Simplicial (Co)homology

Abstract

Simplicial homology was invented by Poincaré in 1899 [162] and its $\mathrm{mod}\, 2$ version, presented in this chapter, was introduced in 1908 by Tietze [196].

Appendices

2.9 Appendix A: An Acyclic Carrier Result

The powerful technique of acyclic carriers was introduced by Eilenberg and MacLane in 1953 [50], after earlier work by Lefschetz. Proposition 2.9.1 below is a very particular example of this technique, adapted to our needs. For a full development of acyclic carriers, see, e.g., [155, Chap. 1,Sect. 13].

Let $(C_*,\partial )$ and $(\bar{C}_*,\bar{\partial })$ be two chain complexes and let $\varphi$: $C_*\rightarrow \bar{C}_*$ be a morphism of chain complexes. We suppose that $C_m$ is equipped with a basis ${\mathcal S}_m$ for each $m$ and denote by ${\mathcal S}$ the union of all ${\mathcal S}_m$. An acyclic carrier $A_*$ for $\varphi$ with respect to the basis ${\mathcal S}$ is a correspondence which associates to each $s\in {\mathcal S}$ a subchain complex $A_*(s)$ of $\bar{C}_*$ such that

• (a) $\varphi (s)\in A_*(s)$.

• (b) $H_0(A_*(s))={\mathbb Z}_2$ and $H_m(A_*(s))=0$ for $m>0$.

• (c) let $s\in {\mathcal S}_m$ and $t\in {\mathcal S}_{m-1}$ such that $t$ occurs in the expression of $\partial \, s$ in the basis ${\mathcal S}_{m-1}$. Then $A_*(t)$ is a subchain complex of $A_*(s)$ and the inclusion $A_*(t)\subset A_*(s)$ induces an isomorphism on $H_0$.

• (d) if $s\in {\mathcal S}_0\subset C_0=Z_0$, then $H_0\varphi (s)\ne 0$ in $H_0(A_*(s))$.

Proposition 2.9.1

Let $\varphi$ and $\varphi '$ be two morphisms of chain complexes from $(C_*,\partial )$ to $(\bar{C}_*,\bar{\partial })$. Suppose that $\varphi$ and $\varphi '$ admit the same acyclic carrier $A_*$ with respect to some basis ${\mathcal S}$ of $C_*$. Then $H_*\varphi =H_*\varphi '$.

Proof

The proof is similar to that of Proposition 2.5.9. By induction on $m$, we shall prove the following property:

Property ${\mathcal H}(m)$: there exists a linear map $D$: $C_m\rightarrow \bar{C}_{m+1}$ such that:

• (i) $\bar{\partial }D(\alpha ) + D(\partial \alpha ) = \varphi (\alpha )+\varphi '(\alpha )$ for all $\alpha \in C_m$.

• (ii) for each $s\in {\mathcal S}_m$, $D(s)\in A_{m+1}(s)$.

Property ${\mathcal H}(m)$ for all $m$ implies that $H_*\varphi =H_*\varphi '$. Indeed, we then have a linear map $D$: $C_*\rightarrow \bar{C}_{*+1}$ satisfying

$$\begin{aligned} \varphi +\varphi '=\bar{\partial }{\scriptstyle \circ } D + D{\scriptstyle \circ } \partial \, . \end{aligned}$$

(2.9.1)

Let $\beta \in Z_*$. By Eq. (2.9.1), one has $\varphi (\beta )+\varphi (\beta )=\bar{\partial }D(\beta )$ which implies that $H_*\varphi ([\beta ])+H_*\varphi '([\beta ])$ in $\bar{H}_*$.

Let us prove ${\mathcal H}(0)$. Let $s\in {\mathcal S}_0$. In $H_0(A_*(s))={\mathbb Z}_2$, one has $H_0\varphi (s)\ne 0$ and $H_0\varphi '(s)\ne 0$. Therefore $H_*\varphi (s)=H_*\varphi '(s)$ in $H_0(A_*(s))$. This implies that $\varphi (s)+\varphi '(s)=\bar{\partial }(\eta _s)$ for some $\eta _s\in A_1(s)$. We set $D(s)=\eta _s$. This procedure, for each $s\in {\mathcal S}_0$, provides a linear map $D$: $C_0\rightarrow \bar{C}_1$, which, as $\partial C_0=0$, satisfies $\varphi (s)+\varphi '(s)=\bar{\partial }D(\alpha )+D(\partial (\alpha ))$.

We now prove that ${\mathcal H}(m-1)$ implies ${\mathcal H}(m)$ for $m\ge 1$. Let $s\in {\mathcal S}_m$. The chain $D(\partial s)$ exists in $A_m(s)$ by ${\mathcal H}(m-1)$. Let $\zeta \in A_{m}(s)$ defined by

$$ \zeta = \varphi (s) + \varphi '(s) + D(\partial s) $$

Using ${\mathcal H}(m-1)$, one checks that $\partial \zeta =0$. Since $H_m(A_*(s))=0$, there exists $\nu \in A_{m+1}(s)$ such that $\zeta =\partial \nu$. Choose such an element $\nu$ and set $D(\sigma )=\nu$. This defines $D$: $C_m\rightarrow \bar{C}_{m+1}$ which satisfies (i) and (ii), proving ${\mathcal H}(m)$. $\Box$

2.10 Appendix B: Ordered Simplicial (Co)homology

This technical section may be skipped in a first reading. It shows that simplicial (co)homology may be defined using larger sets of (co)chains, based on ordered simplexes. This will be used for comparisons between simplicial and singular (co)homology (see 17) and to define the cup and cap products in Chap. 4.

Let $K$ be a simplicial complex. Define

$$ \hat{\mathcal S}_m(K) = \{(v_0,\dots ,v_m)\in V(K)^{m+1} \mid \{v_0,\dots ,v_m\}\in {\mathcal S}(K) \} \, . $$

Observe that $\dim \{v_0,\dots ,v_m\}\le m$ and may be strictly smaller if there are repetitions amongst the $v_i$’s. An element of $\hat{\mathcal S}_m(K)$ is an ordered $m$ -simplex of $K$.

The definitions of ordered (co)chains and (co)homology are the same those for the simplicial case (see Sect. 2.2), replacing the simplexes by the ordered simplexes. We thus set

Definition 2.10.1

(subset definitions)

(a) An ordered $m$ -cochain is a subset of $\hat{\mathcal S}_m(K)$.

(b) An ordered $m$ -chain is a finite subset of $\hat{\mathcal S}_m(K)$.

The set of ordered $m$-cochains of $K$ is denoted by $\hat{C}^m(K)$ and that of ordered $m$-chains by $\hat{C}_m(K)$. As in Sect. 2.2, Definition 2.10.1 are equivalent to

Definition 2.10.2

(colouring definitions)

(a) An ordered $m$ -cochain is a function $a$: $\hat{\mathcal S}_m(K)\rightarrow {\mathbb Z}_2$.

(b) An ordered $m$ -chain is a function $\alpha$: $\hat{\mathcal S}_m(K)\rightarrow {\mathbb Z}_2$ with finite support.

Definition 2.10.2 endow $\hat{C}^m(K)$ and $\hat{C}_m(K)$ with a structure of a ${\mathbb Z}_2$-vector space. The singletons provide a basis of $\hat{C}_m(K)$, in bijection with $\hat{\mathcal S}_m(K)$. Thus, Definition  2.10.2.b is equivalent to

Definition 2.10.3

$\hat{C}_m(K)$ is the ${\mathbb Z}_2$-vector space with basis $\hat{\mathcal S}_m(K)$:

$$ \hat{C}_m(X)=\bigoplus _{\sigma \in \hat{\mathcal S}_m(X)}{\mathbb Z}_2\,\sigma \, . $$

We consider the graded ${\mathbb Z}_2$-vector spaces $\hat{C}_*(K)=\oplus _{m\in {\mathbb N}}\hat{C}_m(K)$ and $\hat{C}^*(K)=\oplus _{m\in {\mathbb N}}\hat{C}^m(K)$. The Kronecker pairing on ordered (co)chains

$$ \hat{C}^m(K)\times \hat{C}_m(K) \xrightarrow {\langle \, ,\,\rangle } {\mathbb Z}_2 $$

is defined, using the various above definitions, by the equivalent formulae

$$\begin{aligned} \begin{array}{rcll} \langle a ,\alpha \rangle &{}=&{} \sharp (a\cap \alpha ) \ (\mathrm{mod\,} 2) &{} \small \mathrm {using\;Definition\;2.10.1a\;and\;b}\\ &{}=&{} \sum \nolimits _{\sigma \in \alpha } a(\sigma ) &{} \small \mathrm {using\;Definitions\;2.10.1a\;and\;2.10.2b}\\ &{}=&{} \sum \nolimits _{\sigma \in S_m(K)} a(\sigma ) \alpha (\sigma ) &{} \small \mathrm {using\;Definitions\;2.10.2a\;and\;b}. \end{array} \end{aligned}$$

(2.10.1)

As in Lemma 2.2.4, we check that the map $\mathbf{k}$: $\hat{C}^m(K)\rightarrow \hat{C}_m(K)^\sharp$, given by $\mathbf{k}(a)=\langle a ,\,\rangle$, is an isomorphism.

The boundary operator $\hat{\partial }$: $\hat{C}_m(K)\rightarrow \hat{C}_{m-1}(K)$ is the ${\mathbb Z}_2$-linear map defined, for $(v_0,\dots ,v_m)\in \hat{\mathcal S}_m(K)$ by

$$\begin{aligned} \hat{\partial }(v_0,\dots ,v_m) = \sum _{i=0}^m (v_0,\dots ,\hat{v}_i,\dots ,v_m) \, , \end{aligned}$$

(2.10.2)

where $(v_0,\dots ,\hat{v}_i,\dots ,v_m)\in \hat{\mathcal S}_{m-1}$ is the $m$-tuple obtained by removing $v_i$. The coboundary operator $\hat{\delta }:C^m(K)\rightarrow C^{m+1}(K)$ is defined by the equation

$$\begin{aligned} \langle \hat{\delta }a ,\alpha \rangle = \langle a ,\hat{\partial }\alpha \rangle \,. \end{aligned}$$

(2.10.3)

With these definition, $(\hat{C}_*(K),\hat{\partial },\hat{C}^*(K),\hat{\delta },\langle \, ,\rangle )$ is a Kronecker pair. We define the vector spaces of ordered cycles $\hat{Z}_*(K)$, ordered boundaries $\hat{B}_*(K)$, ordered cocycles $\hat{Z}^*(K)$, ordered coboundaries $\hat{B}^*(K)$, ordered homology $\hat{H}_*(K)$ and ordered cohomology $\hat{H}^*(K)$ as in Sect. 2.3. By Proposition 2.3.5, the pairing on (co)chain descends to a pairing

$$ H^m(K)\times H_m(K) \xrightarrow {\langle \, ,\,\rangle } {\mathbb Z}_2 $$

so that the map $\mathbf{k}$: $\hat{H}^m\rightarrow \hat{H}_m^\sharp$, given by $\mathbf{k}(a)=\langle a ,\,\rangle$, is an isomorphism (ordered Kronecker duality).

Example 2.10.4

Let $K=pt$ be a point. Then, $\hat{\mathcal S}_m(pt)$ contains one element for each integer $m$, namely the $(m+1)$-tuple $(pt,\dots ,pt)$. Then, $\hat{C}_m(pt)={\mathbb Z}_2$ for all $m\in {\mathbb N}$ and the chain complex looks like

$$ \cdots \xrightarrow {\approx }\hat{C}_{2k+1}(pt)\xrightarrow {0}\hat{C}_{2k}(pt)\xrightarrow {\approx } \hat{C}_{2k-1}(pt) \xrightarrow {0} \cdots \xrightarrow {\approx }\hat{C}_{1}(pt)\xrightarrow {0}\hat{C}_{0}(pt)\rightarrow 0 \, . $$

Therefore,

$$\begin{aligned} \hat{H}^*(pt)\approx \hat{H}_*(pt)\approx \left\{ \begin{array}{lll} 0 &{} \hbox {if *>0} \\ {\mathbb Z}_2 &{} \hbox {if *=0} \, . \end{array}\right. \end{aligned}$$

One sees that, for a simplicial complex reduced to a point, the ordered (co)homology and the simplicial (co)homology are isomorphic.

Example 2.10.5

The unit cochain ${\mathbf 1}\in \hat{C}^0(K)$ is defined as ${\mathbf 1}=\hat{\mathcal S}_0(K)$. It is a cocycle and defines a class ${\mathbf 1}=\hat{H}^0(K)$. If $K$ is non-empty and connected, then $\hat{H}^0(K)\approx {\mathbb Z}_2$ generated by ${\mathbf 1}$. Then $H_0(K)\approx {\mathbb Z}_2$ by Kronecker duality; one has $\hat{Z}_0(K)=\hat{C}_0(K)$ and $\alpha \in \hat{Z}_0(K)$ represents the non-zero element of $H_0(K)$ if and only if $\sharp \alpha$ is odd. The proofs are the same as for Proposition 2.4.1.

Example 2.10.6

Let $L$ be a simplicial complex and $CL$ be the cone on $L$. Then

$$ \hat{H}^*(CL)\approx \hat{H}_*(CL)\approx \left\{ \begin{array}{lll} 0 &{} \hbox {if *>0} \\ {\mathbb Z}_2 &{} \hbox {if *=0} \, . \end{array}\right. $$

The proof is the same as for Proposition 2.4.6, even simpler, since $D$: $\hat{C}_m(CL)\rightarrow \hat{C}_{m+1}(CL)$ is defined, for $(v_0,\dots ,v_m)\in \hat{\mathcal S}_m(CL)$ by the single line formula

$D(v_0,\dots ,v_m)=(\infty ,v_0,\dots ,v_m)$.

Let $f$: $L\rightarrow K$ be a simplicial map. We define $\hat{C}_*f$: $\hat{C}_*(L)\rightarrow \hat{C}_*(K)$ as the degree $0$ linear map such that

$$ \hat{C}_*f(v_0,\dots ,v_m) = (f(v_0),\dots ,f(v_m)) $$

for all $(v_0,\dots ,v_m)\in \hat{\mathcal S}(L)$. The degree $0$ linear map $\hat{C}^*f$: $\hat{C}^*(K)\rightarrow \hat{C}^*(L)$ is defined by

$$\begin{aligned} \langle \hat{C}^*f(a) ,\alpha \rangle =\langle a ,\hat{C}_*f(\alpha )\rangle \, . \end{aligned}$$

By Lemma 2.3.6, $(\hat{C}^*f,\hat{C}_*f)$ is a morphism of Kronecker pairs.

We now construct a functorial isomorphism between the ordered and non-ordered (co)homologies, its existence being suggested by the previous examples. Define $\psi _*$: $\hat{C}_*(K) \rightarrow C_*(K)$ by

$$ \psi _* ((v_0,\dots ,v_{m})) = {\left\{ \begin{array}{ll} \{v_0,\dots ,v_{m}\} &{} \text {if } v_i\ne v_j \ \hbox {for all} \ i\ne j\\ 0 &{} \text {otherwise.} \end{array}\right. } $$

We check that $\psi$ is a morphism of chain complexes. We define $\psi ^* : C^*(K) \rightarrow \hat{C}^*(K)$ by requiring that the equation $\langle {\psi }^*(a) ,\alpha \rangle =\langle a ,{\psi }_*(\alpha )\rangle$ holds for all $a\in C^*(K)$ and all $\alpha \in \hat{C}_*(K)$. By Lemma 2.3.6, $\psi ^*$ is a morphism of cochain complexes and $(\psi _*,\psi ^*)$ is a morphism of Kronecker pairs between $(\hat{C}_*(K),\hat{C}^*(K))$ and $(C_*(K),C^*(K))$. It thus defines a morphism of Kronecker pairs $(H_*{\psi },H^*{\psi })$ between $(\hat{H}_*(K),\hat{H}^*(K))$ and $(H_*(K),H^*(K))$.

To define a morphism of Kronecker pairs in the other direction, choose a simplicial order $\le$ on $K$ (see 2.1.8). Define ${\phi _\le }_*$: $C_*(K) \rightarrow \hat{C}_*(K)$ as the unique linear map such that

$$\begin{aligned} {\phi _\le }_* (\{v_0,\dots ,v_m\}) = (v_0,\dots ,v_m) \ , \end{aligned}$$

where $v_0\le v_1\le \cdots \le v_m$. We check that ${\phi _\le }_*$ is a morphism of chain complexes and define ${\phi _\le }^*$: $\hat{C}^*(K) \rightarrow C^*(K)$ by requiring that the equation $\langle {\phi _\le }^*(a) ,\alpha \rangle =\langle a ,{\phi _\le }_*(\alpha )\rangle$ holds for all $a\in \hat{C}^*(K)$ and all $\alpha \in C_*(K)$. By Lemma 2.3.6, $({\phi _\le }_*,{\phi _\le }^*)$ is a morphism of Kronecker pairs between $(C_*(K),C^*(K))$ and $(\hat{C}_*(K), \hat{C}^*(K))$. It then defines a morphism of Kronecker pairs $(H_*{\phi _\le },H^*{\phi _\le })$ between $(H_*(K),H^*(K))$ and $(\hat{H}_*(K),\hat{H}^*(K))$.

Proposition 2.10.7

$H_*{\psi } {\scriptstyle \circ } H_*{\phi _{\scriptscriptstyle \le }} = \mathrm{id}_{H_*(K)}$ and $H_*{\phi _{\scriptscriptstyle \le }} {\scriptstyle \circ } H_*{\psi } = \mathrm{id}_{\hat{H}_*(K)}$.

Proof

As $\psi _* {\scriptstyle \circ } {\phi _{\scriptscriptstyle \le }}_* = \mathrm{id}_{C_*(K)}$, the first equality follows from Lemma 2.3.7. For the second one, let $(v_0,\dots ,v_m)\in \hat{\mathcal S}_m(K)$. Let $\sigma =\{v_0,\dots ,v_m\}\in {\mathcal S}_k(K)$ with $k\le m$. Clearly, ${\phi _{\scriptscriptstyle \le }}_* {\scriptstyle \circ } \psi _*(v_0,\dots ,v_m)\in \hat{C}_*(\bar{\sigma })$. By what was seen in Examples 2.10.5 and 2.10.6, the correspondence $(v_0,\dots ,v_m)\mapsto \hat{C}_*(\overline{\{v_0,\dots ,v_m\}})$ is an acyclic carrier $A_*$, with respect to the basis $\hat{\mathcal S}_*(K)$, for both $\mathrm{id}_{\hat{C}(K)}$ and ${\phi _{\scriptscriptstyle \le }}_*{\scriptstyle \circ } {\psi }_*$. Therefore, the equality $H_*{\phi _{\scriptscriptstyle \le }} {\scriptstyle \circ } H_*{\psi } = \mathrm{id}_{\hat{H}_*(K)}$ follows by Lemma 2.3.7 and Proposition 2.9.1.

Applying Kronecker duality to Proposition 2.10.7 gives the following

Corollary 2.10.8

$H^*{\psi } {\scriptstyle \circ } H^*{\phi _{\scriptscriptstyle \le }} = \mathrm{id}_{\hat{H}^*(K)}$ and $H^*{\phi _{\scriptscriptstyle \le }} {\scriptstyle \circ } H ^*{\psi } = \mathrm{id}_{H^*(K)}$.

Corollary 2.10.9

$H_*{\psi }$ and $H^*{\psi }$ are isomorphisms.

Corollary 2.10.10

$H_*{\phi _{\scriptscriptstyle \le }}$ and $H^*{\phi _{\scriptscriptstyle \le }}$ are isomorphisms which do not depend on the simplicial order $\le$.

Proof

This follows from Proposition 2.10.7 and Corollary 2.10.8, since $H_*{\psi }$ and $H^*{\psi }$ do not depend on $\le$. $\Box$

We shall see in Sect. 4.1 that $H^*\psi$ and $H^*{\phi _{\scriptscriptstyle \le }}$ are isomorphisms of graded ${\mathbb Z}_2$-algebras. We now prove that they are also natural with respect to simplicial maps. Let $f$: $L\rightarrow K$ be a simplicial map. Let $\hat{C}_*f$: $\hat{C}_*(L)\rightarrow \hat{C}_*(K)$ be the unique linear map such that

$$ \hat{C}_*f((v_0,\dots ,v_m))=(f(v_0),\dots ,f(v_m)) $$

for each $(v_0,\dots ,v_m)\in \hat{\mathcal S}_m(K)$. Doing this for each $m\in {\mathbb N}$ produces a $\mathbf{GrV}$-morphism $\hat{C}_*f$: $\hat{C}_*(L)\rightarrow \hat{C}_*(K)$. The formula $\hat{\partial }{\scriptstyle \circ } \hat{C}_*f = \hat{C}_*f {\scriptstyle \circ } \hat{\partial }$ is straightforward (much easier than that for non-ordered chains). Hence, we get a $\mathbf{GrV}$-morphism $\hat{H}_*f$: $\hat{H}_*(L)\rightarrow \hat{H}_*(K)$. A $\mathbf{GrV}$-morphism $\hat{C}^*f$: $\hat{C}^*(K)\rightarrow \hat{C}^*(L)$ is defined by the equation $\langle \hat{C}^*f(a) ,\alpha \rangle =\langle a ,\hat{C}_*f(\alpha )\rangle$ required to hold for all $a\in \hat{C}^m(L)$, $\alpha \in \hat{C}_m(K)$ and all $m\in {\mathbb N}$. It is a cochain map and induces a $\mathbf{GrV}$-morphism $\hat{H}^*f$: $\hat{H}^*(K)\rightarrow \hat{H}^*(L)$, Kronecker dual to $H_*f$.

Proposition 2.10.11

Let $f$: $L\rightarrow K$ be a simplicial map. Let $\le$ be a simplicial order on $K$ and $\le '$ be a simplicial order on $L$. Then the diagrams

are commutative.

Proof

By Kronecker duality, only the homology statement requires a proof. It is enough to prove that $H_*f{\scriptstyle \circ } H_*\psi = H_*\psi {\scriptstyle \circ } \hat{H}_*f$ since the formula $\hat{H}_*f{\scriptstyle \circ } H_*\phi _{\le '} = H_*\phi _{\le } {\scriptstyle \circ } H_*f$ will follow by Corollary 2.10.8. Finally, the formula $\hat{C}_*f{\scriptstyle \circ } C_*\phi _{\le '} = C_*\phi _{\le } {\scriptstyle \circ } C_*f$ is straightforward. $\Box$

The above isomorphism results also work in relative ordered (co)homology. Let $(K,L)$ be a simplicial pair. Denote by $i$: $L\hookrightarrow K$ the simplicial inclusion. We define the ${\mathbb Z}_2$-vector space of relative ordered (co)chain by

$$ \hat{C}^m(K,L)=\ker \big (\hat{C}^m(K)\xrightarrow {\hat{C}^*i} \hat{C}^m(L)\big ) $$

and

$$\begin{aligned} \hat{C}_m(K,L)=\mathrm{coker\,}\big (i_*: \hat{C}_m(L)\hookrightarrow \hat{C}_m(K)\big ) \, . \end{aligned}$$

These inherit (co)boundaries $\hat{\delta }: \hat{C}^*(K,L)\rightarrow \hat{C}^*(K,L)$ and $\hat{\partial }=\hat{C}_*(K,L)\rightarrow \hat{C}_{*-1}(K,L)$ which give rise to the definition of relative ordered (co)homology $\hat{H}^*(k,L)$ and $\hat{H}_*(K,L)$. Connecting homomorphisms $\hat{\delta }_*$: $\hat{H}^*(L)\rightarrow \hat{H}^{*+1}(K,L)$ and $\hat{\partial }_*$: $\hat{H}_*(K,L)\rightarrow \hat{H}_{*-1}(L)$ are defined as in Sect. 2.7, giving rise to long exact sequences. Our homomorphisms $\psi _*$: $\hat{C}_*(K) \rightarrow C_*(K)$ and ${\phi _\le }_*$: $C_*(K) \rightarrow \hat{C}_*(K)$ satisfy $\psi _*(\hat{C}_*(L))\subset C_*(L)$ and ${\phi _\le }_*(C_*(L)\subset \hat{C}_*(L)$, giving rise to homomorphisms on relative (co)chains and relative (co)homology $H_*\psi$: $\hat{H}_*(K,L)\rightarrow H-*(K,L)$, etc. Proposition 2.10.7 and Corollary 2.10.8 and their proofs hold in relative (co)homology. Hence, as for Corollaries 2.10.9 and 2.10.10, we get

Corollary 2.10.12

$H_*{\psi }: \hat{H}_*(K,L)\rightarrow H_*(K,L)$ and $H^*{\psi }: H^*(K,L)\rightarrow \hat{H}^*(K,L)$ are isomorphisms.

Corollary 2.10.13

$H_*{\phi _{\scriptscriptstyle \le }}: H_*(K,L)\rightarrow \hat{H}_*(K,L)$ and $H^*{\phi _{\scriptscriptstyle \le }}: \hat{H}^*(K,L)\rightarrow \hat{H}^*(K,L)$ are isomorphisms which do not depend on the simplicial order $\le$.

2.11 Exercises for Chapter 2

• 2.1. Let ${\mathcal F}_n$ be the full complex on the set $\{0,1,\dots ,n\}$ (see p. 24). What are the $2$-simplexes of the barycentric subdivision ${\mathcal F}_2'$ of ${\mathcal F}_2$? How many $n$-simplexes does ${\mathcal F}_n'$ contain?

• 2.2. Compute the Euler characteristic and the Poincaré polynomial of the $k$-skeleton ${\mathcal F}_n^k$ of ${\mathcal F}_n$.

• 2.3. Let $X$ be a metric space and let $\varepsilon >0$. The Vietoris-Rips complex $X_\varepsilon$ of $X$ is the simplicial complex whose simplexes are the finite non-empty subset of $X$ whose diameter is $<\varepsilon$ (the diameter of $A\subset X$ is the least upper bound of $d(x,y)$ for $x,y\in A$). In particular, $V(X_\varepsilon )=X$.

  • (a) Describe $|X_\varepsilon |$ for various $\varepsilon$ when $X$ is the set of vertices of a cube of edge $1$ in ${\mathbb R}^3$. In particular, if $\sqrt{2}<\varepsilon \le \sqrt{3}$, show that $|X_\varepsilon |$ is homeomorphic to $S^3$.

  • (b) Let $X$ be the space $n$-th roots of unity, with the distance $d(x,y)$ being the minimal length of an arc of the unit circle joining $x$ to $y$. Suppose that $4\pi /n<\varepsilon \le 6\pi /n$.

    • (i) If n = 6, show that $|X_\varepsilon |$ is homeomorphic to $S^2$.

    • (ii) If $n\ge 7$ is odd, show that $|X_\varepsilon |$ is homeomorphic to a Möbius band.

    • (iii) If $n\ge 7$ is even, show that $|X_\varepsilon |$ is homeomorphic to $S^1\times [0,1]$.

  Note: the complex $X_\varepsilon$ was introduced by Vietoris in 1927 [201]. After its re-introduction by E. Rips for studying hyperbolic groups, it has been popularized under the name of Rips complex. For some developments and applications, see [84, 129] and Wikipedia’s page “Vietoris-Rips complex”.

• 2.4. Let $\ell =(\ell _1,\dots ,\ell _n)\in {\mathbb R}_{>0}^n$. A subset $J$ of $\{1,\dots ,n\}$ is called $\ell$ -short (or just short) if $\sum _{i\in J}\ell _i<\sum _{i\notin J}\ell _i$. Show that short subsets are the simplexes of a simplicial complex $\mathrm{Sh}(\ell )$ with $V(\mathrm{Sh}(\ell ))\subset J$ (used in Sect. 10.3). Describe $\mathrm{Sh}(1,1,1,1,3)$, $\mathrm{Sh}(1,1,3,3,3)$ and $\mathrm{Sh}(1,1,1,1,1)$. Compute their Euler characteristics and their Poincaré polynomials.

• 2.5. Let $K$ be the simplicial complex with $V(K)={\mathbb Z}$ and ${\mathcal S}_1(K)=\{\{r,r+1\}\mid r\in {\mathbb Z}\}$ ($|K|\approx {\mathbb R}$). Then ${\mathcal S}_1(K)$ is a $1$-cocycle. Find all the cochains $a\in C^0(K)$ such that ${\mathcal S}_1(K)=\delta (a)$.

• 2.6. Find a simplicial pair $(K,L)$ such that $|K|$ is homeomorphic to $S^1\times I$ and $|L|=\mathrm{Bd\,}|K|$. In the spirit of Sect. 2.4.7, compute the simplicial cohomology of $K$ and of $(K,L)$ and find (co)cycles generating $H_*(K)$, $H_*(K,L)$, $H^*(K)$ and $H^*(K,L)$. Write completely the (co)homology sequence of $(K,L)$.

• 2.7. Same exercise as before with $|K|$ the Möbius band and $|L|=\mathrm{Bd\,}|K|$.

• 2.8. Let $f$: $K\rightarrow L$ be a simplicial map between simplicial complexes. Suppose that $L$ is connected and $K$ is non-empty. Show that $H_0f$ is surjective.

• 2.9. Let $m,n,q$ be positive integers. If $m=nq$, the quotient map ${\mathbb Z}\rightarrow {\mathbb Z}/n{\mathbb Z}$ descends to a map ${\mathbb Z}/m{\mathbb Z}\rightarrow {\mathbb Z}/n{\mathbb Z}$, giving rise to a simplicial map $f$: ${\mathcal P}_m\rightarrow {\mathcal P}_n$ between the simplicial polygons ${\mathcal P}_m$ and ${\mathcal P}_n$ (see Example 2.4.3). Compute $H^*f$.

• 2.10. Let $M$ be an $n$-dimensional pseudomanifold. Let $\sigma$ and $\sigma '$ be two distinct $n$-simplexes of $M$. Find $a\in C^{n-1}(M)$ such that $\delta (a)=\{\sigma ,\sigma '\}$.

• 2.11. Let $M$ be a finite non-empty $n$-dimensional pseudomanifold. Let $\gamma \in Z_{n-1}(M)$ which is a boundary. Prove that $\gamma$ is the boundary of exactly two $n$ chains.

• 2.12. Let $f$: $M\rightarrow N$ be a simplicial map between finite $n$-dimensional pseudomanifolds. Show that the following two conditions are equivalent.

  • (a) $H_nf\ne 0$.

  • (b) There exists $\sigma \in {\mathcal S}(N)$ such that $\sharp f^{-1}(\{\sigma \})$ is odd.

• 2.13. Let $\{\pm 1\}$ be the $0$-dimensional simplicial complex with vertices $-1$ and $1$. Let $K$ be a simplicial complex. The simplicial suspension $\Sigma K$ is the join $K * \{\pm 1\}$.

  • (a) Let ${\mathcal P}_4$ be the polygon complex with $4$-edges (see Example 2.4.3). Show that ${\mathcal P}_4 * K$ is isomorphic to the double suspension $\Sigma (\Sigma K)$. [Hint: show that the join operation is associative: $(K*L)*M \approx K*(L*M)$.]

  • (b) Prove that the suspension of a pseudomanifold is a pseudomanifold.

  • (c) Prove that the correspondence $K\mapsto \Sigma K$ gives a functor from $\mathbf{Simp}$ to itself.

• 2.14. Let $A$ be a finite set. Show that $\dot{\mathcal F}A$ is a pseudomanifold.

• 2.15. Let $M$ be an $n$-dimensional pseudomanifold which is infinite. What is $H_n(M)$?

• 2.16. Let $(K,K_1,K_2)$ be a simplicial triad. Suppose that $K_1$ and $K_2$ are connected and that $K_1\cap K_2$ is not empty. Show that $K$ is connected.

• 2.17. Let $(K,K_1,K_2)$ be a simplicial triad and let $K_0=K_1\cap K_2$.

  • (a) Prove that the homomorphism $H_*(K_1,K_0) \rightarrow H_*(K_,K_2)$ induced by the inclusion is an isomorphism (simplicial excision).

  • (b) Write the commutative diagram involving the homology sequences of $(K_1,K_0)$ and $(K_,K_2)$. Using (a), construct out of this diagram the Mayer-Vietoris sequence for the triad $(K,K_1,K_2)$.

• 2.18. Deduce the additivity formula for the Euler characteristic of Lemma 2.4.10 from the Mayer-Vietoris sequence.

• 2.19. Let $M_1$ and $M_2$ be two finite $n$-dimensional pseudomanifolds. Let $\sigma _i\in {\mathcal S}(M_i)$ and let $h$: $\sigma _1\rightarrow \sigma _2$ be a bijection. The simplicial connected sum $M=M_1\,\sharp \,M_2$ (using $h$) is the simplicial complex defined by

  $$ V(M) = V(M_1)\,\dot{\cup }\, V(M_2)\big /\{v\sim h(v) \hbox { for } v\in \sigma _1\} $$

  and

  $$ {\mathcal S}(M) = \big ({\mathcal S}(M_1) - \{\sigma _1\}\big ) \, \dot{\cup }\, \big ({\mathcal S}(M_2) - \{\sigma _2\}\big ) \, . $$

  Prove that $M$ is a pseudomanifold. Compute $H_*(M)$ in terms of $H_*(M_1)$ and $H_*(M_2)$.