^Q(X;\mathbb{Z}) \cong C_2^{33} \oplus C_4^4 H 8 Q ( X ; Z ) ≅ C 2 33 ⊕ C 4 4
For a finite connected quandle X X X p p p k k k p p p Proposition 5.9 gives a noncanonical graded algebra isomorphism
H R ∗ ( X ; k ) ≅ Sh k ( s H ∗ ( Ω L ; k ) ) , L = Cofib ( B H → B G ) , H_R^*(X;k) \cong\operatorname{Sh}_k(sH^*(\Omega L;k)), \qquad L = \operatorname{Cofib}(BH \to BG), H R ∗ ( X ; k ) ≅ Sh k ( s H ∗ ( Ω L ; k )) , L = Cofib ( B H → B G ) , where Sh k ( V ) \operatorname{Sh}_k(V) Sh k ( V ) V V V s s s H ∗ ( Ω L ; k ) H^*(\Omega L;k) H ∗ ( Ω L ; k )
For a finite connected quandle X X X p p p p p p ( M X ) ( p ) (\mathcal{M}_X)_{(p)} ( M X ) ( p ) E 2 E_2 E 2 p ∤ ∣ X ∣ p \nmid|X| p ∤ ∣ X ∣ Theorem 10.4 ). When p ∥ ∣ X ∣ p \parallel|X| p ∥ ∣ X ∣ ( M X ) ( p ) (\mathcal{M}_X)_{(p)} ( M X ) ( p ) p p p Theorem 10.5 ).
Section 1 sets out the definitions and conventions. Section 2 constructs the integral model, and Section 3 treats localization, completion, and the classification of A ∞ A_\infty A ∞ § 6 and prove Theorems A and B in § 7 and § 8 . § 9 illustrates the tensor and Tor terms with a product Alexander quandle of order 49. § 10 gives local computations for transposition quandles and computes the rack and positive-degree quandle homology of rank-one symplectic quandles over finite fields of order greater than 27, away from the defining characteristic. § 11 is devoted to applications to multiplicative structures and unstable homotopy theory. Appendix A supplies the Hall-word count and the asymptotic estimate used in the degreewise growth bound.
Augmented quandles and their rack spaces
We work in the category of compactly generated spaces and use CW models when necessary. Unless a continuous strict monoid homomorphism is specified, an equivalence of monoids means an equivalence of grouplike A ∞ A_{\infty} A ∞ R R R Z \mathbb{Z} Z Z Z Z Z R Z_R Z R Z Z ( p ) = Z ( p ) Z_{\mathbb{Z}_{(p)}}=Z_{(p)} Z Z ( p ) = Z ( p ) Z Q Z_{\mathbb{Q}} Z Q p p p Z p ∧ Z^{\wedge}_p Z p ∧ p p p C n C_n C n n n n
Right actions and augmentation data
A quandle is a nonempty set X X X x ∗ y x * y x ∗ y x ∗ x = x x * x = x x ∗ x = x S y : x ↦ x ∗ y S_y : x \mapsto x * y S y : x ↦ x ∗ y ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) (x * y) * z = (x * z) * (y * z) ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) x g k = ( x g ) k x^{gk}=(x^g)^k x g k = ( x g ) k Inn ( X ) \operatorname{Inn}(X) Inn ( X ) S y S_y S y y ↦ S y y \mapsto S_y y ↦ S y
As ( X ) = ⟨ e x ( x ∈ X ) ∣ e y − 1 e x e y = e x ∗ y ⟩ . (1) \operatorname{As}(X)=\langle e_x\ (x\in X)\mid e_y^{-1}e_xe_y=e_{x*y}\rangle. \tag*{(1)} As ( X ) = ⟨ e x ( x ∈ X ) ∣ e y − 1 e x e y = e x ∗ y ⟩ . (1) The group homomorphism As ( X ) → Inn ( X ) \operatorname{As}(X)\to\operatorname{Inn}(X) As ( X ) → Inn ( X ) e x e_x e x S x S_x S x
An augmented quandle consists of a group G G G G G G X X X η : X → G \eta:X\to G η : X → G
η ( x g ) = g − 1 η ( x ) g , x ∗ y = x η ( y ) , x η ( x ) = x . (2) \eta(x^g)=g^{-1}\eta(x)g,\qquad x*y=x^{\eta(y)},\qquad x^{\eta(x)}=x. \tag*{(2)} η ( x g ) = g − 1 η ( x ) g , x ∗ y = x η ( y ) , x η ( x ) = x . (2) We mainly use the canonical augmentation G = Inn ( X ) G=\operatorname{Inn}(X) G = Inn ( X ) η ( x ) = S x \eta(x)=S_x η ( x ) = S x § 2 and § 3 , the first part of § 4 , and Subsections 4.1 and 4.2, we work more generally with transitive augmentations for which η ( X ) \eta(X) η ( X ) G G G
Fix x 0 ∈ X x_0\in X x 0 ∈ X H = Stab G ( x 0 ) H=\operatorname{Stab}_G(x_0) H = Stab G ( x 0 ) h = η ( x 0 ) h=\eta(x_0) h = η ( x 0 ) i : H ↪ G i:H\hookrightarrow G i : H ↪ G (2) , we have h ∈ Z ( H ) h\in Z(H) h ∈ Z ( H ) H g ↦ x 0 g Hg\mapsto x_0^g H g ↦ x 0 g H \ G ≅ X H\backslash G\cong X H \ G ≅ X G G G
H g ∗ H k = H g k − 1 h k , η ( H g ) = g − 1 h g . (3) Hg*Hk=Hgk^{-1}hk,\qquad\eta(Hg)=g^{-1}hg. \tag*{(3)} H g ∗ H k = H g k − 1 hk , η ( H g ) = g − 1 h g . (3) Since h ∈ Z ( H ) h\in Z(H) h ∈ Z ( H ) h h h H H H G G G
Remark 1.1. The stabilizer satisfies H ⊆ C G ( h ) H\subseteq C_G(h) H ⊆ C G ( h ) η \eta η
Cubical spaces and the monoid product
For a right G G G S S S B ( S ; X ) n = S × X n B(S;X)_n=S\times X^n B ( S ; X ) n = S × X n i i i
d i 0 ( s ; x 1 , … , x n ) = ( s ; x 1 , … , x i ^ , … , x n ) , (4) d_i^0(s;x_1,\ldots,x_n)=(s;x_1,\ldots,\widehat{x_i},\ldots,x_n), \tag*{(4)} d i 0 ( s ; x 1 , … , x n ) = ( s ; x 1 , … , x i , … , x n ) , (4) d i 1 ( s ; x 1 , … , x n ) = ( s η ( x i ) ; x 1 ∗ x i , … , x i − 1 ∗ x i , x i + 1 , … , x n ) . (5) d_i^1(s;x_1,\ldots,x_n)=(s\eta(x_i);x_1*x_i,\ldots,x_{i-1}*x_i,x_{i+1},\ldots,x_n). \tag*{(5)} d i 1 ( s ; x 1 , … , x n ) = ( sη ( x i ) ; x 1 ∗ x i , … , x i − 1 ∗ x i , x i + 1 , … , x n ) . (5) We also write B ( S ; X ) B(S;X) B ( S ; X ) R X = B ( ∗ ; X ) R_X=B(*;X) R X = B ( ∗ ; X ) Λ \Lambda Λ Λ \Lambda Λ ∂ n \partial_n ∂ n R X R_X R X
C n R ( X ; Λ ) = Λ [ X n ] , ∂ n = ∑ i = 1 n ( − 1 ) i ( d i 0 − d i 1 ) , C 0 R ( X ; Λ ) = Λ . (6) C_n^R(X;\Lambda)=\Lambda[X^n],\qquad\partial_n=\sum_{i=1}^{n}(-1)^i(d_i^0-d_i^1),\qquad C_0^R(X;\Lambda)=\Lambda. \tag*{(6)} C n R ( X ; Λ ) = Λ [ X n ] , ∂ n = i = 1 ∑ n ( − 1 ) i ( d i 0 − d i 1 ) , C 0 R ( X ; Λ ) = Λ. (6) We define C ∗ ( S ; X ; Λ ) C_*(S;X;\Lambda) C ∗ ( S ; X ; Λ ) H n R ( X ; Λ ) = H n ( R X ; Λ ) H_n^R(X;\Lambda)=H_n(R_X;\Lambda) H n R ( X ; Λ ) = H n ( R X ; Λ ) R X R_X R X ∂ 1 = 0 \partial_1=0 ∂ 1 = 0
For the specified augmentation, put M = B ( G ; X ) \mathcal{M}=B(G;X) M = B ( G ; X ) M X = B ( Inn ( X ) ; X ) \mathcal{M}_X=B(\operatorname{Inn}(X);X) M X = B ( Inn ( X ) ; X ) N X = B ( X ; X ) N_X=B(X;X) N X = B ( X ; X ) x g x^g x g M × M → M \mathcal{M}\times\mathcal{M}\to\mathcal{M} M × M → M
[ g ; t 1 , x 1 , … , t m , x m ] [ k ; s 1 , y 1 , … , s n , y n ] = [ g k ; t 1 , x 1 k , … , t m , x m k , s 1 , y 1 , … , s n , y n ] (7) [g;t_1,x_1,\ldots,t_m,x_m]\,[k;s_1,y_1,\ldots,s_n,y_n]
=[gk;t_1,x_1^k,\ldots,t_m,x_m^k,s_1,y_1,\ldots,s_n,y_n] \tag*{(7)} [ g ; t 1 , x 1 , … , t m , x m ] [ k ; s 1 , y 1 , … , s n , y n ] = [ g k ; t 1 , x 1 k , … , t m , x m k , s 1 , y 1 , … , s n , y n ] (7) defines a strictly associative product on M \mathcal{M} M 1 ∈ G 1\in G 1 ∈ G η ( x k ) = k − 1 η ( x ) k \eta(x^k)=k^{-1}\eta(x)k η ( x k ) = k − 1 η ( x ) k ( x ∗ y ) k = x k ∗ y k (x*y)^k=x^k*y^k ( x ∗ y ) k = x k ∗ y k x g k = ( x g ) k x^{gk}=(x^g)^k x g k = ( x g ) k [10 Sections 2.4–2.6] .
The realization has CW type even for an arbitrary set of cubes, and the unit vertex is a closed CW subcomplex; hence M \mathcal{M} M
The vertices form a subgroup G ⊂ M G\subset\mathcal{M} G ⊂ M a ∈ G a\in G a ∈ G [ g ; t , x ] [g;t,x] [ g ; t , x ] [ a g ; t , x ] [ag;t,x] [ a g ; t , x ] M → R X \mathcal{M}\to R_X M → R X B ( S ; X ) ≅ S × G M B(S;X)\cong S\times_G\mathcal{M} B ( S ; X ) ≅ S × G M S S S M \mathcal{M} M M → N X \mathcal{M}\to N_X M → N X
N X ≅ H \ M , [ g ; t , x ] ⟼ [ x 0 g ; t , x ] . (8) N_X\cong H\backslash\mathcal{M},\qquad[g;t,x]\longmapsto[x_0^g;t,x]. \tag*{(8)} N X ≅ H \ M , [ g ; t , x ] ⟼ [ x 0 g ; t , x ] . (8) The covering property follows by lifting a cube with a prescribed initial vertex [Cl, Section 2.5][FR1, Section 3].
The edge [ g ; t , x ] [g;t,x] [ g ; t , x ] g g g g η ( x ) g\eta(x) g η ( x ) M \mathcal{M} M M \mathcal{M} M π 1 ( M ) \pi_1(\mathcal{M}) π 1 ( M ) B Z = ∣ [ n ] ↦ Z n ∣ BZ=|[n]\mapsto Z^n| B Z = ∣ [ n ] ↦ Z n ∣ Z Z Z A ∞ A_\infty A ∞ Z Z Z B Z BZ B Z A ∞ A_\infty A ∞ A ∞ A_\infty A ∞ M ≃ A ∞ Ω B M \mathcal{M}\simeq_{A_\infty}\Omega B\mathcal{M} M ≃ A ∞ Ω B M ( x , y ) (x,y) ( x , y ) R X R_X R X e x e y = e y e x ∗ y e_xe_y=e_ye_{x*y} e x e y = e y e x ∗ y As ( X ) → G \operatorname{As}(X)\to G As ( X ) → G e x ↦ η ( x ) e_x\mapsto\eta(x) e x ↦ η ( x ) (1) and covering space theory that
π 1 ( R X ) = As ( X ) , π 1 ( M ) = ker ( As ( X ) → G ) . (9) \pi_1(R_X)=\operatorname{As}(X),\qquad\pi_1(\mathcal{M})=\ker(\operatorname{As}(X)\to G). \tag*{(9)} π 1 ( R X ) = As ( X ) , π 1 ( M ) = ker ( As ( X ) → G ) . (9) For every n ≥ 2 n\ge2 n ≥ 2 π 1 ( R X ) = As ( X ) \pi_1(R_X)=\operatorname{As}(X) π 1 ( R X ) = As ( X ) π n ( R X ) \pi_n(R_X) π n ( R X ) [12 Proposition 5.2] . This also follows from the Clauwens monoid structure on the universal cover B ( As ( X ) ; X ) B(\operatorname{As}(X);X) B ( As ( X ) ; X ) As ( X ) \operatorname{As}(X) As ( X ) x ↦ e x x\mapsto e_x x ↦ e x [10 Proposition 2.16] . Indeed, deck transformations are left multiplications by vertices, and left multiplication along a path from the unit to a vertex gives a homotopy to the identity compatible with change of basepoint.
Definition 1.2. When H H H p p p p ∤ ∣ H ∣ p \nmid\lvert H\rvert p ∤ ∣ H ∣ p p p
An integral model with its classifying map
We use the generating transitive augmentation and the notation G G G H H H h h h i i i M \mathcal{M} M § 2 , without imposing finiteness.
Define group homomorphisms
a : H × Z ⟶ G , a ( u , m ) = i ( u ) , b h : H × Z ⟶ H , b h ( u , m ) = u h m . (10) a : H \times\mathbb{Z} \longrightarrow G,\qquad a(u,m) = i(u),\qquad b_h : H \times\mathbb{Z} \longrightarrow H,\qquad b_h(u,m) = u h^m. \tag*{(10)} a : H × Z ⟶ G , a ( u , m ) = i ( u ) , b h : H × Z ⟶ H , b h ( u , m ) = u h m . (10) The condition h ∈ Z ( H ) h \in Z(H) h ∈ Z ( H ) b h b_h b h B a Ba B a B b h Bb_h B b h Y Y Y j j j
Y = hocolim [ B G ← B a B ( H × Z ) → B b h B H ] , j : B G ⟶ Y . (11) Y = \operatorname{hocolim}\left[BG \xleftarrow{Ba} B(H \times\mathbb{Z}) \xrightarrow{Bb_h} BH\right],\qquad j : BG \longrightarrow Y. \tag*{(11)} Y = hocolim [ B G B a B ( H × Z ) B b h B H ] , j : B G ⟶ Y . (11) We take the double mapping cylinder as the model for Y Y Y j j j Figure 1 .
Figure 1. Schematic double mapping cylinder Y Y Y A = B ( H × Z ) A = B(H \times\mathbb{Z}) A = B ( H × Z ) j : B G ↪ Y j : BG \hookrightarrow Y j : B G ↪ Y B G BG B G
Theorem 2.1 (Integral model). Let X X X Y Y Y (11) is a simply connected CW complex, and there is a based homotopy equivalence under B G BG B G
B M ≃ Y . B\mathcal{M} \simeq Y. B M ≃ Y . Here the continuous map B G → B M BG \to B\mathcal{M} B G → B M G ⊂ M = B ( G ; X ) G \subset\mathcal{M} = B(G;X) G ⊂ M = B ( G ; X ) A ∞ A_{\infty} A ∞
M ≃ A ∞ Ω Y . (12) \mathcal{M} \simeq_{A_{\infty}} \Omega Y. \tag*{(12)} M ≃ A ∞ Ω Y . (12) For any classifying map γ : R X → B G \gamma: R_X \to BG γ : R X → B G M → R X \mathcal{M} \to R_X M → R X B G BG B G
( R X , γ ) ≃ ( hofib ( j : B G → Y ) , hofib ( j ) ⟶ B G ) . (13) (R_X,\gamma) \simeq\left(\operatorname{hofib}(j : BG \to Y),\ \operatorname{hofib}(j) \longrightarrow BG\right). \tag*{(13)} ( R X , γ ) ≃ ( hofib ( j : B G → Y ) , hofib ( j ) ⟶ B G ) . (13) If G G G Y Y Y
An equivalence over B G BG B G B G BG B G
The proof may be skipped on a first reading: subsequent sections use the model ( Y , j ) (Y,j) ( Y , j )
Proof of Theorem 2.1 . We first describe the simplicial group used in the comparison. For n ≥ 0 n \ge0 n ≥ 0 E n E_n E n G G G u j ( x ) u_j(x) u j ( x ) x ∈ X x \in X x ∈ X 0 ≤ j < n 0 \le j < n 0 ≤ j < n
the relations of G , g − 1 u j ( x ) g = u j ( x g ) . (14) \text{the relations of }G,\qquad g^{-1}u_j(x)g=u_j(x^g). \tag*{(14)} the relations of G , g − 1 u j ( x ) g = u j ( x g ) . (14) In particular, E 0 = G E_0=G E 0 = G d k : E n → E n − 1 d_k:E_n\to E_{n-1} d k : E n → E n − 1 n ≥ 1 n\ge1 n ≥ 1 s k : E n → E n + 1 s_k:E_n\to E_{n+1} s k : E n → E n + 1 n ≥ 0 n\ge0 n ≥ 0 0 ≤ k ≤ n 0\le k\le n 0 ≤ k ≤ n G G G
d k u j ( x ) = { 1 , k = j = 0 , u j − 1 ( x ) , k ≤ j , j > 0 , u j ( x ) , k > j , j < n − 1 , η ( x ) , k = n , j = n − 1 , s k u j ( x ) = { u j + 1 ( x ) , k ≤ j , u j ( x ) , k > j . d_k u_j(x)=
\begin{cases}
1, & k=j=0,\\
u_{j-1}(x), & k\le j,\ j>0,\\
u_j(x), & k>j,\ j<n-1,\\
\eta(x), & k=n,\ j=n-1,
\end{cases}
\qquad
s_k u_j(x)=
\begin{cases}
u_{j+1}(x), & k\le j,\\
u_j(x), & k>j.
\end{cases} d k u j ( x ) = ⎩ ⎨ ⎧ 1 , u j − 1 ( x ) , u j ( x ) , η ( x ) , k = j = 0 , k ≤ j , j > 0 , k > j , j < n − 1 , k = n , j = n − 1 , s k u j ( x ) = { u j + 1 ( x ) , u j ( x ) , k ≤ j , k > j . Equivariance of η \eta η u j ( x ) u_j(x) u j ( x ) Δ [ 1 ] \Delta[1] Δ [ 1 ] 0 0 0 0 , … , j 0,\ldots,j 0 , … , j 1 1 1 j + 1 , … , n j+1,\ldots,n j + 1 , … , n 1 1 1 0 0 0 1 1 1 η ( x ) \eta(x) η ( x ) Δ [ 1 ] \Delta[1] Δ [ 1 ] E_\mathbin{\bullet} . Write
E=\lvert E_\mathbin{\bullet}\rvert
for its geometric realization. The degreewise multiplication induces a topological group structure, and the constant simplicial subgroup G G G G ↪ E G\hookrightarrow E G ↪ E E_\mathbin{\bullet} is the one constructed in [22 Section 2.1] for the augmentation extended to the free group on X X X [22 Theorem 3.1 and its proof] gives a continuous strict monoid homomorphism M → E \mathcal{M}\to E M → E G G G ( g , x ) (g,x) ( g , x ) ( x g − 1 , g ) (x^{g^{-1}},g) ( x g − 1 , g ) x ∈ X x\in X x ∈ X t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] ε x ( t ) = [ 1 ; t , x ] \varepsilon_x(t)=[1;t,x] ε x ( t ) = [ 1 ; t , x ]
ε x ( 0 ) = 1 , ε x ( 1 ) = η ( x ) , ε x ( t ) k = k ε x k ( t ) ( k ∈ G ) , \varepsilon_x(0)=1,\qquad\varepsilon_x(1)=\eta(x),\qquad\varepsilon_x(t)k=k\varepsilon_{x^k}(t)\quad(k\in G), ε x ( 0 ) = 1 , ε x ( 1 ) = η ( x ) , ε x ( t ) k = k ε x k ( t ) ( k ∈ G ) , and
[ g ; t 1 , x 1 , … , t n , x n ] = g ε x 1 ( t 1 ) ⋯ ε x n ( t n ) . [g;t_1,x_1,\ldots,t_n,x_n]=g\varepsilon_{x_1}(t_1)\cdots\varepsilon_{x_n}(t_n). [ g ; t 1 , x 1 , … , t n , x n ] = g ε x 1 ( t 1 ) ⋯ ε x n ( t n ) . These identities recover both face maps in (4) –(5) . Under the indicated identification of edges and vertex actions, the cubical product agrees with the product in Mostovoy’s model.
Let F n F_n F n v 0 , … , v n − 1 v_0,\ldots,v_{n-1} v 0 , … , v n − 1 F 0 = 1 F_0=1 F 0 = 1
G ∗ H ( H × F n ) → ≅ E n , g ↦ g , v j ↦ u j ( x 0 ) . (15) G*_{H}(H\times F_n)\xrightarrow{\cong}E_n,\qquad g\mapsto g,\quad v_j\mapsto u_j(x_0). \tag*{(15)} G ∗ H ( H × F n ) ≅ E n , g ↦ g , v j ↦ u j ( x 0 ) . (15) Indeed, H H H u j ( x 0 ) u_j(x_0) u j ( x 0 ) G G G u j ( x 0 g ) u_j(x_0^g) u j ( x 0 g ) g − 1 v j g g^{-1}v_jg g − 1 v j g v j v_j v j H H H
Use nerve models for classifying spaces, and put A = B ( H × Z ) A=B(H\times\mathbb{Z}) A = B ( H × Z ) s : H → H × Z s:H\to H\times\mathbb{Z} s : H → H × Z s ( u ) = ( u , 0 ) s(u)=(u,0) s ( u ) = ( u , 0 ) B H BH B H A A A
R n = B G ∪ B H A 0 ∪ B H ⋯ ∪ B H A n − 1 , R 0 = B G . (16) R_n=BG\mathop{\cup}_{BH}A_0\mathop{\cup}_{BH}\cdots\mathop{\cup}_{BH}A_{n-1},\qquad R_0=BG. \tag*{(16)} R n = B G ∪ B H A 0 ∪ B H ⋯ ∪ B H A n − 1 , R 0 = B G . (16) where each A j A_j A j A A A B H BH B H B s Bs B s B i Bi B i A j A_j A j Δ [ 1 ] n \Delta[1]_n Δ [ 1 ] n u j ( x 0 ) u_j(x_0) u j ( x 0 ) Δ [ 1 ] \Delta[1] Δ [ 1 ] 1 1 1 0 0 0 B G BG B G B a Ba B a B i ∘ B b h Bi\circ Bb_h B i ∘ B b h B G BG B G B H BH B H a ∘ s = i a\circ s=i a ∘ s = i b h ∘ s = id H b_h\circ s=\operatorname{id}_H b h ∘ s = id H
The homomorphisms ( u , m ) ↦ i ( u ) u j ( x 0 ) m (u,m)\mapsto i(u)u_j(x_0)^m ( u , m ) ↦ i ( u ) u j ( x 0 ) m H × Z H\times\mathbb{Z} H × Z E n E_n E n G ↪ E n G\hookrightarrow E_n G ↪ E n R n → B E n R_n\to BE_n R n → B E n B G BG B G [13 Theorem 1B.11 and Example 1B.12] , applied to the injections H ↪ G H\hookrightarrow G H ↪ G H ↪ H × Z H\hookrightarrow H\times\mathbb{Z} H ↪ H × Z (15) . The CW inclusions in (16) make the ordinary pushout a homotopy pushout. The maps form a simplicial map by the displayed formulas, including d 0 u 0 ( x 0 ) = 1 d_{0}u_{0}(x_{0})=1 d 0 u 0 ( x 0 ) = 1 d n u n − 1 ( x 0 ) = h d_{n}u_{n-1}(x_{0})=h d n u n − 1 ( x 0 ) = h R ∙ R_{\bullet} R ∙ B G BG B G A A A B E ∙ BE_{\bullet} B E ∙ [29 Appendix A, Proposition A.1(ii), (iv)] and compatibility with finite products [20 Theorem 11.5] therefore give
∣ R ∙ ∣ ≃ ∣ B E ∙ ∣ ≅ B E |R_{\bullet}| \simeq|BE_{\bullet}| \cong BE ∣ R ∙ ∣ ≃ ∣ B E ∙ ∣ ≅ B E under B G BG B G
Put I = [ 0 , 1 ] I=[0,1] I = [ 0 , 1 ] ∣ R ∙ ∣ |R_{\bullet}| ∣ R ∙ ∣ Y Y Y B H × I BH \times I B H × I B s Bs B s B G BG B G B G BG B G Δ [ 1 ] \Delta[1] Δ [ 1 ] B i ( B H ) Bi(BH) B i ( B H ) Y → ∣ R ∙ ∣ Y\to|R_{\bullet}| Y → ∣ R ∙ ∣ B G BG B G M M M E E E M → E M\to E M → E B M → B E BM\to BE B M → B E
Y ≃ B E ≃ B M . Y \simeq BE \simeq BM. Y ≃ B E ≃ B M . Since the inclusions of B G BG B G Y Y Y B E BE B E B G BG B G B M ≃ Y BM\simeq Y B M ≃ Y B G BG B G § 2 gives (12) ; since M M M π 1 Y ≅ π 0 M = 0 \pi_{1}Y\cong\pi_{0}M=0 π 1 Y ≅ π 0 M = 0
Let p E G : E G → B G p_{EG}:EG\to BG p E G : E G → B G G G G G G G Z Z Z E G × G Z EG\times_{G}Z E G × G Z E G × Z EG\times Z E G × Z ( ξ g , z ) ∼ ( ξ , g z ) (\xi g,z)\sim(\xi,gz) ( ξ g , z ) ∼ ( ξ , g z ) [ ξ , z ] ↦ p E G ( ξ ) [\xi,z]\mapsto p_{EG}(\xi) [ ξ , z ] ↦ p E G ( ξ ) B G BG B G γ : R X → B G \gamma:R_{X}\to BG γ : R X → B G e : M → E G e:M\to EG e : M → E G
e ( g m ) = e ( m ) g − 1 , p E G e ( m ) = γ ( [ m ] ) . e(gm)=e(m)g^{-1}, \qquad p_{EG}e(m)=\gamma([m]). e ( g m ) = e ( m ) g − 1 , p E G e ( m ) = γ ([ m ]) . Then [ m ] ↦ [ e ( m ) , m ] [m]\mapsto[e(m),m] [ m ] ↦ [ e ( m ) , m ] R X → E G × G M R_{X}\to EG\times_{G}M R X → E G × G M B G BG B G R X R_{X} R X E G EG E G R X R_{X} R X R X R_{X} R X G G G M → E M\to E M → E B G BG B G
R X ≃ E G × G M ≃ E G × G E ≃ hofib ( B G → B E ) ≃ hofib ( j ) R_{X}\simeq EG\times_{G}M\simeq EG\times_{G}E\simeq\operatorname{hofib}(BG\to BE)\simeq\operatorname{hofib}(j) R X ≃ E G × G M ≃ E G × G E ≃ hofib ( B G → B E ) ≃ hofib ( j ) over B G BG B G E G × G E EG\times_{G}E E G × G E E E E G ↪ E G\hookrightarrow E G ↪ E Y Y Y (13) .
Finally, if G G G H H H B G BG B G B H BH B H B ( H × Z ) ≃ B H × S 1 B(H\times\mathbb{Z})\simeq BH\times S^{1} B ( H × Z ) ≃ B H × S 1 Y Y Y □ \square □
Localization, completion, and classification of local monoids
We use the transitive augmentation and marked delooping Y Y Y Theorem 2.1 . For the moment, no finiteness assumption is imposed. Let L L L B i Bi B i i i i ℓ \ell ℓ
L = Cofib ( B H → B i B G ) , ℓ : B G → L . (17) L=\operatorname{Cofib}(BH \xrightarrow{Bi} BG), \qquad\ell:BG\to L. \tag*{(17)} L = Cofib ( B H B i B G ) , ℓ : B G → L . (17) We construct these maps using the nerves of groups and take the quotient L = B G / B H L=BG/BH L = B G / B H B H ⊂ B G BH\subset BG B H ⊂ B G
This homotopy cofiber was used for Alexander quandles in [10 Section 4.2, Definition 4.6, Proposition 4.7] , where H H H
Lemma 3.1. The space L L L R R R Z \mathbb{Z} Z H ~ ∗ ( B H ; R ) = 0 \widetilde{H}_*(BH;R)=0 H ∗ ( B H ; R ) = 0 ℓ \ell ℓ R R R H H H ∣ H ∣ |H| ∣ H ∣ R R R
If G G G L L L H ~ ∗ ( L ; Q ) = 0 \widetilde{H}_*(L;\mathbb{Q})=0 H ∗ ( L ; Q ) = 0 π n ( L ) \pi_n(L) π n ( L ) n ≥ 2 n\ge2 n ≥ 2
Proof. Since the conjugates of h ∈ H h\in H h ∈ H G G G π 1 ( L ) = G / ⟨ ⟨ H ⟩ ⟩ = 1 \pi_1(L)=G/\langle\!\langle H\rangle\!\rangle=1 π 1 ( L ) = G / ⟨ ⟨ H ⟩ ⟩ = 1 G G G B G / B H BG/BH B G / B H [21 Theorem 4.5.4] implies that each π n ( L ) \pi_n(L) π n ( L ) L L L L Q ≃ ∗ L_{\mathbb{Q}}\simeq* L Q ≃ ∗ □ \square □
Theorem 3.2. There is a based map Φ : Y → L ∨ S 2 \Phi:Y\to L\vee S^2 Φ : Y → L ∨ S 2 Φ j \Phi j Φ j B G → ℓ L ↪ L ∨ S 2 BG\xrightarrow{\ell}L\hookrightarrow L\vee S^2 B G ℓ L ↪ L ∨ S 2 R ⊂ Q R\subset\mathbb{Q} R ⊂ Q Z \mathbb{Z} Z H ~ ∗ ( B H ; R ) = 0 \widetilde{H}_*(BH;R)=0 H ∗ ( B H ; R ) = 0 Φ \Phi Φ R R R
Y R ≃ ( S 2 ∨ L ) R . (18) Y_R\simeq(S^2\vee L)_R. \tag*{(18)} Y R ≃ ( S 2 ∨ L ) R . (18) This is an equivalence under B G BG B G B G → j Y → Y R BG\xrightarrow{j}Y\to Y_R B G j Y → Y R B G → ℓ L ↪ S 2 ∨ L → ( S 2 ∨ L ) R BG\xrightarrow{\ell}L\hookrightarrow S^2\vee L\to(S^2\vee L)_R B G ℓ L ↪ S 2 ∨ L → ( S 2 ∨ L ) R M R M_R M R A ∞ A_\infty A ∞ Ω ( Y R ) \Omega(Y_R) Ω ( Y R ) A ∞ A_\infty A ∞
M R ≃ A ∞ Ω ( ( S 2 ∨ L ) R ) . (19) M_R\simeq_{A_\infty}\Omega((S^2\vee L)_R). \tag*{(19)} M R ≃ A ∞ Ω (( S 2 ∨ L ) R ) . (19) In particular, when H H H R = Z ( p ) R=\mathbb{Z}_{(p)} R = Z ( p ) p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ R = Z [ 1 / ∣ H ∣ ] R=\mathbb{Z}[1/|H|] R = Z [ 1/∣ H ∣ ] G G G X X X
Proof. Let r : H × Z → Z r:H\times\mathbb{Z}\to\mathbb{Z} r : H × Z → Z
B G ← B a B ( H × Z ) → B b h B H ℓ ↓ B r ↓ ↓ L ← ∗ B Z → ∗ ∗ \begin{CD}
BG @<{Ba}<< B(H\times\mathbb{Z}) @>{B b_h}>> BH \\
@V{\ell}VV @V{B r}VV @VVV \\
L @<{*}<< B\mathbb{Z} @>{*}>> *
\end{CD} B G ℓ ↓ ⏐ L B a ∗ B ( H × Z ) B r ↓ ⏐ B Z B b h ∗ B H ↓ ⏐ ∗ Indeed, B a Ba B a B H BH B H ℓ \ell ℓ Φ : Y → L ∨ Σ B Z ≃ L ∨ S 2 \Phi:Y\to L\vee\Sigma B\mathbb{Z}\simeq L\vee S^2 Φ : Y → L ∨ Σ B Z ≃ L ∨ S 2 R R R B ( H × Z ) ≃ B H × B Z B(H\times\mathbb{Z})\simeq BH\times B\mathbb{Z} B ( H × Z ) ≃ B H × B Z H ∗ ( B Z ; R ) H_*(B\mathbb{Z};R) H ∗ ( B Z ; R ) Φ \Phi Φ R R R
Since Y Y Y S 2 ∨ L S^2\vee L S 2 ∨ L [21 Theorem 6.1.2] gives (18) . The restriction of Φ \Phi Φ B G BG B G [21 Proposition 6.2.4] give M R ≃ A ∞ Ω ( Y R ) M_R\simeq_{A_\infty}\Omega(Y_R) M R ≃ A ∞ Ω ( Y R ) (18) yields (19) . □ \square □
Identifying the localized homotopy cofiber
In this subsection, G G G H H H X X X B G BG B G p p p B G p ∧ BG^{\wedge}_p B G p ∧
Proposition 3.3. If G G G p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣
L ( p ) ≃ B G p ∧ . (20) L_{(p)}\simeq BG^{\wedge}_p. \tag*{(20)} L ( p ) ≃ B G p ∧ . (20) It may be chosen as an equivalence in the homotopy category under B G BG B G B G → ℓ L → L ( p ) BG \xrightarrow{\ell} L \to L_{(p)} B G ℓ L → L ( p ) B G → B G ^ p BG \to\widehat{BG}_p B G → B G p A ∞ A_\infty A ∞
M ( p ) ≃ A ∞ Ω ( ( S 2 ∨ B G ^ p ) ( p ) ) (21) \mathcal{M}_{(p)} \simeq_{A_\infty} \Omega\left((S^2 \vee\widehat{BG}_p)_{(p)}\right) \tag*{(21)} M ( p ) ≃ A ∞ Ω ( ( S 2 ∨ B G p ) ( p ) ) (21) and if p ∤ ∣ G ∣ p \nmid|G| p ∤ ∣ G ∣ M ( p ) ≃ A ∞ Ω ( S ( p ) 2 ) \mathcal{M}_{(p)} \simeq_{A_\infty} \Omega(S^2_{(p)}) M ( p ) ≃ A ∞ Ω ( S ( p ) 2 )
Proof. Lemma 3.1 and the completion exact sequence [1 Theorem 2.6] give π n ( L ^ p ) ≅ π n ( L ) ⊗ Z ( p ) \pi_n(\widehat{L}_p) \cong\pi_n(L) \otimes\mathbb{Z}_{(p)} π n ( L p ) ≅ π n ( L ) ⊗ Z ( p ) n ≥ 2 n \ge2 n ≥ 2 π 1 ( L ^ p ) = 0 \pi_1(\widehat{L}_p)=0 π 1 ( L p ) = 0 L ^ p \widehat{L}_p L p p p p L → L ( p ) → κ L ^ p L \to L_{(p)} \xrightarrow{\kappa} \widehat{L}_p L → L ( p ) κ L p κ \kappa κ ℓ \ell ℓ p p p
B G ^ p → ℓ ^ p L ^ p ← κ L ( p ) \widehat{BG}_p \xrightarrow{\widehat{\ell}_p} \widehat{L}_p \xleftarrow{\kappa} L_{(p)} B G p ℓ p L p κ L ( p ) [5 Chapter I, Section 5] . By naturality of completion and the factorization involving κ \kappa κ B G BG B G
The map S 2 ∨ L → S 2 ∨ L ( p ) S^2 \vee L \to S^2 \vee L_{(p)} S 2 ∨ L → S 2 ∨ L ( p ) Z ( p ) \mathbb{Z}_{(p)} Z ( p ) Theorem 3.2 give (21) . If p ∤ ∣ G ∣ p \nmid|G| p ∤ ∣ G ∣ H ~ ∗ ( L ; Z ( p ) ) = 0 \widetilde{H}_*(L;\mathbb{Z}_{(p)})=0 H ∗ ( L ; Z ( p ) ) = 0 L ( p ) ≃ ∗ L_{(p)} \simeq* L ( p ) ≃ ∗
Corollary 3.4 (An arithmetic criterion for nontriviality). Let X X X G = Inn ( X ) G=\operatorname{Inn}(X) G = Inn ( X ) H = Stab G ( x 0 ) H=\operatorname{Stab}_G(x_0) H = Stab G ( x 0 ) p ∤ ∣ H ∣ p \nmid|H| p ∤ ∣ H ∣
L ( p ) ≃ ∗ ⟺ B G ^ p ≃ ∗ ⟺ p ∤ ∣ G ∣ ⟺ p ∤ ∣ X ∣ . (22) L_{(p)} \simeq* \quad\Longleftrightarrow\quad\widehat{BG}_p \simeq* \quad\Longleftrightarrow\quad p \nmid|G| \quad\Longleftrightarrow\quad p \nmid|X|. \tag*{(22)} L ( p ) ≃ ∗ ⟺ B G p ≃ ∗ ⟺ p ∤ ∣ G ∣ ⟺ p ∤ ∣ X ∣. (22) Consequently, among the good primes, L ( p ) L_{(p)} L ( p ) p p p ∣ X ∣ |X| ∣ X ∣
Proof. The first equivalence follows from Proposition 3.3. If p ∤ ∣ G ∣ p \nmid|G| p ∤ ∣ G ∣ L ( p ) ≃ ∗ L_{(p)} \simeq* L ( p ) ≃ ∗ p ∣ ∣ G ∣ p \mid|G| p ∣ ∣ G ∣ p p p [17 Proposition 3.3 and its proof] gives H ~ ∗ ( B G ; F p ) ≠ 0 \widetilde{H}^*(BG;\mathbb{F}_p) \ne0 H ∗ ( B G ; F p ) = 0 B G → L BG \to L B G → L p p p L ( p ) L_{(p)} L ( p ) ∣ G ∣ = ∣ H ∣ ∣ X ∣ |G|=|H||X| ∣ G ∣ = ∣ H ∣∣ X ∣ p ∤ ∣ H ∣ p \nmid|H| p ∤ ∣ H ∣
Classification of local monoids and recovery of the completion
At a good prime for a finite connected quandle, the converse to Proposition 3.3 holds: the homotopy type of B G ^ p \widehat{BG}_p B G p A ∞ A_\infty A ∞
Lemma 3.5 (Sphere cancellation). Let V , V ′ V,V' V , V ′ p p p p p p
S ( p ) 2 ∨ V ≃ S ( p ) 2 ∨ V ′ ⟹ V ≃ V ′ . S^2_{(p)} \vee V \simeq S^2_{(p)} \vee V' \quad\Longrightarrow\quad V \simeq V'. S ( p ) 2 ∨ V ≃ S ( p ) 2 ∨ V ′ ⟹ V ≃ V ′ . Moreover, if the image of e ∈ π 2 ( S ( p ) 2 ∨ V ) e \in\pi_2(S^2_{(p)} \vee V) e ∈ π 2 ( S ( p ) 2 ∨ V ) Z ( p ) \mathbb{Z}_{(p)} Z ( p ) S ( p ) 2 → S ( p ) 2 ∨ V S^2_{(p)} \to S^2_{(p)} \vee V S ( p ) 2 → S ( p ) 2 ∨ V e e e V V V
Proof. Put R = Z ( p ) R=\mathbb{Z}_{(p)} R = Z ( p ) p p p π 2 ( S ( p ) 2 ∨ V ) = R ι ⊕ π 2 ( V ) \pi_2(S^2_{(p)} \vee V)=R\iota\oplus\pi_2(V) π 2 ( S ( p ) 2 ∨ V ) = R ι ⊕ π 2 ( V ) f f f ι \iota ι a ι ′ + t a\iota'+t a ι ′ + t a ∈ R × a\in R^\times a ∈ R × t ∈ π 2 ( V ′ ) t\in\pi_2(V') t ∈ π 2 ( V ′ ) E E E V ′ V' V ′ ι ′ \iota' ι ′ a − 1 ( ι ′ − t ) a^{-1}(\iota'-t) a − 1 ( ι ′ − t ) E f Ef E f V ≃ V ′ V\simeq V' V ≃ V ′ e = a ι + t e=a\iota+t e = a ι + t
Theorem 3.6 (Classification of unmarked local monoids). Let X , X ′ X, X' X , X ′ G , G ′ G, G' G , G ′ H , H ′ H, H' H , H ′ p ∤ ∣ H ∣ ∣ H ′ ∣ p \nmid|H||H'| p ∤ ∣ H ∣∣ H ′ ∣
( M X ) ( p ) ≃ A ∞ ( M X ′ ) ( p ) ⟺ B G p ∧ ≃ B G p ′ ∧ . (23) (\mathcal{M}_X)_{(p)} \simeq_{A_\infty} (\mathcal{M}_{X'})_{(p)} \qquad\Longleftrightarrow\qquad BG_p^\wedge\simeq BG_p^{\prime\wedge}. \tag*{(23)} ( M X ) ( p ) ≃ A ∞ ( M X ′ ) ( p ) ⟺ B G p ∧ ≃ B G p ′ ∧ . (23) The space B G p ∧ BG_p^\wedge B G p ∧ S ( p ) 2 → B ( ( M X ) ( p ) ) S^2_{(p)} \to B((\mathcal{M}_X)_{(p)}) S ( p ) 2 → B (( M X ) ( p ) ) Z ( p ) \mathbb{Z}_{(p)} Z ( p ) π 2 ( B ( ( M X ) ( p ) ) ) / torsion \pi_2(B((\mathcal{M}_X)_{(p)}))/\text{torsion} π 2 ( B (( M X ) ( p ) )) / torsion
Proof. By Proposition 3.3 and Lemma 3.1 , B G p ∧ ≃ L ( p ) BG_p^\wedge\simeq L_{(p)} B G p ∧ ≃ L ( p ) p p p Proposition 3.3 gives the reverse implication in (23) . For the forward implication, delooping an A ∞ A_\infty A ∞ S ( p ) 2 ∨ B G p ∧ ≃ S ( p ) 2 ∨ B G p ′ ∧ S^2_{(p)} \vee BG_p^\wedge\simeq S^2_{(p)} \vee BG_p^{\prime\wedge} S ( p ) 2 ∨ B G p ∧ ≃ S ( p ) 2 ∨ B G p ′ ∧ Lemma 3.5 then gives the forward implication and the recovery statement. □
The recovered homotopy type of the completion does not determine the group G G G j j j Theorem 2.1 .
Remark 3.7. The outer localization in (21) is essential. The wedge summand S 2 S^2 S 2 Z ( p ) \mathbb{Z}_{(p)} Z ( p ) π 2 ( Y ( p ) ) \pi_2(Y_{(p)}) π 2 ( Y ( p ) ) Z p \mathbb{Z}_p Z p L L L B G p ∧ BG_p^\wedge B G p ∧
Fundamental groups, universal covers, and rationalization
We apply the integral and localized models to compute π 1 ( M ) \pi_1(\mathcal{M}) π 1 ( M ) R X \mathcal{R}_X R X M \mathcal{M} M Theorem 2.1 ; finiteness assumptions will be stated where needed. Write H n ( G ) = H n ( B G ; Z ) H_n(G) = H_n(BG;\mathbb{Z}) H n ( G ) = H n ( B G ; Z )
Torus classes and the fundamental group
We compute π 1 ( M ) \pi_1(\mathcal{M}) π 1 ( M ) H 2 ( G ) H_2(G) H 2 ( G )
Using the group homomorphism b h : H × Z → H b_h:H \times\mathbb{Z} \to H b h : H × Z → H (10) , define a homomorphism of abelian groups by
τ h : H 1 ( H ) ⟶ H 2 ( G ) , τ h ( v ) = ( B i ) ∗ ( B b h ) ∗ ( v × [ Z ] ) . (24) \tau_h:H_1(H) \longrightarrow H_2(G), \qquad\tau_h(v) = (Bi)_*(Bb_h)_*(v \times[\mathbb{Z}]). \tag*{(24)} τ h : H 1 ( H ) ⟶ H 2 ( G ) , τ h ( v ) = ( B i ) ∗ ( B b h ) ∗ ( v × [ Z ]) . (24) Here [ Z ] ∈ H 1 ( Z ) [\mathbb{Z}] \in H_1(\mathbb{Z}) [ Z ] ∈ H 1 ( Z ) H 1 ( H ) ⊗ H 1 ( Z ) H_1(H) \otimes H_1(\mathbb{Z}) H 1 ( H ) ⊗ H 1 ( Z ) H 2 ( H × Z ) H_2(H \times\mathbb{Z}) H 2 ( H × Z ) u ∈ H u \in H u ∈ H τ h ( [ u ] ) \tau_h([u]) τ h ([ u ])
[ i ( u ) ∣ h ] − [ h ∣ i ( u ) ] . (25) [i(u) \mid h] - [h \mid i(u)]. \tag*{(25)} [ i ( u ) ∣ h ] − [ h ∣ i ( u )] . (25) This is a cycle because i ( u ) i(u) i ( u ) h h h
Proposition 4.1. Put
D = ker ( Z ⟶ G a b , 1 ⟼ [ h ] ) . D = \ker(\mathbb{Z} \longrightarrow G_{\mathrm{ab}},\; 1 \longmapsto[h]). D = ker ( Z ⟶ G ab , 1 ⟼ [ h ]) . There is an exact sequence of abelian groups
0 ⟶ H 2 ( G ) / im τ h ⟶ π 1 ( M ) ⟶ D ⟶ 0 , (26) 0 \longrightarrow H_2(G)/\operatorname{im}\tau_h \longrightarrow\pi_1(\mathcal{M}) \longrightarrow D \longrightarrow0, \tag*{(26)} 0 ⟶ H 2 ( G ) / im τ h ⟶ π 1 ( M ) ⟶ D ⟶ 0 , (26) which splits noncanonically. The group G a b G_{\mathrm{ab}} G ab [ h ] [h] [ h ] [ h ] [h] [ h ] q q q D = q Z D = q\mathbb{Z} D = q Z
π 1 ( M ) ≅ Z ⊕ H 2 ( G ) / im τ h . (27) \pi_1(\mathcal{M}) \cong\mathbb{Z} \oplus H_2(G)/\operatorname{im}\tau_h. \tag*{(27)} π 1 ( M ) ≅ Z ⊕ H 2 ( G ) / im τ h . (27) If [ h ] [h] [ h ] D = 0 D = 0 D = 0 G G G H 2 ( G ) / im τ h H_2(G)/\operatorname{im}\tau_h H 2 ( G ) / im τ h
Proof. Since the conjugates of h h h G G G [ h ] [h] [ h ] G a b G_{\mathrm{ab}} G ab (11) and cancel the direct summands H 2 ( H ) H_2(H) H 2 ( H ) H 1 ( H ) H_1(H) H 1 ( H ) − id - \operatorname{id} − id
H 1 ( H ) → τ h H 2 ( G ) ⟶ H 2 ( Y ) → δ Z → 1 ↦ [ h ] G a b ⟶ 0. H_1(H) \xrightarrow{\tau_h} H_2(G) \longrightarrow H_2(Y) \xrightarrow{\delta} \mathbb{Z} \xrightarrow{1\mapsto[h]} G_{\mathrm{ab}} \longrightarrow0. H 1 ( H ) τ h H 2 ( G ) ⟶ H 2 ( Y ) δ Z 1 ↦ [ h ] G ab ⟶ 0. Now (26) follows from im δ = D \operatorname{im}\delta= D im δ = D H 2 ( Y ) ≅ π 2 ( Y ) ≅ π 1 ( M ) H_2(Y) \cong\pi_2(Y) \cong\pi_1(\mathcal{M}) H 2 ( Y ) ≅ π 2 ( Y ) ≅ π 1 ( M ) Theorem 2.1 . The sequence splits because every subgroup of Z \mathbb{Z} Z D D D G G G H 2 ( G ) H_2(G) H 2 ( G ) ∗ ← B Z → ∗ \ast\leftarrow B\mathbb{Z} \rightarrow\ast ∗ ← B Z → ∗ Y → S 2 Y \rightarrow S^2 Y → S 2 δ \delta δ D D D
We now relate this computation to associated groups and reduced Schur multipliers. For connected X X X (1) identifies As ( X ) a b \operatorname{As}(X)_{\mathrm{ab}} As ( X ) ab Z \mathbb{Z} Z ε : As ( X ) → Z \varepsilon:\operatorname{As}(X)\rightarrow\mathbb{Z} ε : As ( X ) → Z e x ↦ 1 e_x\mapsto1 e x ↦ 1 ψ : As ( X ) → Inn ( X ) \psi:\operatorname{As}(X)\rightarrow\operatorname{Inn}(X) ψ : As ( X ) → Inn ( X )
G ‾ = im ( ε , ψ ) , h ‾ = ( 1 , S x 0 ) . \overline{G}=\operatorname{im}(\varepsilon,\psi), \qquad\overline{h}=(1,S_{x_0}). G = im ( ε , ψ ) , h = ( 1 , S x 0 ) . The action through the second projection and the map x ↦ ( 1 , S x ) x\mapsto(1,S_x) x ↦ ( 1 , S x ) G ‾ a b ≅ Z \overline{G}_{\mathrm{ab}}\cong\mathbb{Z} G ab ≅ Z D = 0 D=0 D = 0 (9) therefore give
ker ( ε , ψ ) ≅ H 2 ( G ‾ ) / im τ h ‾ . \ker(\varepsilon,\psi)\cong H_2(\overline{G})/\operatorname{im}\tau_{\overline{h}}. ker ( ε , ψ ) ≅ H 2 ( G ) / im τ h . where τ h ‾ \tau_{\overline{h}} τ h Stab G ‾ ( x 0 ) \operatorname{Stab}_{\overline{G}}(x_0) Stab G ( x 0 ) [16 Theorem 3.1] , giving a geometric proof of its kernel formula for connected quandles. Ishikawa also determines the central extension class and treats nonconnected racks.
For a finite faithful connected quandle with its canonical augmentation, X X X c c c h h h H = C G ( h ) H=C_G(h) H = C G ( h ) H 2 ( G ) / im τ h H_2(G)/\operatorname{im}\tau_h H 2 ( G ) / im τ h H 2 ( G , c ) H_2(G,c) H 2 ( G , c ) [28 Section 4B] , which is isomorphic to ker ψ ∩ [ As ( X ) , As ( X ) ] \ker\psi\cap[\operatorname{As}(X),\operatorname{As}(X)] ker ψ ∩ [ As ( X ) , As ( X )] [19 Proposition 6.9] . For a nonfaithful quandle, the torus relations use the stabilizer H H H C G ( h ) C_G(h) C G ( h )
Universal covers
Let Y ⟨ 2 ⟩ Y\langle2\rangle Y ⟨ 2 ⟩ Y Y Y Y → K ( π 2 Y , 2 ) Y\rightarrow K(\pi_2Y,2) Y → K ( π 2 Y , 2 ) π 2 \pi_2 π 2
Corollary 4.2. There are ordinary homotopy equivalences
R X ~ ≃ M ~ ≃ Ω ( Y ⟨ 2 ⟩ ) . (28) \widetilde{R_X}\simeq\widetilde{\mathcal{M}}\simeq\Omega(Y\langle2\rangle). \tag*{(28)} R X ≃ M ≃ Ω ( Y ⟨ 2 ⟩) . (28) If M ~ \widetilde{\mathcal{M}} M M \mathcal{M} M A ∞ A_\infty A ∞
Proof. The covering M → R X \mathcal{M}\rightarrow R_X M → R X Ω ( Y ⟨ 2 ⟩ ) \Omega(Y\langle2\rangle) Ω ( Y ⟨ 2 ⟩) Ω Y \Omega Y Ω Y Theorem 2.1 gives (28) . The multiplication on M \mathcal{M} M B M ~ → B M B\widetilde{\mathcal{M}}\rightarrow B\mathcal{M} B M → B M B M ≃ Y B\mathcal{M}\simeq Y B M ≃ Y B M ~ ≃ Y ⟨ 2 ⟩ B\widetilde{\mathcal{M}}\simeq Y\langle2\rangle B M ≃ Y ⟨ 2 ⟩ A ∞ A_\infty A ∞
Under a vanishing assumption on second homology, the localized universal cover has the following description.
Corollary 4.3. For a generating transitive augmentation as in Section 2, suppose that G G G p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ H 2 ( G ; Z ) ⊗ Z ( p ) = 0 H_2(G;\mathbb{Z})\otimes\mathbb{Z}_{(p)}=0 H 2 ( G ; Z ) ⊗ Z ( p ) = 0
( R X ~ ) ( p ) ≃ Ω ( ( S 3 ∨ ( S + 1 ∧ Λ ) ) ( p ) ) . (29) \left(\widetilde{R_X}\right)_{(p)}\simeq\Omega\left(\left(S^3\vee(S^1_+\wedge\Lambda)\right)_{(p)}\right). \tag*{(29)} ( R X ) ( p ) ≃ Ω ( ( S 3 ∨ ( S + 1 ∧ Λ ) ) ( p ) ) . (29) If R ~ X \widetilde{\mathcal R}_X R X M ~ \widetilde{\mathcal M} M A ∞ A_\infty A ∞ S + 1 S^1_+ S + 1
S + 1 ∧ L = ( S 1 × L ) / ( S 1 × { ∗ } ) . S^1_+ \wedge L = (S^1 \times L)/(S^1 \times\{\ast\}). S + 1 ∧ L = ( S 1 × L ) / ( S 1 × { ∗ }) . Proof. Put W = S 2 ∨ L W=S^2\vee L W = S 2 ∨ L c : W → K ( Z , 2 ) c:W\to K(\mathbb{Z},2) c : W → K ( Z , 2 ) S 2 S^2 S 2 L L L E E E S 1 → S 3 S^1\to S^3 S 1 → S 3 S 1 → S 1 × L S^1\to S^1\times L S 1 → S 1 × L
E ≃ hocolim [ S 3 ← S 1 → S 1 × L ] ≃ S 3 ∨ ( S + 1 ∧ L ) . E\simeq\operatorname{hocolim}[S^3\leftarrow S^1\to S^1\times L]\simeq S^3\vee(S^1_+\wedge L). E ≃ hocolim [ S 3 ← S 1 → S 1 × L ] ≃ S 3 ∨ ( S + 1 ∧ L ) . The first arrow is null-homotopic, and the second is the inclusion of the subcomplex S 1 × { ∗ } S^1\times\{\ast\} S 1 × { ∗ } E E E π 2 ( E ) ≅ π 2 ( L ) \pi_2(E)\cong\pi_2(L) π 2 ( E ) ≅ π 2 ( L ) π n ( E ) ≅ π n ( W ) \pi_n(E)\cong\pi_n(W) π n ( E ) ≅ π n ( W ) n ≥ 3 n\geq3 n ≥ 3 Lemma 3.1 ,
π 2 ( E ) ⊗ Z ( p ) ≅ H 2 ( G ; Z ) ⊗ Z ( p ) = 0 \pi_2(E)\otimes\mathbb{Z}_{(p)}\cong H_2(G;\mathbb{Z})\otimes\mathbb{Z}_{(p)}=0 π 2 ( E ) ⊗ Z ( p ) ≅ H 2 ( G ; Z ) ⊗ Z ( p ) = 0 so E ( p ) → W ( p ) E_{(p)}\to W_{(p)} E ( p ) → W ( p ) M → M ( p ) ≃ Ω W ( p ) \mathcal M\to\mathcal M_{(p)}\simeq\Omega W_{(p)} M → M ( p ) ≃ Ω W ( p ) Theorem 3.2 lifts to Ω E ( p ) \Omega E_{(p)} Ω E ( p ) ( M ~ ) ( p ) ≃ Ω E ( p ) (\widetilde{\mathcal M})_{(p)}\simeq\Omega E_{(p)} ( M ) ( p ) ≃ Ω E ( p ) A ∞ A_\infty A ∞ Corollary 4.2 gives the same model for ( R ~ X ) ( p ) (\widetilde{\mathcal R}_X)_{(p)} ( R X ) ( p )
Without the vanishing assumption, we have π 1 ( Ω E ( p ) ) = H 2 ( G ; Z ) ( p ) \pi_1(\Omega E_{(p)})=H_2(G;\mathbb{Z})_{(p)} π 1 ( Ω E ( p ) ) = H 2 ( G ; Z ) ( p )
Remark 4.4 (Rationalization and multiplication). Let X X X G = Inn ( X ) G=\operatorname{Inn}(X) G = Inn ( X ) M = B ( G ; X ) \mathcal M=B(G;X) M = B ( G ; X ) Y Y Y (11) . Since L Q ≃ ∗ L_Q\simeq\ast L Q ≃ ∗ Lemma 3.1 , Theorem 3.2 gives, respectively, a homotopy equivalence and an A ∞ A_\infty A ∞
Y Q ≃ S Q 2 , M Q ≃ A ∞ Ω S Q 2 . (30) Y_{\mathbb{Q}}\simeq S^2_{\mathbb{Q}},\qquad\mathcal M_{\mathbb{Q}}\simeq_{A_\infty}\Omega S^2_{\mathbb{Q}}. \tag*{(30)} Y Q ≃ S Q 2 , M Q ≃ A ∞ Ω S Q 2 . (30) In particular, π j ( M ) ⊗ Q = 0 \pi_j(\mathcal M)\otimes\mathbb{Q}=0 π j ( M ) ⊗ Q = 0 j ≥ 3 j\geq3 j ≥ 3
Under (30) , let α ∈ π 1 ( M Q ) \alpha\in\pi_1(\mathcal M_{\mathbb{Q}}) α ∈ π 1 ( M Q ) β ∈ π 2 ( M Q ) \beta\in\pi_2(\mathcal M_{\mathbb{Q}}) β ∈ π 2 ( M Q ) ι 2 = [ id S 2 ] ∈ π 2 ( S 2 ) \iota_2=[\operatorname{id}_{S^2}]\in\pi_2(S^2) ι 2 = [ id S 2 ] ∈ π 2 ( S 2 ) η ∈ π 3 ( S 2 ) \eta\in\pi_3(S^2) η ∈ π 3 ( S 2 ) [ ι 2 , ι 2 ] = ± 2 η [\iota_2,\iota_2]=\pm2\eta [ ι 2 , ι 2 ] = ± 2 η [13 Section 4.J, immediately before Corollary 4J.4] and the Whitehead–Samelson adjunction give
[ α , α ] S a m = ± 2 β ≠ 0. (31) [\alpha,\alpha]_{\mathrm{Sam}}=\pm2\beta\ne0. \tag*{(31)} [ α , α ] Sam = ± 2 β = 0. (31) From the covering monoid to rack and quandle homology
Throughout this section, X X X G , H , M , N X , L G,H,\mathcal M,\mathcal N_X,L G , H , M , N X , L G G G H n R ( X ; Λ ) = H n ( R X ; Λ ) H_n^R(X;\Lambda)=H_n(R_X;\Lambda) H n R ( X ; Λ ) = H n ( R X ; Λ ) H n R ( X ; Λ ) = H n ( R X ; Λ ) H_n^R(X;\Lambda)=H^n(R_X;\Lambda) H n R ( X ; Λ ) = H n ( R X ; Λ ) H 0 R = H 0 R = Λ H_0^R=H_0^R=\Lambda H 0 R = H 0 R = Λ D [ m ] n = D n − m D[m]_n=D_{n-m} D [ m ] n = D n − m d D [ m ] = ( − 1 ) m d D d_{D[m]}=(-1)^m d_D d D [ m ] = ( − 1 ) m d D
We convert the homology of Ω L \Omega L Ω L
Transfer and the degree shift
Proposition 5.1. Let q : M → N X q:\mathcal M\to\mathcal N_X q : M → N X H H H (8) . Every deck transformation is homotopic to the identity. Consequently, for a commutative ring Λ \Lambda Λ ∣ H ∣ |H| ∣ H ∣ Λ \Lambda Λ
q ∗ : H n ( M ; Λ ) → ≅ H n ( N X ; Λ ) , q ∗ : H n ( N X ; Λ ) → ≅ H n ( M ; Λ ) q_*:H_n(\mathcal M;\Lambda)\xrightarrow{\cong}H_n(\mathcal N_X;\Lambda),\qquad q^*:H^n(\mathcal N_X;\Lambda)\xrightarrow{\cong}H^n(\mathcal M;\Lambda) q ∗ : H n ( M ; Λ ) ≅ H n ( N X ; Λ ) , q ∗ : H n ( N X ; Λ ) ≅ H n ( M ; Λ ) are isomorphisms for all n ≥ 0 n\geq0 n ≥ 0 Λ \Lambda Λ
Proof. The identification (8) exhibits q q q H H H M \mathcal{M} M γ h \gamma_h γ h h ∈ H h \in H h ∈ H ( s , m ) ↦ γ h ( s ) m (s,m) \mapsto\gamma_h(s)m ( s , m ) ↦ γ h ( s ) m h h h tr \operatorname{tr} tr Λ \Lambda Λ
q ∗ tr = ∣ H ∣ id , tr q ∗ = ∑ h ∈ H ( h ⋅ − ) ∗ = ∣ H ∣ id . q_* \operatorname{tr} = |H| \operatorname{id}, \qquad\operatorname{tr} q_* = \sum_{h \in H}(h \cdot-)_* = |H| \operatorname{id}. q ∗ tr = ∣ H ∣ id , tr q ∗ = h ∈ H ∑ ( h ⋅ − ) ∗ = ∣ H ∣ id . It follows that ∣ H ∣ − 1 tr |H|^{-1}\operatorname{tr} ∣ H ∣ − 1 tr q ∗ q_* q ∗ Λ \Lambda Λ
This is Clauwens’ transfer argument [10 Proposition 4.2] . The following degree shift is the chain identification of [10 §2.3] ; its integral homological form is also recorded in [25 Remark 4.1] .
Proposition 5.2. For every commutative coefficient ring Λ \Lambda Λ n ≥ 0 n \ge0 n ≥ 0 Λ \Lambda Λ
H n + 1 R ( X ; Λ ) ≅ H n ( N X ; Λ ) , H R n + 1 ( X ; Λ ) ≅ H n ( N X ; Λ ) (32) H^R_{n+1}(X;\Lambda) \cong H_n(N_X;\Lambda), \qquad H_R^{n+1}(X;\Lambda) \cong H^n(N_X;\Lambda) \tag*{(32)} H n + 1 R ( X ; Λ ) ≅ H n ( N X ; Λ ) , H R n + 1 ( X ; Λ ) ≅ H n ( N X ; Λ ) (32) natural in the coefficients. In particular, if p ∤ ∣ H ∣ p \nmid|H| p ∤ ∣ H ∣
H n + 1 R ( X ; Z ( p ) ) ≅ H n ( M ; Z ( p ) ) , H R n + 1 ( X ; Z ( p ) ) ≅ H n ( M ; Z ( p ) ) . (33) H^R_{n+1}(X;\mathbb{Z}_{(p)}) \cong H_n(\mathcal{M};\mathbb{Z}_{(p)}), \qquad H_R^{n+1}(X;\mathbb{Z}_{(p)}) \cong H^n(\mathcal{M};\mathbb{Z}_{(p)}). \tag*{(33)} H n + 1 R ( X ; Z ( p ) ) ≅ H n ( M ; Z ( p ) ) , H R n + 1 ( X ; Z ( p ) ) ≅ H n ( M ; Z ( p ) ) . (33) The corresponding statements hold with coefficients in any Z ( p ) \mathbb{Z}_{(p)} Z ( p ) p p p
Proof. Reindex the positive-degree integral rack complex as D n = C n + 1 R ( X ; Z ) D_n=C^R_{n+1}(X;\mathbb{Z}) D n = C n + 1 R ( X ; Z ) n ≥ 0 n \ge0 n ≥ 0 D n = 0 D_n=0 D n = 0 n < 0 n<0 n < 0 (4) –(6) , is
ψ n : D n ⟶ C n ( X ; X ; Z ) , ψ n ( x 0 ′ , x 1 , … , x n ) = ( − 1 ) n ( x 0 ′ ; x 1 , … , x n ) . (34) \psi_n:D_n \longrightarrow C_n(X;X;\mathbb{Z}), \qquad\psi_n(x'_0,x_1,\ldots,x_n)=(-1)^n(x'_0;x_1,\ldots,x_n). \tag*{(34)} ψ n : D n ⟶ C n ( X ; X ; Z ) , ψ n ( x 0 ′ , x 1 , … , x n ) = ( − 1 ) n ( x 0 ′ ; x 1 , … , x n ) . (34) Since ∂ 1 = 0 \partial_1=0 ∂ 1 = 0 Λ \Lambda Λ Hom Z ( − , Λ ) \operatorname{Hom}_{\mathbb{Z}}(-,\Lambda) Hom Z ( − , Λ ) (32) for all n ≥ 0 n \ge0 n ≥ 0
For finite connected Alexander quandles whose defining automorphism has order coprime to the quandle order, the integral comparison H n ( M ; Z ) ≅ H n + 1 R ( X ; Z ) H_n(\mathcal{M};\mathbb{Z}) \cong H^R_{n+1}(X;\mathbb{Z}) H n ( M ; Z ) ≅ H n + 1 R ( X ; Z ) n ≥ 1 n \ge1 n ≥ 1 [25 Lemma 5.9] .
Loop homology of a wedge
We recall the algebraic fact needed to use the localized homotopy model. For augmented algebras, ∗ * ∗ T Λ ( V ) = ⨁ m ≥ 0 V ⊗ Λ m T_\Lambda(V)=\bigoplus_{m\ge0}V^{\otimes_\Lambda m} T Λ ( V ) = ⨁ m ≥ 0 V ⊗ Λ m Λ \Lambda Λ V V V T Λ ( Λ u ) T_\Lambda(\Lambda u) T Λ ( Λ u ) T Λ ( u ) T_\Lambda(u) T Λ ( u )
Lemma 5.3. Let U U U V V V k k k
H ∗ ( Ω U ; k ) ∗ H ∗ ( Ω V ; k ) → ≅ H ∗ ( Ω ( U ∨ V ) ; k ) . (35) H_*(\Omega U;k) * H_*(\Omega V;k) \xrightarrow{\cong} H_*(\Omega(U \vee V);k). \tag*{(35)} H ∗ ( Ω U ; k ) ∗ H ∗ ( Ω V ; k ) ≅ H ∗ ( Ω ( U ∨ V ) ; k ) . (35) In particular, for a based simply connected CW space L L L
H ∗ ( Ω ( S 2 ∨ L ) ; k ) ≅ T k ( u ) ∗ H ∗ ( Ω L ; k ) , ∣ u ∣ = 1 , (36) H_*(\Omega(S^2 \vee L);k) \cong T_k(u) * H_*(\Omega L;k), \qquad|u|=1, \tag*{(36)} H ∗ ( Ω ( S 2 ∨ L ) ; k ) ≅ T k ( u ) ∗ H ∗ ( Ω L ; k ) , ∣ u ∣ = 1 , (36) and u u u L L L
Proof. The first assertion is the classical coproduct formula for loop-space homology [9 p. 245, Remark] . The second follows from the Bott–Samelson theorem [4 ] , which gives H ∗ ( Ω S 2 ; k ) ≅ T k ( u ) H_*(\Omega S^2;k) \cong T_k(u) H ∗ ( Ω S 2 ; k ) ≅ T k ( u ) ∣ u ∣ = 1 |u|=1 ∣ u ∣ = 1 u u u
The loop model also determines the Pontryagin Hopf algebra.
Theorem 5.4 (Pontryagin Hopf algebra). Let p p p p ∤ ∣ H ∣ p \nmid|H| p ∤ ∣ H ∣ k k k p p p L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH \to BG) L = Cofib ( B H → B G )
H ∗ ( M ; k ) ≅ T k ( u ) ∗ H ∗ ( Ω L ; k ) , ∣ u ∣ = 1 , Δ u = u ⊗ 1 + 1 ⊗ u . (37) H_{*}(\mathcal{M};k) \cong T_k(u) * H_{*}(\Omega L;k), \qquad|u|=1,\qquad\Delta u=u\otimes1+1\otimes u. \tag*{(37)} H ∗ ( M ; k ) ≅ T k ( u ) ∗ H ∗ ( Ω L ; k ) , ∣ u ∣ = 1 , Δ u = u ⊗ 1 + 1 ⊗ u . (37) Proof. Theorem 3.2 and Lemma 5.3 give (37) , since localization of connected nilpotent loop spaces preserves the homology Hopf algebra with coefficients in k k k □ \square □
Quandle normalization and degenerate summands
This subsection applies to any finite quandle, without assumptions on G G G H H H
We follow [8 §3] for the definition of the quandle complex. For n ≥ 2 n \ge2 n ≥ 2 C n D ( X ; Λ ) C_n^D(X;\Lambda) C n D ( X ; Λ ) C n R ( X ; Λ ) C_n^R(X;\Lambda) C n R ( X ; Λ ) x i = x i + 1 x_i=x_{i+1} x i = x i + 1 i i i C n D = 0 C_n^D=0 C n D = 0 n ≤ 1 n \le1 n ≤ 1 H n D ( X ; Λ ) = H n ( C ∗ D ( X ; Λ ) ) H_n^D(X;\Lambda)=H_n(C_*^D(X;\Lambda)) H n D ( X ; Λ ) = H n ( C ∗ D ( X ; Λ ))
C n Q ( X ; Λ ) = C n R ( X ; Λ ) / C n D ( X ; Λ ) ( n ≥ 1 ) . C_n^Q(X;\Lambda)=C_n^R(X;\Lambda)/C_n^D(X;\Lambda) \qquad(n\ge1). C n Q ( X ; Λ ) = C n R ( X ; Λ ) / C n D ( X ; Λ ) ( n ≥ 1 ) . The differential out of degree one in this positive-degree complex is zero. We denote its positive-degree homology and cohomology by H n Q H_n^Q H n Q H Q n H_Q^n H Q n H 0 Q H_0^Q H 0 Q H 0 R H_0^R H 0 R
Proposition 5.5. Let Λ \Lambda Λ
R i = H i R ( X ; Λ ) ( i ≥ 0 ) , Q j = H j Q ( X ; Λ ) ( j ≥ 1 ) , R 0 = Λ . R_i=H_i^R(X;\Lambda)\quad(i\ge0),\qquad Q_j=H_j^Q(X;\Lambda)\quad(j\ge1),\qquad R_0=\Lambda. R i = H i R ( X ; Λ ) ( i ≥ 0 ) , Q j = H j Q ( X ; Λ ) ( j ≥ 1 ) , R 0 = Λ. For n ≥ 1 n\ge1 n ≥ 1 H n R ( X ; Λ ) ≅ Q n ⊕ H n D ( X ; Λ ) H_n^R(X;\Lambda)\cong Q_n\oplus H_n^D(X;\Lambda) H n R ( X ; Λ ) ≅ Q n ⊕ H n D ( X ; Λ ) Λ \Lambda Λ
0 ⟶ ⨁ i + j = n − 1 i ≥ 0 , j ≥ 1 R i ⊗ Λ Q j ⟶ H n D ( X ; Λ ) ⟶ ⨁ i + j = n − 2 i ≥ 0 , j ≥ 1 Tor 1 Λ ( R i , Q j ) ⟶ 0. (38) 0\longrightarrow\bigoplus_{\substack{i+j=n-1\\i\ge0,\ j\ge1}}R_i\otimes_\Lambda Q_j\longrightarrow H_n^D(X;\Lambda)\longrightarrow\bigoplus_{\substack{i+j=n-2\\i\ge0,\ j\ge1}}\operatorname{Tor}^{\Lambda}_1(R_i,Q_j)\longrightarrow0. \tag*{(38)} 0 ⟶ i + j = n − 1 i ≥ 0 , j ≥ 1 ⨁ R i ⊗ Λ Q j ⟶ H n D ( X ; Λ ) ⟶ i + j = n − 2 i ≥ 0 , j ≥ 1 ⨁ Tor 1 Λ ( R i , Q j ) ⟶ 0. (38) An empty direct sum is understood to be zero.
Proof. The chain-level splitting and tensor-product description are given in [27 Theorems 2.7 and 7.1] . In their conventions, indicated by the superscript PP, tuples in X a + 1 X^{a+1} X a + 1 a a a C N , P P C^{N,\mathrm{PP}} C N , PP C D , P P C^{D,\mathrm{PP}} C D , PP C ^ P P \widehat{C}^{\mathrm{PP}} C PP Λ \Lambda Λ − 1 -1 − 1 Λ [ X ] → Λ \Lambda[X]\to\Lambda Λ [ X ] → Λ x x x 1 − 1 = 0 1-1=0 1 − 1 = 0 ( − 1 ) a (-1)^a ( − 1 ) a a a a
H a ( C ^ P P ) = R a + 1 ( a ≥ − 1 ) , H b ( C N , P P ) = Q b + 1 ( b ≥ 0 ) . H_a(\widehat{C}^{\mathrm{PP}})=R_{a+1}\quad(a\ge-1),\qquad H_b(C^{N,\mathrm{PP}})=Q_{b+1}\quad(b\ge0). H a ( C PP ) = R a + 1 ( a ≥ − 1 ) , H b ( C N , PP ) = Q b + 1 ( b ≥ 0 ) . Apply the cited chain isomorphism C ^ P P [ 2 ] ⊗ Λ C N , P P ≅ C D , P P \widehat{C}^{\mathrm{PP}}[2]\otimes_\Lambda C^{N,\mathrm{PP}}\cong C^{D,\mathrm{PP}} C PP [ 2 ] ⊗ Λ C N , PP ≅ C D , PP i + j = n − 1 i+j=n-1 i + j = n − 1 i + j = n − 2 i+j=n-2 i + j = n − 2 (38) . □ \square □
Corollary 5.6. Over a principal ideal domain Λ \Lambda Λ Λ = k \Lambda=k Λ = k r n = dim k H n R ( X ; k ) r_n=\dim_k H_n^R(X;k) r n = dim k H n R ( X ; k ) q n = dim k H n Q ( X ; k ) q_n=\dim_k H_n^Q(X;k) q n = dim k H n Q ( X ; k ) n ≥ 1 n\ge1 n ≥ 1 r 0 = 1 r_0=1 r 0 = 1
q n = r n − ∑ j = 1 n − 1 r n − 1 − j q j ( n ≥ 1 ) . (39) q_n=r_n-\sum_{j=1}^{n-1}r_{n-1-j}q_j\qquad(n\ge1). \tag*{(39)} q n = r n − j = 1 ∑ n − 1 r n − 1 − j q j ( n ≥ 1 ) . (39) The positive-degree quandle series Q X , + ( t ) = ∑ n ≥ 1 q n t n Q_{X,+}(t) = \sum_{n \ge1} q_n t^n Q X , + ( t ) = ∑ n ≥ 1 q n t n R X ( t ) = ∑ n ≥ 0 r n t n R_X(t) = \sum_{n \ge0} r_n t^n R X ( t ) = ∑ n ≥ 0 r n t n
Q X , + ( t ) = R X ( t ) − 1 1 + t R X ( t ) . (40) Q_{X,+}(t) = \frac{R_X(t)-1}{1+tR_X(t)}. \tag*{(40)} Q X , + ( t ) = 1 + t R X ( t ) R X ( t ) − 1 . (40) We also have q n = dim k H Q n ( X ; k ) q_n = \dim_k H^n_Q(X;k) q n = dim k H Q n ( X ; k )
Proof. Every term Q j Q_j Q j (38) has j < n j<n j < n Λ \Lambda Λ Q n Q_n Q n R n R_n R n Q j Q_j Q j (39) , and summing gives (40) . The dimensions of the cohomology groups follow from the universal coefficient theorem. □ \square □
Additive formulas and torsion orders
We first give a common formula for the additive groups, then specialize it to dimensions over a field and to counts of cyclic summands of each order. All series are interpreted as formal power series.
For a principal ideal domain Λ \Lambda Λ Λ / ( π e ) \Lambda/(\pi^e) Λ/ ( π e ) π e \pi^e π e G Λ \mathcal{G}_{\Lambda} G Λ t t t Z [ t ] \mathbb{Z}[t] Z [ t ] Z [ t ] \mathbb{Z}[t] Z [ t ] 1 1 1 ϵ π , e \epsilon_{\pi,e} ϵ π , e π \pi π Λ \Lambda Λ e ≥ 1 e \ge1 e ≥ 1
ϵ π , e ϵ π , f = ( 1 + t ) ϵ π , min ( e , f ) , ϵ π , e ϵ σ , f = 0 ( π ≁ σ ) . (41) \epsilon_{\pi,e}\epsilon_{\pi,f}=(1+t)\epsilon_{\pi,\min(e,f)},\qquad\epsilon_{\pi,e}\epsilon_{\sigma,f}=0\quad(\pi\not\sim\sigma). \tag*{(41)} ϵ π , e ϵ π , f = ( 1 + t ) ϵ π , m i n ( e , f ) , ϵ π , e ϵ σ , f = 0 ( π ∼ σ ) . (41) For the series considered below, each coefficient involves only finitely many basis symbols. For a finitely generated Λ \Lambda Λ A A A [ A ] [A] [ A ] A A A 1 1 1 ϵ π , e \epsilon_{\pi,e} ϵ π , e Λ / ( π e ) \Lambda/(\pi^e) Λ/ ( π e ) k k k G k = Z [ [ t ] ] \mathcal{G}_k=\mathbb{Z}[[t]] G k = Z [[ t ]] [ A ] = dim k A [A]=\dim_k A [ A ] = dim k A (41) records both tensor and Tor terms:
[ A ] [ B ] = [ A ⊗ Λ B ] + t [ Tor 1 Λ ( A , B ) ] . [A][B]=[A\otimes_{\Lambda}B]+t[\operatorname{Tor}^{\Lambda}_1(A,B)]. [ A ] [ B ] = [ A ⊗ Λ B ] + t [ Tor 1 Λ ( A , B )] . The term multiplied by t t t
Lemma 5.7 (Additive homology formulas). Let Λ \Lambda Λ L L L H n ( Ω L ; Λ ) H_n(\Omega L;\Lambda) H n ( Ω L ; Λ ) n n n
κ ( t ) = ∑ n ≥ 0 [ H n ( Ω L ; Λ ) ] t n , μ ( t ) = ∑ n ≥ 0 [ H n ( Ω ( S 2 ∨ L ) ; Λ ) ] t n . \kappa(t)=\sum_{n\ge0}[H_n(\Omega L;\Lambda)]t^n,\qquad\mu(t)=\sum_{n\ge0}[H_n(\Omega(S^2\vee L);\Lambda)]t^n. κ ( t ) = n ≥ 0 ∑ [ H n ( Ω L ; Λ )] t n , μ ( t ) = n ≥ 0 ∑ [ H n ( Ω ( S 2 ∨ L ) ; Λ )] t n . Then, in G Λ \mathcal{G}_{\Lambda} G Λ
μ ( t ) = ∑ m ≥ 0 t m κ ( t ) m + 1 = κ ( t ) 1 − t κ ( t ) . (42) \mu(t)=\sum_{m\ge0}t^m\kappa(t)^{m+1}=\frac{\kappa(t)}{1-t\kappa(t)}. \tag*{(42)} μ ( t ) = m ≥ 0 ∑ t m κ ( t ) m + 1 = 1 − t κ ( t ) κ ( t ) . (42) For the quandle data of this section, take L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G ) p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ Λ = Z ( p ) \Lambda=\mathbb{Z}_{(p)} Λ = Z ( p ) p p p
ρ ( t ) = ∑ n ≥ 0 [ H n R ( X ; Λ ) ] t n , χ ( t ) = ∑ n ≥ 1 [ H n Q ( X ; Λ ) ] t n \rho(t)=\sum_{n\ge0}[H^R_n(X;\Lambda)]t^n,\qquad\chi(t)=\sum_{n\ge1}[H^Q_n(X;\Lambda)]t^n ρ ( t ) = n ≥ 0 ∑ [ H n R ( X ; Λ )] t n , χ ( t ) = n ≥ 1 ∑ [ H n Q ( X ; Λ )] t n satisfy
ρ ( t ) = 1 1 − t κ ( t ) , χ ( t ) = t κ ( t ) 1 + t − t κ ( t ) . (43) \rho(t)=\frac{1}{1-t\kappa(t)},\qquad\chi(t)=\frac{t\kappa(t)}{1+t-t\kappa(t)}. \tag*{(43)} ρ ( t ) = 1 − t κ ( t ) 1 , χ ( t ) = 1 + t − t κ ( t ) t κ ( t ) . (43) These formulas record module isomorphism classes, including torsion orders; they make no assertion about cup products or homology algebras over a principal ideal domain.
Proof. Choose a 1-reduced simplicial model K K K L L L C = Ω C ∗ ( K ; Λ ) C = \Omega C_*(K;\Lambda) C = Ω C ∗ ( K ; Λ ) C ∗ C_* C ∗ Ω \Omega Ω C 0 = Λ C_0 = \Lambda C 0 = Λ S = Δ [ 2 ] / ∂ Δ [ 2 ] S = \Delta[2]/\partial\Delta[2] S = Δ [ 2 ] / ∂ Δ [ 2 ] K ∨ S K \vee S K ∨ S
Ω C ∗ ( K ∨ S ; Λ ) ≅ C ∗ T Λ ( u ) , ∣ u ∣ = 1 , d u = 0. (44) \Omega C_*(K \vee S;\Lambda) \cong C * T_\Lambda(u), \qquad|u| = 1,\quad du = 0. \tag*{(44)} Ω C ∗ ( K ∨ S ; Λ ) ≅ C ∗ T Λ ( u ) , ∣ u ∣ = 1 , d u = 0. (44) The natural integral Szczarba comparison with chains on the Kan loop group [14 Theorem 3.1 and the following paragraph] has an acyclic bounded-below cone of free abelian groups, hence a contractible cone. It remains a quasi-isomorphism after tensoring with Λ \Lambda Λ (44) models loop chains compatibly with the inclusions.
The differential preserves the number m m m u u u c 0 u c 1 u ⋯ u c m c_0uc_1u\cdots uc_m c 0 u c 1 u ⋯ u c m c i c_i c i
C ∗ T Λ ( u ) ≅ ⨁ m ≥ 0 C ⊗ Λ ( m + 1 ) [ m ] . (45) C * T_\Lambda(u) \cong\bigoplus_{m\ge0} C^{\otimes_\Lambda(m+1)}[m]. \tag*{(45)} C ∗ T Λ ( u ) ≅ m ≥ 0 ⨁ C ⊗ Λ ( m + 1 ) [ m ] . (45) On homogeneous factors, the isomorphism from the m m m
c 0 ⊗ ⋯ ⊗ c m ⟼ ( − 1 ) ∑ i = 0 m ( m − i ) ∣ c i ∣ c 0 u c 1 u ⋯ u c m . c_0 \otimes\cdots\otimes c_m \longmapsto(-1)^{\sum_{i=0}^{m}(m-i)|c_i|}c_0uc_1u\cdots uc_m. c 0 ⊗ ⋯ ⊗ c m ⟼ ( − 1 ) ∑ i = 0 m ( m − i ) ∣ c i ∣ c 0 u c 1 u ⋯ u c m . The split Künneth theorem, recorded by (41) , gives (42) . Only m ≤ n m \le n m ≤ n n n n
For the quandle application, Theorem 3.2 and localization identify H ∗ ( M ; Λ ) H_*(\mathcal{M};\Lambda) H ∗ ( M ; Λ ) H ∗ ( Ω ( S 2 ∨ L ) ; Λ ) H_*(\Omega(S^2 \vee L);\Lambda) H ∗ ( Ω ( S 2 ∨ L ) ; Λ ) H n ( Ω L ; Λ ) H_n(\Omega L;\Lambda) H n ( Ω L ; Λ ) H n ( M ; Λ ) H_n(\mathcal{M};\Lambda) H n ( M ; Λ ) M \mathcal{M} M Proposition 5.2 and the split Künneth formula (38) give, respectively,
ρ = 1 + t μ , ρ − 1 = χ + t ρ χ . \rho= 1 + t\mu,\qquad\rho- 1 = \chi+ t\rho\chi. ρ = 1 + t μ , ρ − 1 = χ + tρ χ . The product in G Λ \mathcal{G}_\Lambda G Λ (43) ; the denominators have constant term 1 1 1 G Λ \mathcal{G}_\Lambda G Λ
For the quandle data above and a field k k k p p p p ∤ ∣ H ∣ p \nmid|H| p ∤ ∣ H ∣
A X ( t ) = ∑ n ≥ 0 dim k H n ( Ω L ; k ) t n , P M ( t ) = ∑ n ≥ 0 dim k H n ( M ; k ) t n . A_X(t) = \sum_{n\ge0}\dim_k H_n(\Omega L;k)t^n,\qquad P_{\mathcal{M}}(t) = \sum_{n\ge0}\dim_k H_n(\mathcal{M};k)t^n. A X ( t ) = n ≥ 0 ∑ dim k H n ( Ω L ; k ) t n , P M ( t ) = n ≥ 0 ∑ dim k H n ( M ; k ) t n . Lemma 5.7 ensures finite-dimensionality. With R X R_X R X Q X , + Q_{X,+} Q X , + (42) and (43) apply with Λ = k \Lambda= k Λ = k
( κ , μ , ρ , χ ) = ( A X , P M , R X , Q X , + ) . (\kappa,\mu,\rho,\chi) = (A_X,P_{\mathcal{M}},R_X,Q_{X,+}). ( κ , μ , ρ , χ ) = ( A X , P M , R X , Q X , + ) . The universal coefficient theorem gives the same dimension series for cohomology. Clauwens obtained the corresponding rack recurrence under a Leray–Hirsch hypothesis and applied it to dihedral quandles [10 Section 4.3, Remark 4.11] .
Corollary 5.8 (Multiplicities of cyclic summands). Retain the quandle data and the good prime p p p Lemma 5.7 . Put R = Z ( p ) R = \mathbb{Z}_{(p)} R = Z ( p ) K = Ω ( L ( p ) ) K = \Omega(L_{(p)}) K = Ω ( L ( p ) ) e ≥ 1 e \ge1 e ≥ 1 p p p A A A ν e ( A ) \nu_e(A) ν e ( A ) Z / p f \mathbb{Z}/p^f Z / p f f ≥ e f \ge e f ≥ e H n ( K ; R ) H_n(K;R) H n ( K ; R ) n ≥ 0 n \ge0 n ≥ 0 H n Q ( X ; R ) H_n^Q(X;R) H n Q ( X ; R ) n ≥ 2 n \ge2 n ≥ 2 p p p H 1 Q ( X ; R ) = R H_1^Q(X;R) = R H 1 Q ( X ; R ) = R
K ≥ e ( t ) = ∑ n ≥ 1 ν e ( H n ( K ; R ) ) t n , Q ≥ e ( t ) = ∑ n ≥ 2 ν e ( H n Q ( X ; R ) ) t n K_{\ge e}(t) = \sum_{n\ge1}\nu_e(H_n(K;R))t^n,\qquad Q_{\ge e}(t) = \sum_{n\ge2}\nu_e(H_n^Q(X;R))t^n K ≥ e ( t ) = n ≥ 1 ∑ ν e ( H n ( K ; R )) t n , Q ≥ e ( t ) = n ≥ 2 ∑ ν e ( H n Q ( X ; R )) t n satisfy
Q ≥ e ( t ) = t ( 1 + t ) K ≥ e ( t ) 1 − t ( 1 + t ) K ≥ e ( t ) . (46) Q_{\ge e}(t) = \frac{t(1+t)K_{\ge e}(t)}{1-t(1+t)K_{\ge e}(t)}. \tag*{(46)} Q ≥ e ( t ) = 1 − t ( 1 + t ) K ≥ e ( t ) t ( 1 + t ) K ≥ e ( t ) . (46) The series counting cyclic direct summands of order exactly p e p^e p e Q ≥ e − Q ≥ e + 1 Q_{\ge e}-Q_{\ge e+1} Q ≥ e − Q ≥ e + 1 F p \mathbb{F}_p F p
A X ( t ) = 1 + ( 1 + t ) K ≥ 1 ( t ) , Q X , + ( t ) = t + ( 1 + t ) Q ≥ 1 ( t ) . (47) A_X(t)=1+(1+t)K_{\ge1}(t),\qquad Q_{X,+}(t)=t+(1+t)Q_{\ge1}(t). \tag*{(47)} A X ( t ) = 1 + ( 1 + t ) K ≥ 1 ( t ) , Q X , + ( t ) = t + ( 1 + t ) Q ≥ 1 ( t ) . (47) Proof. Localization identifies H ∗ ( Ω L ; R ) H_*(\Omega L;R) H ∗ ( Ω L ; R ) H ∗ ( K ; R ) H_*(K;R) H ∗ ( K ; R ) L Q ≃ ∗ L_Q\simeq* L Q ≃ ∗ K K K p p p ϵ p , f ↦ 0 \epsilon_{p,f}\mapsto0 ϵ p , f ↦ 0 κ \kappa κ 1 1 1 (43) , χ \chi χ t t t H n Q ( X ; R ) H_n^Q(X;R) H n Q ( X ; R ) n ≥ 2 n\ge2 n ≥ 2 H 1 Q ( X ; R ) = R H_1^Q(X;R)=R H 1 Q ( X ; R ) = R
For each e ≥ 1 e\ge1 e ≥ 1 ϵ p , f ↦ 1 + t \epsilon_{p,f}\mapsto1+t ϵ p , f ↦ 1 + t f ≥ e f\ge e f ≥ e ϵ p , f ↦ 0 \epsilon_{p,f}\mapsto0 ϵ p , f ↦ 0 t t t 1 1 1 Z [ [ t ] ] \mathbb{Z}[[t]] Z [[ t ]] (41) . It sends κ \kappa κ χ \chi χ 1 + ( 1 + t ) K ≥ e 1+(1+t)K_{\ge e} 1 + ( 1 + t ) K ≥ e t + ( 1 + t ) Q ≥ e t+(1+t)Q_{\ge e} t + ( 1 + t ) Q ≥ e (43) gives (46) . Successive differences count exact orders, and the universal coefficient theorem gives (47) . □
Recovering integral homology. For a finite connected Alexander quandle X X X N N N N N N H n Q ( X ; Z ) H_n^Q(X;\mathbb{Z}) H n Q ( X ; Z ) n ≥ 2 n\ge2 n ≥ 2 [24 Corollary 6.2] . These finitely generated groups are therefore finite and are determined by their localizations at the primes dividing N N N N = p a N=p^a N = p a p p p H 1 Q ( X ; Z ) ≅ Z H_1^Q(X;\mathbb{Z})\cong\mathbb{Z} H 1 Q ( X ; Z ) ≅ Z
The rack cup product
We now use the marked fiber model to determine the ordinary cup product on H R ∗ ( X ; k ) = H ∗ ( R X ; k ) H_R^*(X;k)=H^*(R_X;k) H R ∗ ( X ; k ) = H ∗ ( R X ; k ) [10 Sections 2.8 and 3] .
Proposition 5.9 (Rack cohomology as a shuffle algebra). Let p p p p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ k k k p p p K = Ω L K=\Omega L K = Ω L L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G )
H R ∗ ( X ; k ) ≅ Sh k ( s H ∗ ( K ; k ) ) . (48) H_R^*(X;k)\cong\operatorname{Sh}_k(sH^*(K;k)). \tag*{(48)} H R ∗ ( X ; k ) ≅ Sh k ( s H ∗ ( K ; k )) . (48) Here s s s ( s H ∗ ( K ; k ) ) n = H n − 1 ( K ; k ) (sH^*(K;k))^n=H^{n-1}(K;k) ( s H ∗ ( K ; k ) ) n = H n − 1 ( K ; k ) n ≥ 1 n\ge1 n ≥ 1 Sh k ( V ) \operatorname{Sh}_k(V) Sh k ( V ) ⨁ m ≥ 0 V ⊗ m \bigoplus_{m\ge0}V^{\otimes m} ⨁ m ≥ 0 V ⊗ m V V V H ∗ ( K ; k ) H^*(K;k) H ∗ ( K ; k ) H 0 ( K ; k ) H^0(K;k) H 0 ( K ; k )
Proof. Put W = S 2 ∨ L W=S^2\vee L W = S 2 ∨ L i : L → W i:L\to W i : L → W F = hofib ( i ) F=\operatorname{hofib}(i) F = hofib ( i )
Ω Y ⟶ R X ⟶ B G Ω Φ ↓ g ↓ ℓ ↓ Ω W ⟶ F ⟶ L \begin{array}{ccccc}
\Omega Y & \longrightarrow& R_X & \longrightarrow& BG \\
\Omega\Phi\downarrow& & g\downarrow& & \ell\downarrow\\
\Omega W & \longrightarrow& F & \longrightarrow& L
\end{array} Ω Y ΩΦ ↓ Ω W ⟶ ⟶ R X g ↓ F ⟶ ⟶ B G ℓ ↓ L By Lemma 3.1 and localization of loop spaces, ℓ \ell ℓ Ω Φ \Omega\Phi ΩΦ k k k Y Y Y W W W g g g k k k g ∗ : H ∗ ( F ; k ) → H R ∗ ( X ; k ) g^*:H^*(F;k)\to H_R^*(X;k) g ∗ : H ∗ ( F ; k ) → H R ∗ ( X ; k )
Let r : W → L r:W\to L r : W → L E = hofib ( r ) E=\operatorname{hofib}(r) E = hofib ( r ) [30 Example 3.6 and Remark 3.7] gives
E ≃ ( S 2 × K ) / ( { ∗ } × K ) = S 2 ∧ K + = Σ 2 K + , E\simeq(S^2\times K)/(\{*\}\times K)=S^2\wedge K_+=\Sigma^2K_+, E ≃ ( S 2 × K ) / ({ ∗ } × K ) = S 2 ∧ K + = Σ 2 K + , where K + K_+ K + K K K r i = id L ri=\operatorname{id}_L r i = id L r ∗ i ∗ = id r_*i_*=\operatorname{id} r ∗ i ∗ = id Ω E → Ω W → F \Omega E\to\Omega W\to F Ω E → Ω W → F F ≃ Ω Σ 2 K + F\simeq\Omega\Sigma^2K_+ F ≃ Ω Σ 2 K +
Set V = Σ K + V=\Sigma K_+ V = Σ K + [4 ] gives
H ∗ ( Ω Σ V ; k ) ≅ T k ( H ~ ∗ ( V ; k ) ) = T k ( s H ∗ ( K ; k ) ) . H_*(\Omega\Sigma V;k)\cong T_k(\widetilde{H}_*(V;k))=T_k(sH_*(K;k)). H ∗ ( ΩΣ V ; k ) ≅ T k ( H ∗ ( V ; k )) = T k ( s H ∗ ( K ; k )) . Because V V V Lemma 5.7 , the letter space is finite dimensional in each degree and concentrated in positive degrees. Thus each total degree has only finitely many word lengths, and taking degreewise duals gives Sh k ( s H ∗ ( K ; k ) ) \operatorname{Sh}_k(sH^*(K;k)) Sh k ( s H ∗ ( K ; k )) g ∗ g^* g ∗
Cyclic Alexander quandles and cyclic Sylow subgroups
We prove Theorem A and then extend the local computation to finite connected quandles X X X p p p p p p Inn ( X ) \operatorname{Inn}(X) Inn ( X )
We first record the augmentation data used in the Alexander examples. Let A A A T ∈ Aut ( A ) T \in\operatorname{Aut}(A) T ∈ Aut ( A ) 1 − T 1-T 1 − T X = A X=A X = A x ∗ y = T x + ( 1 − T ) y x*y=Tx+(1-T)y x ∗ y = T x + ( 1 − T ) y ( 1 − T ) A = A (1-T)A=A ( 1 − T ) A = A A A A T T T X X X 0 0 0
G ≅ A ⋊ ⟨ T ⟩ , H ≅ ⟨ T ⟩ = C d , d = ord ( T ) . (49) G \cong A \rtimes\langle T\rangle,\qquad H \cong\langle T\rangle=C_d,\qquad d=\operatorname{ord}(T). \tag*{(49)} G ≅ A ⋊ ⟨ T ⟩ , H ≅ ⟨ T ⟩ = C d , d = ord ( T ) . (49) Our right-action multiplication is opposite to the usual left affine multiplication; inversion identifies the two groups and sends T T T T − 1 T^{-1} T − 1 regular if gcd ( ∣ A ∣ , d ) = 1 \gcd(|A|,d)=1 g cd( ∣ A ∣ , d ) = 1 ∣ H ∣ = d |H|=d ∣ H ∣ = d Proposition 3.3 applies at every prime p ∤ d p\nmid d p ∤ d ∣ A ∣ |A| ∣ A ∣
For the cyclic case, fix an odd prime p p p r ≥ 1 r\geq1 r ≥ 1 ω ∈ ( Z / p r ) × \omega\in(\mathbb{Z}/p^r)^\times ω ∈ ( Z / p r ) × d > 1 d>1 d > 1 p − 1 p-1 p − 1
X p r , ω = Z / p r , x ∗ y = ω x + ( 1 − ω ) y . (50) X_{p^r,\omega}=\mathbb{Z}/p^r,\qquad x*y=\omega x+(1-\omega)y. \tag*{(50)} X p r , ω = Z / p r , x ∗ y = ω x + ( 1 − ω ) y . (50) Since the kernel of reduction modulo p p p p p p ω \omega ω d d d 1 − ω 1-\omega 1 − ω X p r , ω X_{p^r,\omega} X p r , ω (49) , its inner automorphism group and the stabilizer of 0 0 0
G = C p r ⋊ C d , H = C d (51) G=C_{p^r}\rtimes C_d,\qquad H=C_d \tag*{(51)} G = C p r ⋊ C d , H = C d (51) and the augmentation element is h = S 0 h=S_0 h = S 0 C d C_d C d ω \omega ω ω − 1 \omega^{-1} ω − 1 ( G , H ) (G,H) ( G , H ) h h h L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G )
In the case r = 1 r=1 r = 1 d = ord ( ω ) > 1 d=\operatorname{ord}(\omega)>1 d = ord ( ω ) > 1 d d d F p \mathbb{F}_p F p [24 Theorem 1.1, Corollary 5.9] .
Cohomology and higher Bocksteins
The following calculation supplies the input for the loop-space identification in Theorem 6.2 . We use the Bockstein spectral sequence associated with the p p p p p p β 1 \beta_1 β 1 β r \beta_r β r r r r Λ ( − ) \Lambda(-) Λ ( − ) F p \mathbb{F}_p F p
Lemma 6.1. Generators z z z w w w
H ∗ ( G ; F p ) = Λ ( z 2 d − 1 ) ⊗ F p [ w 2 d ] , β r ( z ) = w . (52) H^*(G;\mathbb{F}_p)=\Lambda(z_{2d-1})\otimes\mathbb{F}_p[w_{2d}],\qquad\beta_r(z)=w. \tag*{(52)} H ∗ ( G ; F p ) = Λ ( z 2 d − 1 ) ⊗ F p [ w 2 d ] , β r ( z ) = w . (52) All Bockstein differentials before β r \beta_r β r
Proof. For P = C p r P=C_{p^r} P = C p r H ∗ ( P ; F p ) = Λ ( a 1 ) ⊗ F p [ b 2 ] H^*(P;\mathbb{F}_p)=\Lambda(a_1)\otimes\mathbb{F}_p[b_2] H ∗ ( P ; F p ) = Λ ( a 1 ) ⊗ F p [ b 2 ] 0 0 0 p r p^r p r r r r
and generators may be chosen such that β r ( a ) = b \beta_r(a)=b β r ( a ) = b β r ( b ) = 0 \beta_r(b)=0 β r ( b ) = 0 p ∤ d p\nmid d p ∤ d
H ∗ ( G ; F p ) ≅ H ∗ ( P ; F p ) C d . H^*(G;\mathbb{F}_p)\cong H^*(P;\mathbb{F}_p)^{C_d}. H ∗ ( G ; F p ) ≅ H ∗ ( P ; F p ) C d . By naturality of H 1 ( P ; F p ) = Hom ( P , F p ) H^1(P;\mathbb{F}_p)=\operatorname{Hom}(P,\mathbb{F}_p) H 1 ( P ; F p ) = Hom ( P , F p ) β r \beta_r β r C d C_d C d a , b a,b a , b d d d b j d b^{jd} b j d j ≥ 0 j\geq0 j ≥ 0 a b j d − 1 ab^{jd-1} a b j d − 1 j ≥ 1 j\geq1 j ≥ 1 z = a b d − 1 z=ab^{d-1} z = a b d − 1 w = b d w=b^d w = b d p p p z 2 = 0 z^2=0 z 2 = 0 β r ( z ) = w \beta_r(z)=w β r ( z ) = w
The loop space of the homotopy cofiber
For odd m > 1 m>1 m > 1 N ≥ 1 N\geq1 N ≥ 1 S m { N } S^m\{N\} S m { N } N N N [ N ] : S m → S m [N]:S^m\to S^m [ N ] : S m → S m
S m { N } ( p ) ≃ hofib ( [ N ] ( p ) : S ( p ) m → S ( p ) m ) . (53) S^m\{N\}_{(p)}\simeq\operatorname{hofib}\left([N]_{(p)}:S^m_{(p)}\to S^m_{(p)}\right). \tag*{(53)} S m { N } ( p ) ≃ hofib ( [ N ] ( p ) : S ( p ) m → S ( p ) m ) . (53) For an odd prime p p p d > 1 d>1 d > 1 p − 1 p-1 p − 1 S 2 d − 1 { p r } S^{2d-1}\{p^r\} S 2 d − 1 { p r } p p p Z Z Z p p p Ω Z ≃ S 2 d − 1 { p r } \Omega Z\simeq S^{2d-1}\{p^r\} Ω Z ≃ S 2 d − 1 { p r } [7 Corollary 1.2 and Lemma 2.1] ]. This characterization requires the classes to survive to the r r r d d d
Theorem 6.2. For X = X p r , ω X=X_{p^r,\omega} X = X p r , ω (50) , there is an ordinary homotopy equivalence
Ω L ( p ) ≃ S 2 d − 1 { p r } ( p ) (54) \Omega L_{(p)}\simeq S^{2d-1}\{p^r\}_{(p)} \tag*{(54)} Ω L ( p ) ≃ S 2 d − 1 { p r } ( p ) (54) and M ( p ) \mathcal{M}_{(p)} M ( p ) A ∞ A_\infty A ∞
M ( p ) ≃ A ∞ Ω ( ( S 2 ∨ B ( C p r ⋊ C d ) ∧ ) ( p ) ) . (55) \mathcal{M}_{(p)}\simeq_{A_\infty}\Omega\left((S^2\vee B(C_{p^r}\rtimes C_d)^\wedge)_{(p)}\right). \tag*{(55)} M ( p ) ≃ A ∞ Ω ( ( S 2 ∨ B ( C p r ⋊ C d ) ∧ ) ( p ) ) . (55) Moreover,
π j ( Ω L ( p ) ) = 0 ( 1 ≤ j < 2 d − 2 ) , π 2 d − 2 ( Ω L ( p ) ) ≅ C p r . (56) \pi_j(\Omega L_{(p)})=0\quad(1\leq j<2d-2),\qquad\pi_{2d-2}(\Omega L_{(p)})\cong C_{p^r}. \tag*{(56)} π j ( Ω L ( p ) ) = 0 ( 1 ≤ j < 2 d − 2 ) , π 2 d − 2 ( Ω L ( p ) ) ≅ C p r . (56) Proof. By Proposition 3.3, L ( p ) ≃ ( B G ) p ∧ L_{(p)}\simeq(BG)^\wedge_p L ( p ) ≃ ( B G ) p ∧ p p p B G → L ( p ) BG\to L_{(p)} B G → L ( p ) p p p p r p^r p r [7 Section 1.1] ], giving (54) . The homotopy exact sequence of (53) gives (56) . Proposition 3.3 gives (55) . □
Integral input for rack and quandle homology
Henceforth, put R = Z ( p ) R=\mathbb{Z}_{(p)} R = Z ( p ) R R R R / p r R R/p^rR R / p r R C p r C_{p^r} C p r
Lemma 6.3. Let m = 2 d − 1 m=2d-1 m = 2 d − 1 F = S m { p r } ( p ) F=S^m\{p^r\}_{(p)} F = S m { p r } ( p )
H j ( F ; R ) ≅ { R , j = 0 , C p r , j = k ( 2 d − 2 ) for some k ≥ 1 , 0 , otherwise. (57) H_j(F;R)\cong
\begin{cases}
R, & j=0,\\
C_{p^r}, & j=k(2d-2)\text{ for some }k\geq1,\\
0, & \text{otherwise.}
\end{cases} \tag*{(57)} H j ( F ; R ) ≅ ⎩ ⎨ ⎧ R , C p r , 0 , j = 0 , j = k ( 2 d − 2 ) for some k ≥ 1 , otherwise. (57) In particular, these are also the groups H j ( Ω L ; R ) H_j(\Omega L;R) H j ( Ω L ; R ) X = X p r , ω X=X_{p^r,\omega} X = X p r , ω F p \mathbb{F}_p F p
A X ( t ) = 1 + t 2 d − 1 1 − t 2 d − 2 . (58) A_X(t)=\frac{1+t^{2d-1}}{1-t^{2d-2}}. \tag*{(58)} A X ( t ) = 1 − t 2 d − 2 1 + t 2 d − 1 . (58) Proof. Formula (57) is the p p p [7 Section 1.1] ]. By Theorem 6.2 and invariance of homology with coefficients in R R R H j ( Ω L ; R ) H_j(\Omega L;R) H j ( Ω L ; R ) A X ( t ) = 1 + ( 1 + t ) t 2 d − 2 / ( 1 − t 2 d − 2 ) A_X(t)=1+(1+t)t^{2d-2}/(1-t^{2d-2}) A X ( t ) = 1 + ( 1 + t ) t 2 d − 2 / ( 1 − t 2 d − 2 ) (58) . □
For any field k k k p p p H ∗ ( Ω B G p ∧ ; k ) H_*(\Omega BG^{\wedge}_p;k) H ∗ ( Ω B G p ∧ ; k ) A ∞ A_\infty A ∞ (58) [2 Theorem 1.3 and Section 5] .
Theorem 6.4. For X = X p r , ω X=X_{p^r,\omega} X = X p r , ω d > 1 d>1 d > 1 p − 1 p-1 p − 1
H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ ( C p r ) b n ( n ≥ 2 ) , (59) H^Q_1(X;\mathbb{Z})\cong\mathbb{Z},\qquad H^Q_n(X;\mathbb{Z})\cong(C_{p^r})^{b_n}\quad(n\geq2), \tag*{(59)} H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ ( C p r ) b n ( n ≥ 2 ) , (59) where the multiplicities are determined by
∑ n ≥ 2 b n t n = t 2 d − 1 ( 1 + t ) 1 − t 2 d − 2 − t 2 d − 1 − t 2 d . (60) \sum_{n\geq2} b_n t^n=\frac{t^{2d-1}(1+t)}{1-t^{2d-2}-t^{2d-1}-t^{2d}}. \tag*{(60)} n ≥ 2 ∑ b n t n = 1 − t 2 d − 2 − t 2 d − 1 − t 2 d t 2 d − 1 ( 1 + t ) . (60) Proof. In Corollary 5.8, Lemma 6.3 gives
K ≥ 1 ( t ) = ⋯ = K ≥ r ( t ) = t 2 d − 2 1 − t 2 d − 2 , K ≥ r + 1 ( t ) = 0. K_{\geq1}(t)=\cdots=K_{\geq r}(t)=\frac{t^{2d-2}}{1-t^{2d-2}},\qquad K_{\geq r+1}(t)=0. K ≥ 1 ( t ) = ⋯ = K ≥ r ( t ) = 1 − t 2 d − 2 t 2 d − 2 , K ≥ r + 1 ( t ) = 0. Thus Q ≥ 1 = ⋯ = Q ≥ r Q_{\geq1}=\cdots=Q_{\geq r} Q ≥ 1 = ⋯ = Q ≥ r (60) , and Q ≥ r + 1 = 0 Q_{\geq r+1}=0 Q ≥ r + 1 = 0 C p r C_{p^r} C p r H 1 Q ( X ; R ) = R H^Q_1(X;R)=R H 1 Q ( X ; R ) = R X X X p r p^r p r
Corollary 6.5. For X = X p r , ω X=X_{p^r,\omega} X = X p r , ω p p p
P M ( t ) = 1 + t 2 d − 1 1 − t − t 2 d − 2 − t 2 d . (61) P_M(t)=\frac{1+t^{2d-1}}{1-t-t^{2d-2}-t^{2d}}. \tag*{(61)} P M ( t ) = 1 − t − t 2 d − 2 − t 2 d 1 + t 2 d − 1 . (61) R X ( t ) = 1 − t 2 d − 2 1 − t − t 2 d − 2 − t 2 d . (62) R_X(t)=\frac{1-t^{2d-2}}{1-t-t^{2d-2}-t^{2d}}. \tag*{(62)} R X ( t ) = 1 − t − t 2 d − 2 − t 2 d 1 − t 2 d − 2 . (62) The same series give the corresponding cohomology dimensions. Let b n b_n b n Theorem 6.4 , with b 1 = 0 b_1=0 b 1 = 0
dim F p H n Q ( X ; F p ) = dim F p H Q n ( X ; F p ) = { 1 , n = 1 , b n + b n − 1 , n ≥ 2. \dim_{\mathbb{F}_p}H^Q_n(X;\mathbb{F}_p)=\dim_{\mathbb{F}_p}H^n_Q(X;\mathbb{F}_p)=
\begin{cases}
1, & n=1,\\
b_n+b_{n-1}, & n\geq2.
\end{cases} dim F p H n Q ( X ; F p ) = dim F p H Q n ( X ; F p ) = { 1 , b n + b n − 1 , n = 1 , n ≥ 2. Proof. The two series follow by substituting (58) into Lemma 5.7 with Λ = F p \Lambda=\mathbb{F}_p Λ = F p Theorem 6.4 and the universal coefficient theorem. □
Example 6.6 (The dihedral family and parameter lifts). Let p p p d > 1 d>1 d > 1 p − 1 p-1 p − 1 ω ˉ ∈ F p × \bar{\omega}\in\mathbb{F}_p^\times ω ˉ ∈ F p × d d d r ≥ 1 r\geq1 r ≥ 1 ω ˉ \bar{\omega} ω ˉ u d − 1 u^d-1 u d − 1 ω r ∈ ( Z / p r ) × \omega_r\in(\mathbb{Z}/p^r)^\times ω r ∈ ( Z / p r ) × d d d X r = X p r , ω r X_r=X_{p^r,\omega_r} X r = X p r , ω r Theorem 6.4 gives
H n Q ( X r ; Z ) = 0 ( 2 ≤ n ≤ 2 d − 2 ) , H 2 d − 1 Q ( X r ; Z ) ≅ H 2 d Q ( X r ; Z ) ≅ C p r . H^Q_n(X_r;\mathbb{Z})=0\quad(2\leq n\leq2d-2),\qquad H^Q_{2d-1}(X_r;\mathbb{Z})\cong H^Q_{2d}(X_r;\mathbb{Z})\cong C_{p^r}. H n Q ( X r ; Z ) = 0 ( 2 ≤ n ≤ 2 d − 2 ) , H 2 d − 1 Q ( X r ; Z ) ≅ H 2 d Q ( X r ; Z ) ≅ C p r . Thus the first degree at least two in which torsion occurs is 2 d − 1 2d-1 2 d − 1 F p \mathbb{F}_p F p 2 d − 1 2d-1 2 d − 1 2 d 2d 2 d
The case d = 2 d=2 d = 2 ω r = − 1 \omega_r=-1 ω r = − 1
Q X r , + ( t ) = t + t 4 1 − t 2 − t 3 − t 4 . Q_{X_r,+}(t)=\frac{t+t^4}{1-t^2-t^3-t^4}. Q X r , + ( t ) = 1 − t 2 − t 3 − t 4 t + t 4 . The positive-degree dimensions begin 1 , 0 , 1 , 2 , 2 , 3 , 5 , 7 , 10 , … 1,0,1,2,2,3,5,7,10,\ldots 1 , 0 , 1 , 2 , 2 , 3 , 5 , 7 , 10 , …
For the prime-power family X p r , ω X_{p^r,\omega} X p r , ω
Proposition 6.7. For fixed p p p d d d p p p r r r Ω L ( p ) \Omega L_{(p)} Ω L ( p ) M ( p ) \mathcal{M}_{(p)} M ( p ) r r r m = 2 d − 1 m=2d-1 m = 2 d − 1
π m − 1 ( M ( p ) ) ≅ π m ( S 2 ) ( p ) ⊕ C p r . (63) \pi_{m-1}(\mathcal{M}_{(p)}) \cong\pi_m(S^2)_{(p)} \oplus C_{p^r}. \tag*{(63)} π m − 1 ( M ( p ) ) ≅ π m ( S 2 ) ( p ) ⊕ C p r . (63) The graded algebras H ∗ ( C p r ⋊ C d ; F p ) H^*(C_{p^r}\rtimes C_d;\mathbb{F}_p) H ∗ ( C p r ⋊ C d ; F p ) r r r r r r
Proof. The assertions about the dimensions and Bocksteins follow from Corollary 6.5 and Lemma 6.1 . By (56) , L ( p ) L_{(p)} L ( p ) ( m − 1 ) (m-1) ( m − 1 ) π m ( L ( p ) ) ≅ C p r \pi_m(L_{(p)})\cong C_{p^r} π m ( L ( p ) ) ≅ C p r L ( p ) L_{(p)} L ( p ) m m m ( S 2 × L ( p ) , S 2 ∨ L ( p ) ) (S^2\times L_{(p)},S^2\vee L_{(p)}) ( S 2 × L ( p ) , S 2 ∨ L ( p ) ) m + 2 m+2 m + 2 π m \pi_m π m p p p Theorem 3.2 gives (63) . Since the torsion subgroup of π m ( S 2 ) ( p ) \pi_m(S^2)_{(p)} π m ( S 2 ) ( p ) r r r (63) determines r r r Ω L ( p ) \Omega L_{(p)} Ω L ( p ) r r r (56) . □
Reduction to cyclic Sylow subgroups
The same computation extends to the case of cyclic Sylow subgroups. The inertial index of the Sylow subgroup then plays the role of d d d
Theorem 6.8. Let p p p X X X G = Inn ( X ) G=\operatorname{Inn}(X) G = Inn ( X ) H H H p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ p p p P P P G G G p r p^r p r r ≥ 1 r\geq1 r ≥ 1
e = ∣ N G ( P ) : C G ( P ) ∣ . (64) e=\lvert N_G(P):C_G(P)\rvert. \tag*{(64)} e = ∣ N G ( P ) : C G ( P )∣ . (64) Then p p p e > 1 e>1 e > 1 e ∣ p − 1 e\mid p-1 e ∣ p − 1 L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G )
Ω L ( p ) ≃ S { p r } ( p ) 2 e − 1 . (65) \Omega L_{(p)}\simeq S^{2e-1}_{\{p^r\}_{(p)}}. \tag*{(65)} Ω L ( p ) ≃ S { p r } ( p ) 2 e − 1 . (65) If ω ∈ ( Z / p r ) × \omega\in(\mathbb{Z}/p^r)^\times ω ∈ ( Z / p r ) × e e e A ∞ A_\infty A ∞
( M X ) ( p ) ≃ A ∞ ( M X p r , ω ) ( p ) . (66) (\mathcal{M}_X)_{(p)}\simeq_{A_\infty}(\mathcal{M}_{X_{p^r,\omega}})_{(p)}. \tag*{(66)} ( M X ) ( p ) ≃ A ∞ ( M X p r , ω ) ( p ) . (66) The p p p X X X
H 1 Q ( X ; Z ( p ) ) ≅ Z ( p ) , H n Q ( X ; Z ( p ) ) ≅ ( C p r ) b n ( e ) ( n ≥ 2 ) . (67) H_1^Q(X;\mathbb{Z}_{(p)})\cong\mathbb{Z}_{(p)},\qquad H_n^Q(X;\mathbb{Z}_{(p)})\cong(C_{p^r})^{b_n(e)}\quad(n\geq2). \tag*{(67)} H 1 Q ( X ; Z ( p ) ) ≅ Z ( p ) , H n Q ( X ; Z ( p ) ) ≅ ( C p r ) b n ( e ) ( n ≥ 2 ) . (67) Here b n ( e ) b_n(e) b n ( e ) (60) with d = e d=e d = e p p p d = e d=e d = e
Proof. The cyclic Sylow reduction in [2 Section 1] gives B G p ∧ ≃ B Γ p ∧ BG_p^\wedge\simeq B\Gamma_p^\wedge B G p ∧ ≃ B Γ p ∧ Γ = C p r ⋊ C e \Gamma=C_{p^r}\rtimes C_e Γ = C p r ⋊ C e e ∣ p − 1 e\mid p-1 e ∣ p − 1 C e C_e C e C p r C_{p^r} C p r Proposition 3.3 , B G p ∧ BG_p^\wedge B G p ∧ e = 1 e=1 e = 1 B Γ p ∧ ≃ B C p r B\Gamma_p^\wedge\simeq BC_{p^r} B Γ p ∧ ≃ B C p r p p p p p p r ≥ 1 r\geq1 r ≥ 1 e > 1 e>1 e > 1 p p p
Choose ω ∈ ( Z / p r ) × \omega\in(\mathbb{Z}/p^r)^\times ω ∈ ( Z / p r ) × e e e Γ ≅ Inn ( X p r , ω ) \Gamma\cong\operatorname{Inn}(X_{p^r,\omega}) Γ ≅ Inn ( X p r , ω ) C e C_e C e Proposition 3.3 and Theorem 6.2 give (65) and (66) . The monoid equivalence and Proposition 5.2 identify the rack homology groups over Z ( p ) \mathbb{Z}_{(p)} Z ( p ) X p r , ω X_{p^r,\omega} X p r , ω Theorem 6.4 for X p r , ω X_{p^r,\omega} X p r , ω (67) . Over F p \mathbb{F}_p F p Proposition 5.2 give the P M P_M P M R X R_X R X (67) gives its quandle dimensions. □
Corollary 6.9 (Regular cyclic Alexander quandles). Let N > 1 N > 1 N > 1 ω ∈ ( Z / N ) × \omega\in(\mathbb{Z}/N)^\times ω ∈ ( Z / N ) × D = ord N ( ω ) D = \operatorname{ord}_N(\omega) D = ord N ( ω )
gcd ( N , 1 − ω ) = gcd ( N , D ) = 1 , \gcd(N,1-\omega)=\gcd(N,D)=1, g cd( N , 1 − ω ) = g cd( N , D ) = 1 , and equip X = Z / N X=\mathbb{Z}/N X = Z / N x ∗ y = ω x + ( 1 − ω ) y x*y=\omega x+(1-\omega)y x ∗ y = ω x + ( 1 − ω ) y N = ∏ p ∣ N p r p N=\prod_{p\mid N}p^{r_p} N = ∏ p ∣ N p r p d p = ord p r p ( ω ) d_p=\operatorname{ord}_{p^{r_p}}(\omega) d p = ord p r p ( ω ) p ∣ N p\mid N p ∣ N d p > 1 d_p>1 d p > 1 p − 1 p-1 p − 1
H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ ⨁ p ∣ N ( C p r p ) b n ( d p ) ( n ≥ 2 ) . (68) H_1^Q(X;\mathbb{Z})\cong\mathbb{Z},\qquad H_n^Q(X;\mathbb{Z})\cong\bigoplus_{p\mid N}(C_{p^{r_p}})^{b_n(d_p)}\quad(n\geq2). \tag*{(68)} H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ p ∣ N ⨁ ( C p r p ) b n ( d p ) ( n ≥ 2 ) . (68) where b n ( d ) b_n(d) b n ( d ) (60) . In particular, for the dihedral quandle R N R_N R N N > 1 N>1 N > 1
H n Q ( R N ; Z ) ≅ ( C N ) b n ( 2 ) ( n ≥ 2 ) . (69) H_n^Q(R_N;\mathbb{Z})\cong(C_N)^{b_n(2)}\quad(n\geq2). \tag*{(69)} H n Q ( R N ; Z ) ≅ ( C N ) b n ( 2 ) ( n ≥ 2 ) . (69) Proof. The first coprimality assumption makes X X X (49) gives G = C N ⋊ C D G=C_N\rtimes C_D G = C N ⋊ C D H = C D H=C_D H = C D p ∣ N p\mid N p ∣ N p ∤ D p\nmid D p ∤ D P = C p r p P=C_{p^{r_p}} P = C p r p p p p p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ C D C_D C D Aut ( P ) \operatorname{Aut}(P) Aut ( P ) d p d_p d p d p d_p d p p p p p p p d p d_p d p (68) . For ω = − 1 \omega=-1 ω = − 1 N N N d p d_p d p (69) .
Example 6.10. For X = Z / 15 X=\mathbb{Z}/15 X = Z /15 x ∗ y = 2 x − y x*y=2x-y x ∗ y = 2 x − y ω = 2 \omega=2 ω = 2 d 3 = 2 d_3=2 d 3 = 2 d 5 = 4 d_5=4 d 5 = 4
H n Q ( X ; Z ) ≅ ( C 3 ) b n ( 2 ) ⊕ ( C 5 ) b n ( 4 ) ( n ≥ 2 ) . H_n^Q(X;\mathbb{Z})\cong(C_3)^{b_n(2)}\oplus(C_5)^{b_n(4)}\quad(n\geq2). H n Q ( X ; Z ) ≅ ( C 3 ) b n ( 2 ) ⊕ ( C 5 ) b n ( 4 ) ( n ≥ 2 ) . The first 3-primary torsion occurs in degree 3, and the first 5-primary torsion occurs in degree 7.
The tetrahedral quandle
We prove Theorem B by identifying a homotopy-fiber model and computing its homology over Z ( 2 ) \mathbb{Z}_{(2)} Z ( 2 )
Let F 4 = F 2 [ ω ] / ( ω 2 + ω + 1 ) \mathbb{F}_4=\mathbb{F}_2[\omega]/(\omega^2+\omega+1) F 4 = F 2 [ ω ] / ( ω 2 + ω + 1 )
x ∗ y = ω x + ( 1 − ω ) y . x*y=\omega x+(1-\omega)y. x ∗ y = ω x + ( 1 − ω ) y . Choosing the other primitive element gives an isomorphic quandle via the Frobenius automorphism. Denote this quandle by X X X
G = Inn ( X ) , H = Stab G ( 0 ) , M = B ( G ; X ) , L = Cofib ( B H ⟶ B G ) . G=\operatorname{Inn}(X),\qquad H=\operatorname{Stab}_G(0),\qquad\mathcal{M}=B(G;X),\qquad L=\operatorname{Cofib}(BH\longrightarrow BG). G = Inn ( X ) , H = Stab G ( 0 ) , M = B ( G ; X ) , L = Cofib ( B H ⟶ B G ) . By (49) , G G G ω \omega ω
G = F 4 ⋊ ⟨ ω ⟩ ≅ A 4 , H = ⟨ ω ⟩ ≅ C 3 . (70) G=\mathbb{F}_4\rtimes\langle\omega\rangle\cong A_4,\qquad H=\langle\omega\rangle\cong C_3. \tag*{(70)} G = F 4 ⋊ ⟨ ω ⟩ ≅ A 4 , H = ⟨ ω ⟩ ≅ C 3 . (70) Lemma 3.1 shows that L L L
Let c : R P 3 → S 3 c:\mathbb{R}P^3\to S^3 c : R P 3 → S 3
K t e t = hofib ( R P ( 2 ) 3 → [ 4 ] ( 2 ) ∘ c ( 2 ) S ( 2 ) 3 ) . (71) K_{\mathrm{tet}}=\operatorname{hofib}\left(\mathbb{R}P^3_{(2)}\xrightarrow{[4]_{(2)}\circ c_{(2)}}S^3_{(2)}\right). \tag*{(71)} K tet = hofib ( R P ( 2 ) 3 [ 4 ] ( 2 ) ∘ c ( 2 ) S ( 2 ) 3 ) . (71) Here the homotopy fiber is based, and the degree 4 refers to the action on H 3 H_3 H 3 π 3 \pi_3 π 3
The fiber sequence ( S 3 ) 2 ∧ → ( B A 4 ) 2 ∧ → B S O ( 3 ) 2 ∧ (S^3)^\wedge_2\to(BA_4)^\wedge_2\to BSO(3)^\wedge_2 ( S 3 ) 2 ∧ → ( B A 4 ) 2 ∧ → B S O ( 3 ) 2 ∧ Ω ( ( B A 4 ) 2 ∧ ) \Omega((BA_4)^\wedge_2) Ω (( B A 4 ) 2 ∧ ) F 2 \mathbb{F}_2 F 2 [3 Example 13.3] . Below, we identify the degree-4 map geometrically and compare it with the quandle model.
Theorem 7.1. There is a homotopy equivalence of underlying spaces
Ω L ( 2 ) ≃ K t e t . (72) \Omega L_{(2)} \simeq K_{\mathrm{tet}}. \tag*{(72)} Ω L ( 2 ) ≃ K tet . (72) The space K t e t K_{\mathrm{tet}} K tet 2 2 2
π 1 K t e t ≅ C 2 , π 2 K t e t ≅ C 8 . (73) \pi_1K_{\mathrm{tet}} \cong C_2,\qquad\pi_2K_{\mathrm{tet}} \cong C_8. \tag*{(73)} π 1 K tet ≅ C 2 , π 2 K tet ≅ C 8 . (73) In the notation of (53) , its universal cover has the ordinary homotopy type
K ~ t e t ≃ S 3 { 8 } ( 2 ) . (74) \widetilde K_{\mathrm{tet}} \simeq S^3\{8\}_{(2)}. \tag*{(74)} K tet ≃ S 3 { 8 } ( 2 ) . (74) The loop space comparison for the covering monoid is the A ∞ A_\infty A ∞
M ( 2 ) ≃ A ∞ Ω ( ( S 2 ∨ L ) ( 2 ) ) . (75) \mathcal{M}_{(2)} \simeq_{A_\infty} \Omega\bigl((S^2\vee L)_{(2)}\bigr). \tag*{(75)} M ( 2 ) ≃ A ∞ Ω ( ( S 2 ∨ L ) ( 2 ) ) . (75) An A ∞ A_\infty A ∞ K t e t K_{\mathrm{tet}} K tet (72) .
Proof. Realize A 4 A_4 A 4
Q = S O ( 3 ) / A 4 ≅ S 3 / ( 2 T ) , ∣ 2 T ∣ = 24. Q=SO(3)/A_4\cong S^3/(2T),\qquad|2T|=24. Q = S O ( 3 ) / A 4 ≅ S 3 / ( 2 T ) , ∣2 T ∣ = 24. For the binary tetrahedral group 2 T ≅ Q 8 ⋊ C 3 2T\cong Q_8\rtimes C_3 2 T ≅ Q 8 ⋊ C 3 C 3 C_3 C 3 Q 8 / [ Q 8 , Q 8 ] ≅ C 2 2 Q_8/[Q_8,Q_8]\cong C_2^2 Q 8 / [ Q 8 , Q 8 ] ≅ C 2 2 ( 2 T ) a b ≅ C 3 (2T)_{\mathrm{ab}}\cong C_3 ( 2 T ) ab ≅ C 3 Q Q Q 3 3 3 F 2 \mathbb{F}_2 F 2 1 1 1 e : Q → S 3 e:Q\to S^3 e : Q → S 3 F 2 \mathbb{F}_2 F 2 e 2 ∧ : Q 2 ∧ → ( S 3 ) 2 ∧ e_2^\wedge:Q_2^\wedge\to(S^3)_2^\wedge e 2 ∧ : Q 2 ∧ → ( S 3 ) 2 ∧
The inclusion A 4 ⊂ S O ( 3 ) A_4\subset SO(3) A 4 ⊂ S O ( 3 )
Q ⟶ B A 4 ⟶ B S O ( 3 ) . (76) Q\longrightarrow BA_4\longrightarrow BSO(3). \tag*{(76)} Q ⟶ B A 4 ⟶ B S O ( 3 ) . (76) Since B S O ( 3 ) BSO(3) B S O ( 3 ) 2 2 2 [5 Chapter II, §5] applies both to this sequence and to the path-space fibration of B S O ( 3 ) BSO(3) B S O ( 3 ) Ω ( ( B S O ( 3 ) ) 2 ∧ ) ≃ S O ( 3 ) 2 ∧ \Omega((BSO(3))_2^\wedge)\simeq SO(3)_2^\wedge Ω (( B S O ( 3 ) ) 2 ∧ ) ≃ S O ( 3 ) 2 ∧ S O ( 3 ) ≃ Ω B S O ( 3 ) SO(3)\simeq\Omega BSO(3) S O ( 3 ) ≃ Ω B S O ( 3 ) q : S O ( 3 ) → Q q:SO(3)\to Q q : S O ( 3 ) → Q e 2 ∧ e_2^\wedge e 2 ∧ Ω ( ( B A 4 ) 2 ∧ ) \Omega((BA_4)_2^\wedge) Ω (( B A 4 ) 2 ∧ ) ( e ∘ q ) 2 ∧ (e\circ q)_2^\wedge ( e ∘ q ) 2 ∧ S O ( 3 ) ≅ R P 3 SO(3)\cong\mathbb{R}P^3 S O ( 3 ) ≅ R P 3 q q q 12 12 12 [ R P 3 , S 3 ] ≅ H 3 ( R P 3 ; Z ) ≅ Z [\mathbb{R}P^3,S^3]\cong H^3(\mathbb{R}P^3;\mathbb{Z})\cong\mathbb{Z} [ R P 3 , S 3 ] ≅ H 3 ( R P 3 ; Z ) ≅ Z e ∘ q ≃ [ 12 ] ∘ c ≃ [ 3 ] ∘ [ 4 ] ∘ c e\circ q\simeq[12]\circ c\simeq[3]\circ[4]\circ c e ∘ q ≃ [ 12 ] ∘ c ≃ [ 3 ] ∘ [ 4 ] ∘ c 3 3 3 S 3 S^3 S 3
Ω ( ( B A 4 ) 2 ∧ ) ≃ hofib ( ( R P 3 ) 2 ∧ → [ 4 ] ∘ c ) 2 ∧ ( S 3 ) 2 ∧ ) . (77) \Omega((BA_4)_2^\wedge)\simeq\operatorname{hofib}\left((\mathbb{R}P^3)_2^\wedge\xrightarrow{[4]\circ c)_2^\wedge}(S^3)_2^\wedge\right). \tag*{(77)} Ω (( B A 4 ) 2 ∧ ) ≃ hofib ( ( R P 3 ) 2 ∧ [ 4 ] ∘ c ) 2 ∧ ( S 3 ) 2 ∧ ) . (77) For the localized fiber, let ρ : S 3 → R P 3 \rho:S^3\to\mathbb{R}P^3 ρ : S 3 → R P 3 f = [ 4 ] ( 2 ) ∘ c ( 2 ) f=[4]_{(2)}\circ c_{(2)} f = [ 4 ] ( 2 ) ∘ c ( 2 ) R P 3 ≅ S O ( 3 ) \mathbb{R}P^3\cong SO(3) R P 3 ≅ S O ( 3 ) ρ ( 2 ) \rho_{(2)} ρ ( 2 ) R P ( 2 ) 3 \mathbb{R}P^3_{(2)} R P ( 2 ) 3 f ∘ ρ ( 2 ) f\circ\rho_{(2)} f ∘ ρ ( 2 ) 8 8 8 π 1 K t e t ≅ C 2 \pi_1K_{\mathrm{tet}}\cong C_2 π 1 K tet ≅ C 2 π 2 K t e t ≅ Z ( 2 ) / 8 Z ( 2 ) ≅ C 8 \pi_2K_{\mathrm{tet}}\cong\mathbb{Z}_{(2)}/8\mathbb{Z}_{(2)}\cong C_8 π 2 K tet ≅ Z ( 2 ) /8 Z ( 2 ) ≅ C 8 K t e t → R P ( 2 ) 3 K_{\mathrm{tet}}\to\mathbb{R}P^3_{(2)} K tet → R P ( 2 ) 3 ρ ( 2 ) \rho_{(2)} ρ ( 2 ) K ~ t e t \widetilde K_{\mathrm{tet}} K tet 8 8 8 (74) .
Since f f f R P ( 2 ) 3 \mathbb{R}P^3_{(2)} R P ( 2 ) 3 K t e t K_{\mathrm{tet}} K tet [21 Proposition 4.4.1(i)] . The universal-cover description, injectivity of multiplication by 8 8 8 Z ( 2 ) \mathbb{Z}_{(2)} Z ( 2 ) π n ( S 3 ) \pi_n(S^3) π n ( S 3 ) n > 3 n>3 n > 3 K t e t K_{\mathrm{tet}} K tet 2 2 2 K t e t K_{\mathrm{tet}} K tet 2 2 2 [1 Theorem 2.6(1)] . Since S ( 2 ) 3 S^3_{(2)} S ( 2 ) 3 2 2 2 ( K t e t ) 2 ∧ (K_{\mathrm{tet}})_2^\wedge ( K tet ) 2 ∧ f 2 ∧ f_2^\wedge f 2 ∧ 2 2 2 R P 3 \mathbb{R}P^3 R P 3 S 3 S^3 S 3 2 2 2 2 2 2 K t e t K_{\mathrm{tet}} K tet K t e t K_{\mathrm{tet}} K tet (77) . Finally, ∣ H ∣ = 3 |H|=3 ∣ H ∣ = 3 L ( 2 ) ≃ ( B A 4 ) 2 ∧ L_{(2)}\simeq(BA_4)_2^\wedge L ( 2 ) ≃ ( B A 4 ) 2 ∧ (72) , and Theorem 3.2 gives (75) .
Remark 7.2. The universal-cover computation agrees with [7 Proposition 2.2] . This geometric comparison also retains the map on R P 3 \mathbb{R}P^3 R P 3 Ω L ( 2 ) \Omega L_{(2)} Ω L ( 2 ) π 2 ( R X ) ≅ Z ⊕ C 2 ⊕ C 8 \pi_2(R_X)\cong\mathbb{Z}\oplus C_2\oplus C_8 π 2 ( R X ) ≅ Z ⊕ C 2 ⊕ C 8 [26 Proposition 7.10, equation (18)] .
Fiber homology and integral quandle homology
We compute the homology of the fiber with coefficients in Z ( 2 ) \mathbb{Z}_{(2)} Z ( 2 )
Lemma 7.3. Put R = Z ( 2 ) R = \mathbb{Z}_{(2)} R = Z ( 2 ) K t e t K_{\mathrm{tet}} K tet (71) is
H 0 ( K t e t ; R ) = R , H 2 j ( K t e t ; R ) ≅ C 4 ( j ≥ 1 ) , H 2 j + 1 ( K t e t ; R ) ≅ C 2 ( j ≥ 0 ) . (78) H_0(K_{\mathrm{tet}};R)=R,\qquad H_{2j}(K_{\mathrm{tet}};R)\cong C_4\quad(j\geq1),\qquad H_{2j+1}(K_{\mathrm{tet}};R)\cong C_2\quad(j\geq0). \tag*{(78)} H 0 ( K tet ; R ) = R , H 2 j ( K tet ; R ) ≅ C 4 ( j ≥ 1 ) , H 2 j + 1 ( K tet ; R ) ≅ C 2 ( j ≥ 0 ) . (78) Proof. The defining map f = [ 4 ] ( 2 ) ∘ c ( 2 ) f=[4]_{(2)}\circ c_{(2)} f = [ 4 ] ( 2 ) ∘ c ( 2 )
Ω S ( 2 ) 3 ⟶ K t e t ⟶ R P ( 2 ) 3 . (79) \Omega S^3_{(2)}\longrightarrow K_{\mathrm{tet}}\longrightarrow\mathbb{R}P^3_{(2)}. \tag*{(79)} Ω S ( 2 ) 3 ⟶ K tet ⟶ R P ( 2 ) 3 . (79) The local coefficient action factors through π 1 ( S ( 2 ) 3 ) = 0 \pi_1(S^3_{(2)})=0 π 1 ( S ( 2 ) 3 ) = 0
E a , b 2 = H a ( R P ( 2 ) 3 ; R ) ⊗ R H b ( Ω S ( 2 ) 3 ; R ) . E^2_{a,b}=H_a(\mathbb{R}P^3_{(2)};R)\otimes_R H_b(\Omega S^3_{(2)};R). E a , b 2 = H a ( R P ( 2 ) 3 ; R ) ⊗ R H b ( Ω S ( 2 ) 3 ; R ) . The nonzero columns are a = 0 , 1 , 3 a=0,1,3 a = 0 , 1 , 3 R R R C 2 C_2 C 2 R R R d 2 = 0 d_2=0 d 2 = 0
d 3 : E 3 , 2 j 3 ⟶ E 0 , 2 j + 2 3 ( j ≥ 0 ) . d_3:E^3_{3,2j}\longrightarrow E^3_{0,2j+2}\qquad(j\geq0). d 3 : E 3 , 2 j 3 ⟶ E 0 , 2 j + 2 3 ( j ≥ 0 ) . In the path-space fibration over S ( 2 ) 3 S^3_{(2)} S ( 2 ) 3 d 3 d_3 d 3 4 4 4 H 3 H_3 H 3 4 4 4 C 4 C_4 C 4 C 2 C_2 C 2
Proof of Theorem B. Put R = Z ( 2 ) R=\mathbb{Z}_{(2)} R = Z ( 2 ) Theorem 7.1 and Lemma 7.3 give the following input to Corollary 5.8 :
K ≥ 1 ( t ) = t 1 − t , K ≥ 2 ( t ) = t 2 1 − t 2 , K ≥ e ( t ) = 0 ( e ≥ 3 ) . (80) K_{\geq1}(t)=\frac{t}{1-t},\qquad K_{\geq2}(t)=\frac{t^2}{1-t^2},\qquad K_{\geq e}(t)=0\quad(e\geq3). \tag*{(80)} K ≥ 1 ( t ) = 1 − t t , K ≥ 2 ( t ) = 1 − t 2 t 2 , K ≥ e ( t ) = 0 ( e ≥ 3 ) . (80) Since ∣ H ∣ = 3 |H|=3 ∣ H ∣ = 3 R R R (46) gives
Q ≥ 1 ( t ) = t 2 ( 1 + t ) 1 − t − t 2 − t 3 , Q ≥ 2 ( t ) = t 3 1 − t − t 3 , Q ≥ e ( t ) = 0 ( e ≥ 3 ) . Q_{\geq1}(t)=\frac{t^2(1+t)}{1-t-t^2-t^3},\qquad Q_{\geq2}(t)=\frac{t^3}{1-t-t^3},\qquad Q_{\geq e}(t)=0\quad(e\geq3). Q ≥ 1 ( t ) = 1 − t − t 2 − t 3 t 2 ( 1 + t ) , Q ≥ 2 ( t ) = 1 − t − t 3 t 3 , Q ≥ e ( t ) = 0 ( e ≥ 3 ) . Thus the only torsion summands are C 2 C_2 C 2 C 4 C_4 C 4 a n a_n a n b n b_n b n n ≥ 2 n\geq2 n ≥ 2 t n t^n t n Q ≥ 1 − Q ≥ 2 Q_{\geq1}-Q_{\geq2} Q ≥ 1 − Q ≥ 2 Q ≥ 2 Q_{\geq2} Q ≥ 2 X X X
The universal coefficient theorem also gives H 1 Q ( X ; Z ) ≅ Z H^Q_1(X;\mathbb{Z})\cong\mathbb{Z} H 1 Q ( X ; Z ) ≅ Z H 2 Q ( X ; Z ) = 0 H^Q_2(X;\mathbb{Z})=0 H 2 Q ( X ; Z ) = 0 H n Q ( X ; Z ) ≅ C 2 a n − 1 ⊕ C 4 b n − 1 H^Q_n(X;\mathbb{Z})\cong C_2^{a_{n-1}}\oplus C_4^{b_{n-1}} H n Q ( X ; Z ) ≅ C 2 a n − 1 ⊕ C 4 b n − 1 n ≥ 3 n\geq3 n ≥ 3
The groups through degree 7 agree with the computations of Niebrzydowski–Przytycki [23 Example 13(ii)] , who denote the tetrahedral quandle by S 4 S_4 S 4
Theorem B proves their conjectured recurrence b n = b n − 1 + b n − 3 b_n=b_{n-1}+b_{n-3} b n = b n − 1 + b n − 3 n ≥ 4 n\geq4 n ≥ 4 C 4 C_4 C 4 H 8 Q ( X ; Z ) ≅ C 2 33 ⊕ C 4 4 H^Q_8(X;\mathbb{Z})\cong C_2^{33}\oplus C_4^4 H 8 Q ( X ; Z ) ≅ C 2 33 ⊕ C 4 4 C 2 32 ⊕ C 4 4 C_2^{32}\oplus C_4^4 C 2 32 ⊕ C 4 4 C 2 C_2 C 2 2 2 2
A product Alexander quandle
The following example illustrates the tensor and Tor terms in the integral calculation when the Sylow subgroup is not cyclic.
Example 8.1. Let X = ( F 7 2 , T ) X=(\mathbb{F}_7^2,T) X = ( F 7 2 , T ) T = diag ( − 1 , 2 ) T=\operatorname{diag}(-1,2) T = diag ( − 1 , 2 ) x ∗ y = T x + ( 1 − T ) y x*y=Tx+(1-T)y x ∗ y = T x + ( 1 − T ) y X X X
H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ C 7 b n ( n ≥ 2 ) , H^Q_1(X;\mathbb{Z})\cong\mathbb{Z},\qquad H^Q_n(X;\mathbb{Z})\cong C_7^{b_n}\quad(n\geq2), H 1 Q ( X ; Z ) ≅ Z , H n Q ( X ; Z ) ≅ C 7 b n ( n ≥ 2 ) , where, on putting
U ( t ) = t 2 1 − t 2 + t 4 1 − t 4 + ( 1 + t ) 6 ( 1 − t 2 ) ( 1 − t 4 ) , U(t)=\frac{t^2}{1-t^2}+\frac{t^4}{1-t^4}+\frac{(1+t)^6}{(1-t^2)(1-t^4)}, U ( t ) = 1 − t 2 t 2 + 1 − t 4 t 4 + ( 1 − t 2 ) ( 1 − t 4 ) ( 1 + t ) 6 , the multiplicities are determined by
∑ n ≥ 2 b n t n = t ( 1 + t ) U ( t ) 1 − t ( 1 + t ) U ( t ) = t 3 + t 4 + 2 t 5 + 3 t 6 + 4 t 7 + 8 t 8 + O ( t 9 ) . \sum_{n\geq2}b_nt^n=\frac{t(1+t)U(t)}{1-t(1+t)U(t)}=t^3+t^4+2t^5+3t^6+4t^7+8t^8+O(t^9). n ≥ 2 ∑ b n t n = 1 − t ( 1 + t ) U ( t ) t ( 1 + t ) U ( t ) = t 3 + t 4 + 2 t 5 + 3 t 6 + 4 t 7 + 8 t 8 + O ( t 9 ) . In particular, H n Q ( X ; Z ) H^Q_n(X;\mathbb{Z}) H n Q ( X ; Z ) n ≥ 2 n\geq2 n ≥ 2
Proof. Write X = X 1 × X 2 X=X_1\times X_2 X = X 1 × X 2 X 1 = ( F 7 , − 1 ) X_1=(\mathbb{F}_7,-1) X 1 = ( F 7 , − 1 ) X 2 = ( F 7 , 2 ) X_2=(\mathbb{F}_7,2) X 2 = ( F 7 , 2 ) ⟨ T ⟩ = ⟨ − 1 ⟩ × ⟨ 2 ⟩ \langle T\rangle=\langle-1\rangle\times\langle2\rangle ⟨ T ⟩ = ⟨ − 1 ⟩ × ⟨ 2 ⟩ (49) therefore gives
G = Inn ( X ) ≅ ( C 7 ⋊ C 2 ) × ( C 7 ⋊ C 3 ) , H = Stab G ( 0 ) ≅ C 6 . G=\operatorname{Inn}(X)\cong(C_7\rtimes C_2)\times(C_7\rtimes C_3),\qquad H=\operatorname{Stab}_G(0)\cong C_6. G = Inn ( X ) ≅ ( C 7 ⋊ C 2 ) × ( C 7 ⋊ C 3 ) , H = Stab G ( 0 ) ≅ C 6 . Put G i = Inn ( X i ) G_i=\operatorname{Inn}(X_i) G i = Inn ( X i ) H i = Stab G i ( 0 ) H_i=\operatorname{Stab}_{G_i}(0) H i = Stab G i ( 0 ) L i = Cofib ( B H i → B G i ) L_i=\operatorname{Cofib}(BH_i\to BG_i) L i = Cofib ( B H i → B G i ) L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G ) Proposition 3.3 , the product map B G 1 × B G 2 → ( L 1 ) ( 7 ) × ( L 2 ) ( 7 ) BG_1\times BG_2\to(L_1)_{(7)}\times(L_2)_{(7)} B G 1 × B G 2 → ( L 1 ) ( 7 ) × ( L 2 ) ( 7 ) 7 7 7 ( G , H ) (G,H) ( G , H ) L ( 7 ) ≃ ( L 1 ) ( 7 ) × ( L 2 ) ( 7 ) L_{(7)}\simeq(L_1)_{(7)}\times(L_2)_{(7)} L ( 7 ) ≃ ( L 1 ) ( 7 ) × ( L 2 ) ( 7 ) Theorem 6.2 yields the ordinary homotopy equivalence
K : = Ω L ( 7 ) ≃ S 3 { 7 } ( 7 ) × S 5 { 7 } ( 7 ) . K:=\Omega L_{(7)}\simeq S^3\{7\}_{(7)}\times S^5\{7\}_{(7)}. K := Ω L ( 7 ) ≃ S 3 { 7 } ( 7 ) × S 5 { 7 } ( 7 ) . Put R = Z ( 7 ) R=\mathbb{Z}_{(7)} R = Z ( 7 ) K K K C 7 C_7 C 7 U ( t ) U(t) U ( t ) Tor 1 R ( C 7 , C 7 ) ≅ C 7 \operatorname{Tor}^{R}_1(C_7,C_7)\cong C_7 Tor 1 R ( C 7 , C 7 ) ≅ C 7 K ≥ 1 = U K_{\geq1}=U K ≥ 1 = U K ≥ e = 0 K_{\geq e}=0 K ≥ e = 0 e ≥ 2 e\geq2 e ≥ 2 Corollary 5.8 . That corollary and the recovery of integral homology give the asserted integral groups. □
For any field k k k
A X ( t ) = ∑ n ≥ 0 dim k H n ( Ω L ; k ) t n = 1 + ( 1 + t ) U ( t ) = ( 1 + t 3 ) ( 1 + t 5 ) ( 1 − t 2 ) ( 1 − t 4 ) . A_X(t)=\sum_{n\geq0}\dim_k H_n(\Omega L;k)t^n=1+(1+t)U(t)=\frac{(1+t^3)(1+t^5)}{(1-t^2)(1-t^4)}. A X ( t ) = n ≥ 0 ∑ dim k H n ( Ω L ; k ) t n = 1 + ( 1 + t ) U ( t ) = ( 1 − t 2 ) ( 1 − t 4 ) ( 1 + t 3 ) ( 1 + t 5 ) . Substituting κ = A X \kappa=A_X κ = A X (43) with Λ = k \Lambda=k Λ = k k k k
Two non-Alexander families
The same local calculation applies to two families of non-Alexander quandles. For an integer e ≥ 2 e \ge2 e ≥ 2 b n ( e ) b_n(e) b n ( e ) Theorem 6.8 and define a n ( e ) a_n(e) a n ( e )
∑ n ≥ 1 a n ( e ) t n = t 2 e − 1 ( 1 − t ) ( 1 − t − t 2 e − 2 − t 2 e ) , ∑ n ≥ 2 b n ( e ) t n = t 2 e − 1 ( 1 + t ) 1 − t 2 e − 2 − t 2 e − 1 − t 2 e . (81) \begin{aligned}
\sum_{n\ge1} a_n(e)t^n &= \frac{t^{2e-1}}{(1-t)(1-t-t^{2e-2}-t^{2e})},\\
\sum_{n\ge2} b_n(e)t^n &= \frac{t^{2e-1}(1+t)}{1-t^{2e-2}-t^{2e-1}-t^{2e}}. \tag*{(81)}
\end{aligned} n ≥ 1 ∑ a n ( e ) t n n ≥ 2 ∑ b n ( e ) t n = ( 1 − t ) ( 1 − t − t 2 e − 2 − t 2 e ) t 2 e − 1 , = 1 − t 2 e − 2 − t 2 e − 1 − t 2 e t 2 e − 1 ( 1 + t ) . (81) Let X X X G = Inn ( X ) G=\operatorname{Inn}(X) G = Inn ( X ) H = Stab G ( x 0 ) H=\operatorname{Stab}_G(x_0) H = Stab G ( x 0 ) L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G ) ℓ \ell ℓ ℓ ∤ ∣ H ∣ \ell\nmid|H| ℓ ∤ ∣ H ∣
Ω L ( ℓ ) ≃ S 2 e − 1 { ℓ r } ( ℓ ) , r ≥ 1. \Omega L_{(\ell)} \simeq S^{2e-1}\{\ell^r\}_{(\ell)}, \qquad r\ge1. Ω L ( ℓ ) ≃ S 2 e − 1 { ℓ r } ( ℓ ) , r ≥ 1. The positive-degree integral homology of this fiber is C ℓ r C_{\ell^r} C ℓ r 2 e − 2 2e-2 2 e − 2 [7 Section 1.1] . Substitution into Lemma 5.7 and Corollary 5.8 therefore gives
H n R ( X ; Z ( ℓ ) ) ≅ Z ( ℓ ) ⊕ ( C ℓ r ) a n ( e ) ( n ≥ 1 ) , H 1 Q ( X ; Z ( ℓ ) ) ≅ Z ( ℓ ) , H n Q ( X ; Z ( ℓ ) ) ≅ ( C ℓ r ) b n ( e ) ( n ≥ 2 ) . (82) \begin{aligned}
H_n^R(X;\mathbb{Z}_{(\ell)}) &\cong\mathbb{Z}_{(\ell)}\oplus(C_{\ell^r})^{a_n(e)} && (n\ge1),\\
H_1^Q(X;\mathbb{Z}_{(\ell)}) &\cong\mathbb{Z}_{(\ell)},\\
H_n^Q(X;\mathbb{Z}_{(\ell)}) &\cong(C_{\ell^r})^{b_n(e)} && (n\ge2). \tag*{(82)}
\end{aligned} H n R ( X ; Z ( ℓ ) ) H 1 Q ( X ; Z ( ℓ ) ) H n Q ( X ; Z ( ℓ ) ) ≅ Z ( ℓ ) ⊕ ( C ℓ r ) a n ( e ) ≅ Z ( ℓ ) , ≅ ( C ℓ r ) b n ( e ) ( n ≥ 1 ) , ( n ≥ 2 ) . (82) To verify that the following examples are not Alexander, we use a common criterion. In a finite connected Alexander quandle, 1 − T 1-T 1 − T y ↦ x ∗ y y\mapsto x*y y ↦ x ∗ y
Example 9.1 (The conjugation quandle of transpositions). Let ℓ ≥ 5 \ell\ge5 ℓ ≥ 5 m ∈ { ℓ , ℓ + 1 } m\in\{\ell,\ell+1\} m ∈ { ℓ , ℓ + 1 } T m T_m T m S m S_m S m x ∗ y = y − 1 x y x*y=y^{-1}xy x ∗ y = y − 1 x y T m T_m T m
H n R ( T m ; Z ( ℓ ) ) ≅ Z ( ℓ ) ⊕ ( C ℓ ) a n ( ℓ − 1 ) ( n ≥ 1 ) , H 1 Q ( T m ; Z ( ℓ ) ) ≅ Z ( ℓ ) , H n Q ( T m ; Z ( ℓ ) ) ≅ ( C ℓ ) b n ( ℓ − 1 ) ( n ≥ 2 ) . (83) \begin{aligned}
H_n^R(T_m;\mathbb{Z}_{(\ell)}) &\cong\mathbb{Z}_{(\ell)}\oplus(C_\ell)^{a_n(\ell-1)} && (n\ge1),\\
H_1^Q(T_m;\mathbb{Z}_{(\ell)}) &\cong\mathbb{Z}_{(\ell)},\\
H_n^Q(T_m;\mathbb{Z}_{(\ell)}) &\cong(C_\ell)^{b_n(\ell-1)} && (n\ge2). \tag*{(83)}
\end{aligned} H n R ( T m ; Z ( ℓ ) ) H 1 Q ( T m ; Z ( ℓ ) ) H n Q ( T m ; Z ( ℓ ) ) ≅ Z ( ℓ ) ⊕ ( C ℓ ) a n ( ℓ − 1 ) ≅ Z ( ℓ ) , ≅ ( C ℓ ) b n ( ℓ − 1 ) ( n ≥ 1 ) , ( n ≥ 2 ) . (83) In particular,
H n Q ( T m ; Z ( ℓ ) ) = 0 ( 2 ≤ n ≤ 2 ℓ − 4 ) , H_n^Q(T_m;\mathbb{Z}_{(\ell)})=0 \qquad(2\le n\le2\ell-4), H n Q ( T m ; Z ( ℓ ) ) = 0 ( 2 ≤ n ≤ 2 ℓ − 4 ) , H 2 ℓ − 3 Q ( T m ; Z ( ℓ ) ) ≅ H 2 ℓ − 2 Q ( T m ; Z ( ℓ ) ) ≅ C ℓ . H_{2\ell-3}^Q(T_m;\mathbb{Z}_{(\ell)})\cong H_{2\ell-2}^Q(T_m;\mathbb{Z}_{(\ell)})\cong C_\ell. H 2 ℓ − 3 Q ( T m ; Z ( ℓ ) ) ≅ H 2 ℓ − 2 Q ( T m ; Z ( ℓ ) ) ≅ C ℓ . The torsion subgroups of rack homology in these last two degrees are C ℓ C_\ell C ℓ ( C ℓ ) 2 (C_\ell)^2 ( C ℓ ) 2
Proof. Transpositions generate S m S_m S m T m T_m T m
G = Inn ( T m ) ≅ S m , H = C S m ( ( 12 ) ) ≅ C 2 × S m − 2 . G=\operatorname{Inn}(T_m)\cong S_m,\qquad H=C_{S_m}((12))\cong C_2\times S_{m-2}. G = Inn ( T m ) ≅ S m , H = C S m (( 12 )) ≅ C 2 × S m − 2 . Since m = ℓ m=\ell m = ℓ ℓ + 1 \ell+1 ℓ + 1 ℓ ∤ ∣ H ∣ \ell\nmid|H| ℓ ∤ ∣ H ∣ ℓ \ell ℓ P ≅ C ℓ P\cong C_\ell P ≅ C ℓ F ℓ \mathbb{F}_\ell F ℓ P P P x ↦ a x x\mapsto ax x ↦ a x a ∈ F ℓ × a\in\mathbb{F}_\ell^\times a ∈ F ℓ × P P P ∣ N G ( P ) : C G ( P ) ∣ = ℓ − 1 |N_G(P):C_G(P)|=\ell-1 ∣ N G ( P ) : C G ( P ) ∣ = ℓ − 1 Theorem 6.8 and (82) , with e = ℓ − 1 e=\ell-1 e = ℓ − 1 r = 1 r=1 r = 1 (83) . The first nonzero groups follow by expanding (81) . Finally, ( 12 ) ∗ ( 12 ) = ( 12 ) ∗ ( 34 ) = ( 12 ) (12)*(12)=(12)*(34)=(12) ( 12 ) ∗ ( 12 ) = ( 12 ) ∗ ( 34 ) = ( 12 ) ( 12 ) (12) ( 12 ) T m T_m T m
Example 9.2 (Rank-one symplectic quandles). Let q = p f > 27 q=p^f>27 q = p f > 27 p p p f ≥ 1 f\ge1 f ≥ 1
X q = F q 2 ∖ { 0 } , v ∗ w = v + det ( v , w ) w , X_q=\mathbb{F}_q^2\setminus\{0\},\qquad v*w=v+\det(v,w)w, X q = F q 2 ∖ { 0 } , v ∗ w = v + det ( v , w ) w , is connected and is not Alexander. Put R = Z [ 1 / p ] R=\mathbb{Z}[1/p] R = Z [ 1/ p ] N = q 2 − 1 N=q^2-1 N = q 2 − 1
H n R ( X q ; R ) ≅ R ⊕ ( C N ) a n ( 2 ) ( n ≥ 1 ) , H 1 Q ( X q ; R ) ≅ R , H n Q ( X q ; R ) ≅ ( C N ) b n ( 2 ) ( n ≥ 2 ) . (84) \begin{aligned}
H_n^R(X_q;R)&\cong R\oplus(C_N)^{a_n(2)} &&(n\ge1),\\
H_1^Q(X_q;R)&\cong R,\\
H_n^Q(X_q;R)&\cong(C_N)^{b_n(2)} &&(n\ge2). \tag*{(84)}
\end{aligned} H n R ( X q ; R ) H 1 Q ( X q ; R ) H n Q ( X q ; R ) ≅ R ⊕ ( C N ) a n ( 2 ) ≅ R , ≅ ( C N ) b n ( 2 ) ( n ≥ 1 ) , ( n ≥ 2 ) . (84) Here
∑ n ≥ 1 a n ( 2 ) t n = t 3 ( 1 − t ) ( 1 − t − t 2 − t 4 ) , ∑ n ≥ 2 b n ( 2 ) t n = t 3 1 − t − t 3 . \sum_{n\ge1}a_n(2)t^n=\frac{t^3}{(1-t)(1-t-t^2-t^4)},\qquad
\sum_{n\ge2}b_n(2)t^n=\frac{t^3}{1-t-t^3}. n ≥ 1 ∑ a n ( 2 ) t n = ( 1 − t ) ( 1 − t − t 2 − t 4 ) t 3 , n ≥ 2 ∑ b n ( 2 ) t n = 1 − t − t 3 t 3 . For example,
H 2 Q ( X q ; R ) = 0 , H 3 Q ( X q ; R ) ≅ H 4 Q ( X q ; R ) ≅ H 5 Q ( X q ; R ) ≅ C N , H_2^Q(X_q;R)=0,\qquad H_3^Q(X_q;R)\cong H_4^Q(X_q;R)\cong H_5^Q(X_q;R)\cong C_N, H 2 Q ( X q ; R ) = 0 , H 3 Q ( X q ; R ) ≅ H 4 Q ( X q ; R ) ≅ H 5 Q ( X q ; R ) ≅ C N , H 6 Q ( X q ; R ) ≅ ( C N ) 2 . H_6^Q(X_q;R)\cong(C_N)^2. H 6 Q ( X q ; R ) ≅ ( C N ) 2 . Thus (84) determines all primary components away from p p p
Proof. The linear extension of a right translation is a transvection. Replacing its direction vector w w w c w cw c w c 2 c^2 c 2 SL 2 ( F q ) \operatorname{SL}_2(\mathbb{F}_q) SL 2 ( F q )
G = Inn ( X q ) ≅ SL 2 ( F q ) , H = Stab G ( ( 1 , 0 ) ) ≅ ( F q , + ) ≅ ( C p ) f . G=\operatorname{Inn}(X_q)\cong\operatorname{SL}_2(\mathbb{F}_q),\qquad H=\operatorname{Stab}_G((1,0))\cong(\mathbb{F}_q,+)\cong(C_p)^f. G = Inn ( X q ) ≅ SL 2 ( F q ) , H = Stab G (( 1 , 0 )) ≅ ( F q , + ) ≅ ( C p ) f . In particular, X q X_q X q ℓ ≠ p \ell\ne p ℓ = p
For each such prime, we claim that there is an ordinary homotopy equivalence
Ω L ( ℓ ) ≃ ( S 3 { N } ) ( ℓ ) . (85) \Omega L_{(\ell)}\simeq(S^3\{N\})_{(\ell)}. \tag*{(85)} Ω L ( ℓ ) ≃ ( S 3 { N } ) ( ℓ ) . (85) If ℓ \ell ℓ N N N ℓ \ell ℓ P P P G G G q − 1 q-1 q − 1 q + 1 q+1 q + 1 P P P ℓ v ℓ ( N ) \ell^{v_\ell(N)} ℓ v ℓ ( N ) v ℓ ( N ) v_\ell(N) v ℓ ( N ) ℓ \ell ℓ N N N P P P C 2 C_2 C 2 P P P N G ( P ) N_G(P) N G ( P ) C G ( P ) C_G(P) C G ( P ) N G ( P ) / C G ( P ) ≅ C 2 N_G(P)/C_G(P)\cong C_2 N G ( P ) / C G ( P ) ≅ C 2 (85) . If q q q ℓ = 2 \ell=2 ℓ = 2 r = v 2 ( N ) ≥ 3 r=v_2(N)\ge3 r = v 2 ( N ) ≥ 3 Ω ( ( B SL 2 ( F q ) ) 2 ∧ ) \Omega((B\operatorname{SL}_2(\mathbb{F}_q))^\wedge_2) Ω (( B SL 2 ( F q ) ) 2 ∧ ) S 3 { 2 r } S^3\{2^r\} S 3 { 2 r } [7 Section 1.1 and Lemma 2.1] ; Proposition 3.3 then gives (85) at 2. If ℓ ∤ p N \ell\nmid pN ℓ ∤ pN (85) are contractible, since ∣ G ∣ = q N |G|=qN ∣ G ∣ = q N
For ℓ ∣ N \ell\mid N ℓ ∣ N (82) with e = 2 e=2 e = 2 r = v ℓ ( N ) r=v_\ell(N) r = v ℓ ( N ) ℓ ≠ p \ell\ne p ℓ = p (84) . Finally, if c ∈ F q × ∖ { 1 } c\in\mathbb{F}_q^\times\setminus\{1\} c ∈ F q × ∖ { 1 } v ∗ v = v ∗ ( c v ) = v v*v=v*(cv)=v v ∗ v = v ∗ ( c v ) = v X q X_q X q
Multiplicative structures and unstable homotopy
Using the local model of § 4 and the chain model of § 6 , we construct direct summands by mixed Samelson products and determine whether the specified multiplication admits an E 2 E_2 E 2 X X X G = Inn ( X ) G=\operatorname{Inn}(X) G = Inn ( X ) H = Stab G ( x 0 ) H=\operatorname{Stab}_G(x_0) H = Stab G ( x 0 ) L = Cofib ( B H → B G ) L=\operatorname{Cofib}(BH\to BG) L = Cofib ( B H → B G ) M = M X \mathcal{M}=\mathcal{M}_X M = M X p p p ∣ X ∣ |X| ∣ X ∣ p p p
Definition 10.1. We say that a specified grouplike A ∞ A_\infty A ∞ Z Z Z E 2 E_2 E 2 Z Z Z A ∞ A_\infty A ∞ A ∞ A_\infty A ∞ E 2 E_2 E 2 A ∞ A_\infty A ∞ M = B ( G ; X ) \mathcal{M}=B(G;X) M = B ( G ; X ) M ~ \widetilde{\mathcal{M}} M
We compute the homotopy nilpotency of a grouplike A ∞ A_\infty A ∞ Z Z Z [ x , y ] = x y x − 1 y − 1 [x,y]=xyx^{-1}y^{-1} [ x , y ] = x y x − 1 y − 1 r ≥ 2 r\geq2 r ≥ 2 r r r Z r → Z Z^r\to Z Z r → Z
The quotient from a finite product of positive-dimensional spheres to their smash product admits a homotopy right inverse after suspension. Since Z ≃ Ω B Z Z\simeq\Omega BZ Z ≃ Ω B Z Z Z Z
Spherical classes and iterated Samelson products
For a coefficient ring R R R W W W q ≥ 1 q\geq1 q ≥ 1 h R : π q ( W ) → H q ( W ; R ) h_R:\pi_q(W)\to H_q(W;R) h R : π q ( W ) → H q ( W ; R ) R R R [ a , b ] = a b − ( − 1 ) ∣ a ∣ ∣ b ∣ b a [a,b]=ab-(-1)^{|a||b|}ba [ a , b ] = ab − ( − 1 ) ∣ a ∣∣ b ∣ ba i , j ≥ 1 i,j\geq1 i , j ≥ 1 S i → M S^i\to\mathcal{M} S i → M S j → M S^j\to\mathcal{M} S j → M S i ∧ S j S^i\wedge S^j S i ∧ S j ad a ( b ) = [ a , b ] \operatorname{ad}_a(b)=[a,b] ad a ( b ) = [ a , b ] A ∞ A_\infty A ∞
Lemma 10.2. The group π j ( M ) \pi_j(\mathcal{M}) π j ( M ) j ≥ 1 j\geq1 j ≥ 1 j ≥ 3 j\geq3 j ≥ 3 π j ( M ( p ) ) \pi_j(\mathcal{M}_{(p)}) π j ( M ( p ) ) p p p j ≥ 3 j\geq3 j ≥ 3
Proof. By Theorem 2.1 , Y Y Y [21 Theorem 4.5.4] therefore shows that π j ( M ) ≅ π j + 1 ( Y ) \pi_j(\mathcal{M})\cong\pi_{j+1}(Y) π j ( M ) ≅ π j + 1 ( Y ) j ≥ 1 j\geq1 j ≥ 1 π j ( M ) ⊗ Q = 0 \pi_j(\mathcal{M})\otimes\mathbb{Q}=0 π j ( M ) ⊗ Q = 0 j ≥ 3 j\geq3 j ≥ 3 p p p
Proposition 10.3 (Integral splittings by mixed Samelson products). Assume p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ L ( p ) ≄ ∗ L_{(p)}\not\simeq* L ( p ) ≃ ∗
K = Ω L ( p ) , m = min { j ≥ 1 : π j ( K ) ≠ 0 } , A = π m ( K ) . K=\Omega L_{(p)},\qquad m=\min\{j\geq1:\pi_j(K)\ne0\},\qquad A=\pi_m(K). K = Ω L ( p ) , m = min { j ≥ 1 : π j ( K ) = 0 } , A = π m ( K ) . Then A A A p p p Theorem 3.2 , let α ∈ π 1 ( M ( p ) ) \alpha\in\pi_1(\mathcal{M}_{(p)}) α ∈ π 1 ( M ( p ) ) 1 ∈ π 1 ( Ω S ( p ) 2 ) ≅ Z ( p ) 1\in\pi_1(\Omega S^2_{(p)})\cong\mathbb{Z}_{(p)} 1 ∈ π 1 ( Ω S ( p ) 2 ) ≅ Z ( p ) i : K → M ( p ) i:K\to\mathcal{M}_{(p)} i : K → M ( p ) n ≥ 0 n\geq0 n ≥ 0
ψ n : A ⟶ π m + n ( M ( p ) ) , ψ n ( β ) = ad α n ( i ∗ β ) \psi_n:A\longrightarrow\pi_{m+n}(\mathcal{M}_{(p)}),\qquad\psi_n(\beta)=\operatorname{ad}_{\alpha}^{n}(i_*\beta) ψ n : A ⟶ π m + n ( M ( p ) ) , ψ n ( β ) = ad α n ( i ∗ β ) is a split injection. Consequently, these iterated Samelson products realize A A A π j ( M ( p ) ) \pi_j(\mathcal{M}_{(p)}) π j ( M ( p ) ) j ≥ m j\geq m j ≥ m M ( p ) \mathcal{M}_{(p)} M ( p ) π j ( R X ) ⊗ Z ( p ) ≠ 0 \pi_j(R_X)\otimes\mathbb{Z}_{(p)}\ne0 π j ( R X ) ⊗ Z ( p ) = 0 j ≥ max { m , 2 } j\geq\max\{m,2\} j ≥ max { m , 2 }
Proof. Put R = Z ( p ) R = \mathbb{Z}_{(p)} R = Z ( p ) Lemma 3.1 and the Hurewicz isomorphism in the first nonzero degree show that A A A p p p A ≅ H m ( K ; R ) A \cong H_m(K;R) A ≅ H m ( K ; R ) m = 1 m=1 m = 1 T R ( u ) ∗ D T_R(u)*D T R ( u ) ∗ D (44) , where ∣ u ∣ = 1 |u|=1 ∣ u ∣ = 1 d u = 0 du=0 d u = 0 D D D R R R K K K D 0 = R D_0=R D 0 = R D + = ker ( D → R ) D^+=\ker(D\to R) D + = ker ( D → R ) u u u h R ( α ) h_R(\alpha) h R ( α ) D = R ⊕ D + D=R\oplus D^+ D = R ⊕ D + (45) shows that the words u n b u^n b u n b b ∈ D + b\in D^+ b ∈ D + d ( u n b ) = ( − 1 ) n u n d D b d(u^n b)=(-1)^n u^n d_D b d ( u n b ) = ( − 1 ) n u n d D b u n u^n u n
pr n : H m + n ( M ( p ) ; R ) ⟶ H m ( K ; R ) ≅ A . (86) \operatorname{pr}_n:H_{m+n}(\mathcal{M}_{(p)};R)\longrightarrow H_m(K;R)\cong A. \tag*{(86)} pr n : H m + n ( M ( p ) ; R ) ⟶ H m ( K ; R ) ≅ A . (86) For b β = h R ( β ) b_\beta=h_R(\beta) b β = h R ( β ) u n D + u^nD^+ u n D + h R ψ n ( β ) = ad u n ( b β ) h_R\psi_n(\beta)=\operatorname{ad}_u^n(b_\beta) h R ψ n ( β ) = ad u n ( b β ) u n b β u^n b_\beta u n b β u u u
( pr n ∘ h R ) ψ n = id A . (\operatorname{pr}_n\circ h_R)\psi_n=\operatorname{id}_A. ( pr n ∘ h R ) ψ n = id A . Samelson products are additive in these positive degrees, including m = 1 m=1 m = 1 M → R X \mathcal{M}\to\mathcal{R}_X M → R X
E2 refinements of the specified multiplication and universal covers
The specified products on M \mathcal{M} M M Q \mathcal{M}_{\mathbb{Q}} M Q M ( p ) \mathcal{M}_{(p)} M ( p ) p p p x x x π 1 ( M ) \pi_1(\mathcal{M}) π 1 ( M ) (31) shows that [ x , x ] [x,x] [ x , x ] A ∞ A_\infty A ∞ E 2 E_2 E 2 E 2 E_2 E 2
Theorem 10.4 (Criterion for an E 2 E_2 E 2 Assume p ∤ ∣ H ∣ p\nmid|H| p ∤ ∣ H ∣ ( M ~ ) ( p ) (\widetilde{\mathcal{M}})_{(p)} ( M ) ( p ) E 2 E_2 E 2 p ∤ ∣ X ∣ p\nmid|X| p ∤ ∣ X ∣ If L ( p ) ≃ ∗ L_{(p)}\simeq* L ( p ) ≃ ∗ A ∞ A_\infty A ∞
( M ~ ) ( p ) ≃ A ∞ Ω ( S ( p ) 3 ) ≃ A ∞ Ω 2 ( ( B S 3 ) ( p ) ) . (87) (\widetilde{\mathcal{M}})_{(p)}\simeq_{A_\infty}\Omega(S^3_{(p)})\simeq_{A_\infty}\Omega^2((BS^3)_{(p)}). \tag*{(87)} ( M ) ( p ) ≃ A ∞ Ω ( S ( p ) 3 ) ≃ A ∞ Ω 2 (( B S 3 ) ( p ) ) . (87) Here S 3 S^3 S 3 B S 3 BS^3 B S 3 L ( p ) ≄ ∗ L_{(p)}\not\simeq* L ( p ) ≃ ∗ ( M ~ ) ( p ) (\widetilde{\mathcal{M}})_{(p)} ( M ) ( p ) m , A m,A m , A A A A π m + 2 n + 1 ( ( M ~ ) ( p ) ) \pi_{m+2n+1}((\widetilde{\mathcal{M}})_{(p)}) π m + 2 n + 1 (( M ) ( p ) )
Proof. By Corollary 4.2, ( M ~ ) ( p ) (\widetilde{\mathcal{M}})_{(p)} ( M ) ( p ) A ∞ A_\infty A ∞ M ( p ) \mathcal{M}_{(p)} M ( p ) 2 2 2 Y Y Y
If L ( p ) ≃ ∗ L_{(p)}\simeq* L ( p ) ≃ ∗ M ( p ) ≃ A ∞ Ω ( S ( p ) 2 ) \mathcal{M}_{(p)}\simeq_{A_\infty}\Omega(S^2_{(p)}) M ( p ) ≃ A ∞ Ω ( S ( p ) 2 ) S 3 → S 2 → K ( Z , 2 ) S^3\to S^2\to K(\mathbb{Z},2) S 3 → S 2 → K ( Z , 2 ) 2 2 2 S ( p ) 2 S^2_{(p)} S ( p ) 2 S 3 ≃ A ∞ Ω B S 3 S^3\simeq_{A_\infty}\Omega BS^3 S 3 ≃ A ∞ Ω B S 3 (87) and hence an E 2 E_2 E 2
Suppose L ( p ) ≄ ∗ L_{(p)}\not\simeq* L ( p ) ≃ ∗ α , i , R , u , pr j \alpha,i,R,u,\operatorname{pr}_j α , i , R , u , pr j γ ∈ π 2 ( M ( p ) ) \gamma\in\pi_2(\mathcal{M}_{(p)}) γ ∈ π 2 ( M ( p ) ) β ∈ A \beta\in A β ∈ A β ′ = [ α , i ∗ β ] \beta'=[\alpha,i_*\beta] β ′ = [ α , i ∗ β ]
h R ( γ ) = u 2 . (88) h_R(\gamma)=u^2. \tag*{(88)} h R ( γ ) = u 2 . (88) Indeed, let α 0 ∈ π 1 ( Ω S 2 ) \alpha_0\in\pi_1(\Omega S^2) α 0 ∈ π 1 ( Ω S 2 ) u 0 = h Z ( α 0 ) u_0=h_{\mathbb{Z}}(\alpha_0) u 0 = h Z ( α 0 ) (31) , together with h Z [ α 0 , α 0 ] = 2 u 0 2 h_{\mathbb{Z}}[\alpha_0,\alpha_0]=2u_0^2 h Z [ α 0 , α 0 ] = 2 u 0 2 H 2 ( Ω S 2 ; Z ) ≅ Z H_2(\Omega S^2;\mathbb{Z})\cong\mathbb{Z} H 2 ( Ω S 2 ; Z ) ≅ Z p = 2 p=2 p = 2 γ \gamma γ β ′ \beta' β ′ 2 2 2 m + 1 m+1 m + 1 γ ~ \widetilde{\gamma} γ β ~ ′ \widetilde{\beta}' β ′ m = 1 m=1 m = 1 b β = h R ( β ) b_{\beta}=h_R(\beta) b β = h R ( β ) (86) to h R ( ad γ ~ n ( β ~ ′ ) ) = ad u 2 n ( [ u , b β ] ) h_R(\operatorname{ad}_{\widetilde{\gamma}}^n(\widetilde{\beta}'))=\operatorname{ad}_{u^2}^n([u,b_{\beta}]) h R ( ad γ n ( β ′ )) = ad u 2 n ([ u , b β ]) u u u u 2 n + 1 b β u^{2n+1}b_{\beta} u 2 n + 1 b β
pr 2 n + 1 h R ( ad γ ~ n ( β ~ ′ ) ) = β ( n ≥ 0 ) . (89) \operatorname{pr}_{2n+1}h_R(\operatorname{ad}_{\widetilde{\gamma}}^n(\widetilde{\beta}'))=\beta\qquad(n\geq0). \tag*{(89)} pr 2 n + 1 h R ( ad γ n ( β ′ )) = β ( n ≥ 0 ) . (89) For each n ≥ 0 n\geq0 n ≥ 0 β ↦ ad γ ~ n ( β ~ ′ ) \beta\mapsto\operatorname{ad}_{\widetilde{\gamma}}^n(\widetilde{\beta}') β ↦ ad γ n ( β ′ ) pr 2 n + 1 \operatorname{pr}_{2n+1} pr 2 n + 1 E 2 E_2 E 2
Exponential growth of torsion homotopy groups
The first nonzero group also gives an exponential lower bound for cyclic direct-summand multiplicities in each sufficiently large homotopy degree. The tensor products below are over Z \mathbb{Z} Z p p p Z ( p ) \mathbb{Z}_{(p)} Z ( p ) Z / p r \mathbb{Z}/p^r Z / p r [6 Definition 2] .
Theorem 10.5 (Exponential growth in each homotopy degree). Assume the hypotheses of Proposition 10.3 and retain the notation K , m , A K,m,A K , m , A i , j ≥ 1 i,j\geq1 i , j ≥ 1 c i , j c_{i,j} c i , j i i i j j j p p p π i + ( m + 1 ) j ( M ( p ) ) \pi_{i+(m+1)j}(\mathcal{M}_{(p)}) π i + ( m + 1 ) j ( M ( p ) )
( A ⊗ j ) c i , j . (90) \left(A^{\otimes j}\right)^{c_{i,j}}. \tag*{(90)} ( A ⊗ j ) c i , j . (90) In particular, writing ℓ n = c n , n \ell_n=c_{n,n} ℓ n = c n , n π ( m + 2 ) n ( M ( p ) ) \pi_{(m+2)n}(\mathcal{M}_{(p)}) π ( m + 2 ) n ( M ( p ) ) ( A ⊗ n ) ℓ n \left(A^{\otimes n}\right)^{\ell_n} ( A ⊗ n ) ℓ n C p ℓ n h n C_{p}^{\ell_n h^n} C p ℓ n h n A ≅ C p h A\cong C_p^h A ≅ C p h h ≥ 1 h\geq1 h ≥ 1 c i , j c_{i,j} c i , j ℓ n \ell_n ℓ n
Suppose that A A A C p r C_{p^r} C p r r ≥ 1 r\geq1 r ≥ 1 a = ν r ( A ) a=\nu_r(A) a = ν r ( A ) p r p^r p r s N ( r ) s_N(r) s N ( r ) C p r C_{p^r} C p r π N ( R X ) ⊗ Z ( p ) \pi_N(RX)\otimes\mathbb{Z}_{(p)} π N ( R X ) ⊗ Z ( p ) N ≥ 3 N\geq3 N ≥ 3 s N ( r ) > 0 s_N(r)>0 s N ( r ) > 0 N N N
lim inf N → ∞ log s N ( r ) N ≥ log ( 4 a ) m + 2 > 0. (91) \liminf_{N\to\infty}\frac{\log s_N(r)}{N}\geq\frac{\log(4a)}{m+2}>0. \tag*{(91)} N → ∞ lim inf N log s N ( r ) ≥ m + 2 log ( 4 a ) > 0. (91) The same bound holds for the cumulative multiplicity T N ( r ) T_N(r) T N ( r ) C p r C_{p^r} C p r π j ( R X ) ⊗ Z ( p ) \pi_j(RX)\otimes\mathbb{Z}_{(p)} π j ( R X ) ⊗ Z ( p ) 2 ≤ j ≤ N 2\leq j\leq N 2 ≤ j ≤ N R X RX R X Z / p r \mathbb{Z}/p^r Z / p r lim inf N → ∞ log T N ( r ) / N > 0 \liminf_{N\to\infty}\log T_N(r)/N>0 lim inf N → ∞ log T N ( r ) / N > 0
Proof. Put W = S 2 ∨ L ( p ) W=S^2\vee L_{(p)} W = S 2 ∨ L ( p ) E E E W → L ( p ) W\to L_{(p)} W → L ( p )
E ≃ S 2 ∧ K + ≃ S 2 ∨ Σ 2 K = Σ ( S 1 ∨ Σ K ) , (92) E\simeq S^2\wedge K_+\simeq S^2\vee\Sigma^2K=\Sigma(S^1\vee\Sigma K), \tag*{(92)} E ≃ S 2 ∧ K + ≃ S 2 ∨ Σ 2 K = Σ ( S 1 ∨ Σ K ) , (92) where K + K_+ K + K K K L ( p ) → W L_{(p)}\to W L ( p ) → W M ( p ) ≃ Ω ( W ( p ) ) \mathcal{M}_{(p)}\simeq\Omega(W_{(p)}) M ( p ) ≃ Ω ( W ( p ) ) π q ( Ω E ) ⊗ Z ( p ) \pi_q(\Omega E)\otimes\mathbb{Z}_{(p)} π q ( Ω E ) ⊗ Z ( p ) π q ( M ( p ) ) \pi_q(\mathcal{M}_{(p)}) π q ( M ( p ) ) q ≥ 1 q\geq1 q ≥ 1
The Hilton–Milnor decomposition for the connected spaces S 1 S^1 S 1 Σ K \Sigma K Σ K [18 Theorem 5.8] gives one factor
Ω Σ ( ( S 1 ) ∧ i ∧ ( Σ K ) ∧ j ) (93) \Omega\Sigma\left((S^1)^{\wedge i}\wedge(\Sigma K)^{\wedge j}\right) \tag*{(93)} ΩΣ ( ( S 1 ) ∧ i ∧ ( Σ K ) ∧ j ) (93) for each Hall word of multidegree ( i , j ) (i,j) ( i , j ) K K K ( m − 1 ) (m-1) ( m − 1 ) H m ( K ; Z ) = A H_m(K;\mathbb{Z})=A H m ( K ; Z ) = A m = 1 m=1 m = 1 π 1 ( K ) \pi_1(K) π 1 ( K ) d = i + ( m + 1 ) j d=i+(m+1)j d = i + ( m + 1 ) j (93) is ( d − 1 ) (d-1) ( d − 1 ) H d = A ⊗ j H_d=A^{\otimes j} H d = A ⊗ j (93) as A ⊗ j A^{\otimes j} A ⊗ j d d d p p p A ⊗ j A^{\otimes j} A ⊗ j (90) . The ambient groups are finite by Lemma 10.2, since d ≥ 3 d \ge3 d ≥ 3
To obtain the growth estimate, fix a cyclic direct-sum decomposition of A A A C p r C_{p^r} C p r A ⊗ n A^{\otimes n} A ⊗ n a a a p r p^r p r C p r ⊗ C p s ≅ C p r C_{p^r} \otimes C_{p^s} \cong C_{p^r} C p r ⊗ C p s ≅ C p r s ≥ r s \ge r s ≥ r ( C p r ) a n − 1 (C_{p^r})^{a^{n-1}} ( C p r ) a n − 1 N N N N = ( m + 2 ) n + δ N = (m + 2)n + \delta N = ( m + 2 ) n + δ 0 ≤ δ ≤ m + 1 0 \le\delta\le m + 1 0 ≤ δ ≤ m + 1 ( i , j ) = ( n + δ , n ) (i,j) = (n + \delta,n) ( i , j ) = ( n + δ , n ) (90) and using the covering M → R X \mathcal{M} \to\mathcal{R}_X M → R X
s N ( r ) ≥ c n + δ , n a n − 1 . s_N(r) \ge c_{n+\delta,n}a^{n-1}. s N ( r ) ≥ c n + δ , n a n − 1 . Lemma A.1 gives log c n + δ , n / n → log 4 \log c_{n+\delta,n}/n \to\log4 log c n + δ , n / n → log 4 δ \delta δ N / n → m + 2 N/n \to m + 2 N / n → m + 2 (91) in all sufficiently large degrees. Finally, T N ( r ) ≥ s N ( r ) T_N(r) \ge s_N(r) T N ( r ) ≥ s N ( r ) □ \square □
Corollary 10.6 (The cyclic family and cyclic Sylow subgroups). Under the hypotheses of Theorem 6.8, let p r p^r p r p p p e e e j ≥ 2 e − 2 j \ge2e - 2 j ≥ 2 e − 2 π j ( M ( p ) ) ≅ π j ( R X ) ⊗ Z ( p ) \pi_j(\mathcal{M}_{(p)}) \cong\pi_j(\mathcal{R}_X) \otimes\mathbb{Z}_{(p)} π j ( M ( p ) ) ≅ π j ( R X ) ⊗ Z ( p ) C p r C_{p^r} C p r
Moreover, π 2 e n ( M ( p ) ) \pi_{2en}(\mathcal{M}_{(p)}) π 2 e n ( M ( p ) ) ( C p r ) ℓ n (C_{p^r})^{\ell_n} ( C p r ) ℓ n n ≥ 1 n \ge1 n ≥ 1
lim inf N → ∞ log s N ( r ) N ≥ log 4 2 e . (94) \liminf_{N\to\infty}\frac{\log s_N(r)}{N} \ge\frac{\log4}{2e}. \tag*{(94)} N → ∞ lim inf N log s N ( r ) ≥ 2 e log 4 . (94) The same bound holds for T N ( r ) T_N(r) T N ( r ) R X \mathcal{R}_X R X Z / p r \mathbb{Z}/p^r Z / p r X p r , ω X_{p^r,\omega} X p r , ω § 7 with e = d e=d e = d
Proof. The homotopy exact sequence of (65) gives m = 2 e − 2 ≥ 2 m = 2e - 2 \ge2 m = 2 e − 2 ≥ 2 A = C p r A = C_{p^r} A = C p r a = 1 a = 1 a = 1 M → R X \mathcal{M} \to\mathcal{R}_X M → R X □ \square □
For the tetrahedral quandle, Theorem 7.1 gives m = 1 m = 1 m = 1 A = C 2 A = C_2 A = C 2 C 2 C_2 C 2 π j ( M ( 2 ) ) \pi_j(\mathcal{M}_{(2)}) π j ( M ( 2 ) ) j ≥ 1 j \ge1 j ≥ 1 π j ( R X ) ⊗ Z ( 2 ) \pi_j(\mathcal{R}_X) \otimes\mathbb{Z}_{(2)} π j ( R X ) ⊗ Z ( 2 ) j ≥ 2 j \ge2 j ≥ 2
lim inf N → ∞ log s N ( 1 ) N ≥ log 4 3 . \liminf_{N\to\infty}\frac{\log s_N(1)}{N} \ge\frac{\log4}{3}. N → ∞ lim inf N log s N ( 1 ) ≥ 3 log 4 . The splittings used in the growth argument concern the underlying spaces and their homotopy groups; they do not assert product decompositions preserving the specified multiplication.
Appendix A. Counting Hall words and asymptotic estimates
Lemma A.1. Take a Hall basis of the ordinary, ungraded free Lie algebra on two letters, and let c i , j c_{i,j} c i , j i i i j j j i , j ≥ 1 i,j \ge1 i , j ≥ 1
c i , j = 1 i + j ∑ d ∣ gcd ( i , j ) μ ( d ) ( ( i + j ) / d j / d ) . (95) c_{i,j} = \frac{1}{i+j}\sum_{d\mid\gcd(i,j)}\mu(d)\binom{(i+j)/d}{j/d}. \tag*{(95)} c i , j = i + j 1 d ∣ g c d ( i , j ) ∑ μ ( d ) ( j / d ( i + j ) / d ) . (95) Here μ \mu μ ℓ n = c n , n \ell_n = c_{n,n} ℓ n = c n , n
ℓ n = 1 2 n ∑ e ∣ n μ ( e ) ( 2 n / e n / e ) ( n ≥ 1 ) . (96) \ell_n = \frac{1}{2n}\sum_{e\mid n}\mu(e)\binom{2n/e}{n/e}\qquad(n\ge1). \tag*{(96)} ℓ n = 2 n 1 e ∣ n ∑ μ ( e ) ( n / e 2 n / e ) ( n ≥ 1 ) . (96) For every fixed integer δ ≥ 0 \delta\ge0 δ ≥ 0
c n + δ , n ∼ 2 δ 4 n 2 π n 3 / 2 ( n → ∞ ) . (97) c_{n+\delta,n} \sim\frac{2^\delta4^n}{2\sqrt{\pi}\,n^{3/2}} \qquad(n \to\infty). \tag*{(97)} c n + δ , n ∼ 2 π n 3/2 2 δ 4 n ( n → ∞ ) . (97) Consequently, log c n + δ , n / n → log 4 \log c_{n+\delta,n}/n \to\log4 log c n + δ , n / n → log 4
Proof. Including c 1 , 0 = c 0 , 1 = 1 c_{1,0}=c_{0,1}=1 c 1 , 0 = c 0 , 1 = 1
( 1 − x − y ) − 1 = ∏ i + j > 0 ( 1 − x i y j ) − c i , j . (1-x-y)^{-1}=\prod_{i+j>0}(1-x^i y^j)^{-c_{i,j}}. ( 1 − x − y ) − 1 = i + j > 0 ∏ ( 1 − x i y j ) − c i , j . Taking formal logarithms and comparing coefficients gives
1 i + j ( i + j j ) = ∑ d ∣ gcd ( i , j ) c i / d , j / d d . \frac{1}{i+j}\binom{i+j}{j}=\sum_{d\mid\gcd(i,j)}\frac{c_{i/d,j/d}}{d}. i + j 1 ( j i + j ) = d ∣ g c d ( i , j ) ∑ d c i / d , j / d . Multiplication by i + j i+j i + j i = n + δ i=n+\delta i = n + δ j = n j=n j = n k = 2 n + δ k=2n+\delta k = 2 n + δ d = 1 d=1 d = 1
k c n + δ , n = ( 2 n + δ n ) + R n , ∣ R n ∣ ≤ k 2 k / 2 . k c_{n+\delta,n}=\binom{2n+\delta}{n}+R_n,\qquad|R_n|\le k2^{k/2}. k c n + δ , n = ( n 2 n + δ ) + R n , ∣ R n ∣ ≤ k 2 k /2 . Indeed, each remaining divisor d d d d ≥ 2 d\ge2 d ≥ 2 2 k / d ≤ 2 k / 2 2^{k/d}\le2^{k/2} 2 k / d ≤ 2 k /2 k k k δ \delta δ
( 2 n + δ n ) ∼ 2 δ 4 n π n . \binom{2n+\delta}{n}\sim\frac{2^\delta4^n}{\sqrt{\pi n}}. ( n 2 n + δ ) ∼ π n 2 δ 4 n . The error is exponentially smaller than this term; division by k ∼ 2 n k\sim2n k ∼ 2 n
Acknowledgements
The author thanks Katsumi Ishikawa for helpful correspondence concerning his spectral-sequence approach to the homology of connected dihedral quandles. The author used ChatGPT to assist with editing and reviewing the manuscript and takes responsibility for its final content.
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