Course 421: Hilary Term 2003 Homology Theory

David R. Wilkins

Copyright \begin{matrix}c \\ ◯ \end{matrix} David R. Wilkins 1988–2003

Contents

5 Simplicial Homology Groups 2 5.1 The Chain Groups of a Simplicial Complex . . . . . . . . . . . 2 5.2 Boundary Homomorphisms . . . . . . . . . . . . . . . . . . . . 5 5.3 The Homology Groups of a Simplicial Complex . . . . . . . . 6 5.4 Simplicial Maps and Induced Homomorphisms . . . . . . . . . 8 5.5 Connectedness and H_{0}(K) . . . . . . . . . . . . . . . . . . . . 8

6 Introduction to Homological Algebra 11 6.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2 Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.3 The Mayer-Vietoris Sequence . . . . . . . . . . . . . . . . . . 18

7 The Topological Invariance of Simplicial Homology Groups 19 7.1 Contiguous Simplicial Maps . . . . . . . . . . . . . . . . . . . 19 7.2 The Homology of Barycentric Subdivisions . . . . . . . . . . . 21 7.3 Continuous Maps and Induced Homomorphisms . . . . . . . . 23 7.4 Homotopy Equivalence . . . . . . . . . . . . . . . . . . . . . . 26

5 Simplicial Homology Groups

5.1 The Chain Groups of a Simplicial Complex

Let K be a simplicial complex. For each non-negative integer q, let Δ_{q}(K) be the additive group consisting of all formal sums of the form

n_{1}(v_{0}^{1}, v_{1}^{1}, …, v_{q}^{1}) + n_{2}(v_{0}^{2}, v_{1}^{2}, …, v_{q}^{2}) + ⋯ + n_{s}(v_{0}^{s}, v_{1}^{s}, …, v_{q}^{s}),

where n_{1}, n_{2}, …, n_{s} are integers and v_{0}^{r}, v_{1}^{r}, …, v_{q}^{r} are (not necessarily distinct) vertices of K that span a simplex of K for r = 1, 2, …, s. (In more formal language, the group Δ_{q}(K) is the free Abelian group generated by the set of all (q + 1)-tuples of the form (v_{0}, v_{1}, …, v_{q}), where v_{0}, v_{1}, …, v_{q} span a simplex of K.)

We recall some basic facts concerning permutations. A permutation of a set S is a bijection mapping S onto itself. The set of all permutations of some set S is a group; the group multiplication corresponds to composition of permutations. A transposition is a permutation of a set S which interchanges two elements of S, leaving the remaining elements of the set fixed. If S is finite and has more than one element then any permutation of S can be expressed as a product of transpositions. In particular any permutation of the set {0, 1, …, q} can be expressed as a product of transpositions (j − 1, j) that interchange j − 1 and j for some j.

Associated to any permutation π of a finite set S is a number ϵ_{π}, known as the parity or signature of the permutation, which can take on the values ±1. If π can be expressed as the product of an even number of transpositions, then ϵ_{π} = +1; if π can be expressed as the product of an odd number of transpositions then ϵ_{π} = −1. The function π ↦ ϵ_{π} is a homomorphism from the group of permutations of a finite set S to the multiplicative group {+1, −1} (i.e., ϵ_{πρ} = ϵ_{π}ϵ_{ρ} for all permutations π and ρ of the set S). Note in particular that the parity of any transposition is −1.

Definition The qth chain group C_{q}(K) of the simplicial complex K is defined to be the quotient group Δ_{q}(K)/Δ_{q}^{0}(K), where Δ_{q}^{0}(K) is the subgroup of Δ_{q}(K) generated by elements of the form (v_{0}, v_{1}, …, v_{q}) where v_{0}, v_{1}, …, v_{q} are not all distinct, and by elements of the form

(v_{π(0)}, v_{π(1)}, …, v_{π(q)}) − ϵ_{π}(v_{0}, v_{1}, …, v_{q})

where π is some permutation of {0, 1, …, q} with parity ϵ_{π}. For convenience, we define C_{q}(K) = {0} when q < 0 or q > dim K, where dim K is the dimension of the simplicial complex K. An element of the chain group C_{q}(K) is referred to as q-chain of the simplicial complex K.

We denote by ⟨v_{0}, v_{1}, …, v_{q}⟩ the element Δ_{q}^{0}(K) + (v_{0}, v_{1}, …, v_{q}) of C_{q}(K) corresponding to (v_{0}, v_{1}, …, v_{q}). The following results follow immediately from the definition of C_{q}(K).

Lemma 5.1 Let v_{0}, v_{1}, …, v_{q} be vertices of a simplicial complex K that span a simplex of K. Then

• ⟨v_{0}, v_{1}, …, v_{q}⟩ = 0 if v_{0}, v_{1}, …, v_{q} are not all distinct,

• ⟨v_{π(0)}, v_{π(1)}, …, v_{π(q)}⟩ = ϵ_{π}⟨v_{0}, v_{1}, …, v_{q}⟩ for any permutation π of the set {0, 1, …, q}.

Example If v_{0} and v_{1} are the endpoints of some line segment then

⟨v_{0}, v_{1}⟩ = −⟨v_{1}, v_{0}⟩.

If v_{0}, v_{1} and v_{2} are the vertices of a triangle in some Euclidean space then

⟨v_{0}, v_{1}, v_{2}⟩ = ⟨v_{1}, v_{2}, v_{0}⟩ = ⟨v_{2}, v_{0}, v_{1}⟩ = −⟨v_{2}, v_{1}, v_{0}⟩
= −⟨v_{0}, v_{2}, v_{1}⟩ = −⟨v_{1}, v_{0}, v_{2}⟩.

Definition An oriented q-simplex is an element of the chain group C_{q}(K) of the form ±⟨v_{0}, v_{1}, …, v_{q}⟩, where v_{0}, v_{1}, …, v_{q} are distinct and span a simplex of K.

An oriented simplex of K can be thought of as consisting of a simplex of K (namely the simplex spanned by the prescribed vertices), together with one of two possible ‘orientations’ on that simplex. Any ordering of the vertices determines an orientation of the simplex; any even permutation of the ordering of the vertices preserves the orientation on the simplex, whereas any odd permutation of this ordering reverses orientation.

Any q-chain of a simplicial complex K can be expressed as a sum of the form

n_{1}σ_{1} + n_{2}σ_{2} + ⋯ + n_{s}σ_{s}

where n_{1}, n_{2}, …, n_{s} are integers and σ_{1}, σ_{2}, …, σ_{s} are oriented q-simplices of K. If we reverse the orientation on one of these simplices σ_{i} then this reverses the sign of the corresponding coefficient n_{i}. If σ_{1}, σ_{2}, …, σ_{s} represent distinct simplices of K then the coefficients n_{1}, n_{2}, …, n_{s} are uniquely determined.

Example Let v_{0}, v_{1} and v_{2} be the vertices of a triangle in some Euclidean space. Let K be the simplicial complex consisting of this triangle, together with its edges and vertices. Every 0-chain of K can be expressed uniquely in the form

n_{0}⟨v_{0}⟩ + n_{1}⟨v_{1}⟩ + n_{2}⟨v_{2}⟩

for some n_{0}, n_{1}, n_{2} ∈ ℤ. Similarly any 1-chain of K can be expressed uniquely in the form

m_{0}⟨v_{1}, v_{2}⟩ + m_{1}⟨v_{2}, v_{0}⟩ + m_{2}⟨v_{0}, v_{1}⟩

for some m_{0}, m_{1}, m_{2} ∈ ℤ, and any 2-chain of K can be expressed uniquely as n⟨v_{0}, v_{1}, v_{2}⟩ for some integer n.

Lemma 5.2 Let K be a simplicial complex, and let A be an additive group. Suppose that, to each (q + 1)-tuple (v_{0}, v_{1}, …, v_{q}) of vertices spanning a simplex of K, there corresponds an element α(v_{0}, v_{1}, …, v_{q}) of A, where

• α(v_{0}, v_{1}, …, v_{q}) = 0 unless v_{0}, v_{1}, …, v_{q} are all distinct,

• α(v_{0}, v_{1}, …, v_{q}) changes sign on interchanging any two adjacent vertices v_{j−1} and v_{j}.

Then there exists a well-defined homomorphism from C_{q}(K) to A which sends ⟨v_{0}, v_{1}, …, v_{q}⟩ to α(v_{0}, v_{1}, …, v_{q}) whenever v_{0}, v_{1}, …, v_{q} span a simplex of K. This homomorphism is uniquely determined.

Proof The given function defined on (q + 1)-tuples of vertices of K extends to a well-defined homomorphism α: Δ_{q}(K) → A given by

α (∑_{r=1}^{s} n_{r}(v_{0}^{r}, v_{1}^{r}, …, v_{q}^{r})) = ∑_{r=1}^{s} n_{r}α(v_{0}^{r}, v_{1}^{r}, …, v_{q}^{r})

for all ∑_{r=1}^{s} n_{r}(v_{0}^{r}, v_{1}^{r}, …, v_{q}^{r}) ∈ Δ_{q}(K). Moreover (v_{0}, v_{1}, …, v_{q}) ∈ ker α

unless v_{0}, v_{1}, …, v_{q} are all distinct. Also

(v_{π(0)}, v_{π(1)}, …, v_{π(q)}) − ε_{π}(v_{0}, v_{1}, …, v_{q}) ∈ ker α

for all permutations π of {0, 1, …, q}, since the permutation π can be expressed as a product of transpositions (j − 1, j) that interchange j − 1 with j for some j and leave the rest of the set fixed, and the parity ε_{π} of π is given by ε_{π} = +1 when the number of such transpositions is even, and by ε_{π} = −1 when the number of such transpositions is odd. Thus the generators of Δ_{q}^{0}(K) are contained in ker α, and hence Δ_{q}^{0}(K) ⊂ ker α. The required homomorphism α̃: C_{q}(K) → A is then defined by the formula

α̃ (∑_{r=1}^{s} n_{r}⟨v_{0}^{r}, v_{1}^{r}, …, v_{q}^{r}⟩) = ∑_{r=1}^{s} n_{r}α(v_{0}^{r}, v_{1}^{r}, …, v_{q}^{r}).

5.2 Boundary Homomorphisms

Let K be a simplicial complex. We introduce below boundary homomorphisms ∂_{q}: C_{q}(K) → C_{q−1}(K) between the chain groups of K. If σ is an oriented q-simplex of K then ∂_{q}(σ) is a (q − 1)-chain which is a formal sum of the (q − 1)-faces of σ, each with an orientation determined by the orientation of σ.

Let σ be a q-simplex with vertices v_{0}, v_{1}, …, v_{q}. For each integer j between 0 and q we denote by ⟨v_{0}, …, v̂_{j}, …, v_{q}⟩ the oriented (q − 1)-face

⟨v_{0}, …, v_{j−1}, v_{j+1}, …, v_{q}⟩

of the simplex σ obtained on omitting v_{j} from the set of vertices of σ. In particular

⟨v̂_{0}, v_{1}, …, v_{q}⟩ ≡ ⟨v_{1}, …, v_{q}⟩, ⟨v_{0}, …, v_{q−1}, v̂_{q}⟩ ≡ ⟨v_{0}, …, v_{q−1}⟩.

Similarly if j and k are integers between 0 and q, where j < k, we denote by

⟨v_{0}, …, v̂_{j}, …, v̂_{k}, … v_{q}⟩

the oriented (q−2)-face ⟨v_{0}, …, v_{j−1}, v_{j+1}, …, v_{k−1}, v_{k+1}, …, v_{q}⟩ of the simplex σ obtained on omitting v_{j} and v_{k} from the set of vertices of σ.

We now define a ‘boundary homomorphism’ ∂_{q}: C_{q}(K) → C_{q−1}(K) for each integer q. Define ∂_{q} = 0 if q ≤ 0 or q > dim K. (In this case one or other of the groups C_{q}(K) and C_{q−1}(K) is trivial.) Suppose then that 0 < q ≤ dim K. Given vertices v_{0}, v_{1}, …, v_{q} spanning a simplex of K, let

α(v_{0}, v_{1}, …, v_{q}) = ∑_{j=0}^{q}(−1)^{j}⟨v_{0}, …, v̂_{j}, …, v_{q}⟩.

Inspection of this formula shows that α(v_{0}, v_{1}, …, v_{q}) changes sign whenever two adjacent vertices v_{i−1} and v_{i} are interchanged.

Suppose that v_{j} = v_{k} for some j and k satisfying j < k. Then

α(v_{0}, v_{1}, …, v_{q}) = (−1)^{j}⟨v_{0}, …, v̂_{j}, …, v_{q}⟩ + (−1)^{k}⟨v_{0}, …, v̂_{k}, …, v_{q}⟩,

since the remaining terms in the expression defining α(v_{0}, v_{1}, …, v_{q}) contain both v_{j} and v_{k}. However (v_{0}, …, v̂_{k}, …, v_{q}) can be transformed to (v_{0}, …, v̂_{j}, …, v_{q}) by making k − j − 1 transpositions which interchange v_{j} successively with the vertices v_{j+1}, v_{j+2}, …, v_{k−1}. Therefore

⟨v_{0}, …, v̂_{k}, …, v_{q}⟩ = (−1)^{k−j−1}⟨v_{0}, …, v̂_{j}, …, v_{q}⟩.

Thus α(v_{0}, v_{1}, …, v_{q}) = 0 unless v_{0}, v_{1}, …, v_{q} are all distinct. It now follows immediately from Lemma 5.2 that there is a well-defined homomorphism ∂_{q}: C_{q}(K) → C_{q−1}(K), characterized by the property that

∂_{q} (⟨v_{0}, v_{1}, …, v_{q}⟩) = ∑_{j=0}^{q}(−1)^{j}⟨v_{0}, …, v̂_{j}, …, v_{q}⟩

whenever v_{0}, v_{1}, …, v_{q} span a simplex of K.

Lemma 5.3 ∂_{q−1} ∘ ∂_{q} = 0 for all integers q.

Proof The result is trivial if q < 2, since in this case ∂_{q−1} = 0. Suppose that q ≥ 2. Let v_{0}, v_{1}, …, v_{q} be vertices spanning a simplex of K. Then

∂_{q−1}∂_{q} (⟨v_{0}, v_{1}, …, v_{q}⟩) = ∑_{j=0}^{q}(−1)^{j}∂_{q−1} (⟨v_{0}, …, v̂_{j}, …, v_{q}⟩)

= ∑_{j=0}^{q} ∑_{k=0}^{j−1}(−1)^{j+k}⟨v_{0}, …, v̂_{k}, …, v̂_{j}, …, v_{q}⟩

+ ∑_{j=0}^{q} ∑_{k=j+1}^{q}(−1)^{j+k−1}⟨v_{0}, …, v̂_{j}, …, v̂_{k}, …, v_{q}⟩

= 0

(since each term in this summation over j and k cancels with the corresponding term with j and k interchanged). The result now follows from the fact that the homomorphism ∂_{q−1} ∘ ∂_{q} is determined by its values on all oriented q-simplices of K.

5.3 The Homology Groups of a Simplicial Complex

Let K be a simplicial complex. A q-chain z is said to be a q-cycle if ∂_{q}z = 0. A q-chain b is said to be a q-boundary if b = ∂_{q+1}c′ for some (q+1)-chain c′. The group of q-cycles of K is denoted by Z_{q}(K), and the group of q-boundaries of K is denoted by B_{q}(K). Thus Z_{q}(K) is the kernel of the boundary homomorphism ∂_{q}: C_{q}(K) → C_{q−1}(K), and B_{q}(K) is the image of the boundary homomorphism ∂_{q+1}: C_{q+1}(K) → C_{q}(K). However ∂_{q} ∘ ∂_{q+1} = 0, by Lemma 5.3. Therefore B_{q}(K) ⊂ Z_{q}(K). But these groups are subgroups of the Abelian group C_{q}(K). We can therefore form the quotient group H_{q}(K), where H_{q}(K) = Z_{q}(K)/B_{q}(K). The group H_{q}(K) is referred to as the qth homology group of the simplicial complex K. Note that H_{q}(K) = 0 if q < 0 or q > dim K (since Z_{q}(K) = 0 and B_{q}(K) = 0 in these cases). It can be shown that the homology groups of a simplicial complex are topological invariants of the polyhedron of that complex.

The element [z] ∈ H_{q}(K) of the homology group H_{q}(K) determined by z ∈ Z_{q}(K) is referred to as the homology class of the q-cycle z. Note that [z_{1} + z_{2}] = [z_{1}] + [z_{2}] for all z_{1}, z_{2} ∈ Z_{q}(K), and [z_{1}] = [z_{2}] if and only if z_{1} − z_{2} = ∂_{q+1}c for some (q + 1)-chain c.

Proposition 5.4 Let K be a simplicial complex. Suppose that there exists a vertex w of K with the following property:

• if vertices v_{0}, v_{1}, …, v_{q} span a simplex of K then so do w, v_{0}, v_{1}, …, v_{q}.

Then H_{0}(K) ≅ ℤ, and H_{q}(K) is the zero group for all q > 0.

Proof Using Lemma 5.2, we see that there is a well-defined homomorphism D_{q}: C_{q}(K) → C_{q+1}(K) characterized by the property that

D_{q}(⟨v_{0}, v_{1}, …, v_{q}⟩) = ⟨w, v_{0}, v_{1}, …, v_{q}⟩

whenever v_{0}, v_{1}, …, v_{q} span a simplex of K. Now ∂_{1}(D_{0}(v)) = v − w for all vertices v of K. It follows that

∑_{r=1}^{s} n_{r}⟨v_{r}⟩ − (∑_{r=1}^{s} n_{r}) ⟨w⟩ = ∑_{r=1}^{s} n_{r}(⟨v_{r}⟩ − ⟨w⟩) ∈ B_{0}(K)

for all ∑_{r=1}^{s} n_{r}⟨v_{r}⟩ ∈ C_{0}(K). But Z_{0}(K) = C_{0}(K) (since ∂_{0} = 0 by definition),

and thus H_{0}(K) = C_{0}(K)/B_{0}(K). It follows that there is a well-defined surjective homomorphism from H_{0}(K) to ℤ induced by the homomorphism

from C_{0}(K) to ℤ that sends ∑_{r=1}^{s} n_{r}⟨v_{r}⟩ ∈ C_{0}(K) to ∑_{r=1}^{s} n_{r}. Moreover this

induced homomorphism is an isomorphism from H_{0}(K) to ℤ.

Now let q > 0. Then

∂_{q+1}(D_{q}(⟨v_{0}, v_{1}, …, v_{q}⟩))

= ∂_{q+1}(⟨w, v_{0}, v_{1}, …, v_{q}⟩)
= ⟨v_{0}, v_{1}, …, v_{q}⟩ + ∑_{j=0}^{q}(−1)^{j+1}⟨w, v_{0}, …, v̂_{j}, …, v_{q}⟩
= ⟨v_{0}, v_{1}, …, v_{q}⟩ − D_{q−1}(∂_{q}(⟨v_{0}, v_{1}, …, v_{q}⟩))

whenever v_{0}, v_{1}, …, v_{q} span a simplex of K. Thus

∂_{q+1}(D_{q}(c)) + D_{q−1}(∂_{q}(c)) = c

for all c ∈ C_{q}(K). In particular z = ∂_{q+1}(D_{q}(z)) for all z ∈ Z_{q}(K), and hence Z_{q}(K) = B_{q}(K). It follows that H_{q}(K) is the zero group for all q > 0, as required.

Example The hypotheses of the proposition are satisfied for the complex K_{σ} consisting of a simplex σ together with all of its faces: we can choose w to be any vertex of the simplex σ. They are also satisfied for the first barycentric subdivision K′_{σ} of K_{σ}: in this case we must choose w to be the barycentre σ̂ of the simplex σ. Thus the groups H_{0}(K_{σ}) and H_{0}(K′_{σ}) are both isomorphic of ℤ, and the groups H_{q}(K_{σ}) and H_{q}(K′_{σ}) are the zero group for all q > 0.

5.4 Simplicial Maps and Induced Homomorphisms

Any simplicial map φ: K → L between simplicial complexes K and L induces well-defined homomorphisms φ_{q}: C_{q}(K) → C_{q}(L) of chain groups, where

φ_{q}(⟨v_{0}, v_{1}, …, v_{q}⟩) = ⟨φ(v_{0}), φ(v_{1}), …, φ(v_{q})⟩

whenever v_{0}, v_{1}, …, v_{q} span a simplex of K. (The existence of these induced homomorphisms follows from a straightforward application of Lemma 5.2.) Note that φ_{q} (⟨v_{0}, v_{1}, …, v_{q}⟩) = 0 unless φ(v_{0}), φ(v_{1}), …, φ(v_{q}) are all distinct.

Now φ_{q−1} ∘ ∂_{q} = ∂_{q} ∘ φ_{q} for each integer q. Therefore φ_{q}(Z_{q}(K)) ⊂ Z_{q}(L) and φ_{q}(B_{q}(K)) ⊂ B_{q}(L) for all integers q. It follows that any simplicial map φ: K → L induces well-defined homomorphisms φ_{∗}: H_{q}(K) → H_{q}(L) of homology groups, where φ_{∗}([z]) = [φ_{q}(z)] for all q-cycles z ∈ Z_{q}(K). It is a trivial exercise to verify that if K, L and M are simplicial complexes and if φ: K → L and ψ: L → M are simplicial maps then the induced homomorphisms of homology groups satisfy (ψ ∘ φ)_{∗} = ψ_{∗} ∘ φ_{∗}.

5.5 Connectedness and H_{0}(K)

Lemma 5.5 Let K be a simplicial complex. Then K can be partitioned into pairwise disjoint subcomplexes K_{1}, K_{2}, …, K_{r} whose polyhedra are the connected components of the polyhedron |K| of K.

Proof Let X_{1}, X_{2}, …, X_{r} be the connected components of the polyhedron of K, and, for each j, let K_{j} be the collection of all simplices σ of K for which σ ⊂ X_{j}. If a simplex belongs to K_{j} for all j then so do all its faces. Therefore K_{1}, K_{2}, …, K_{r} are subcomplexes of K. These subcomplexes are pairwise disjoint since the connected components X_{1}, X_{2}, …, X_{r} of |K| are pairwise disjoint. Moreover, if σ ∈ K then σ ⊂ X_{j} for some j, since σ is a connected subset of |K|, and any connected subset of a topological space is contained in some connected component. But then σ ∈ K_{j}. It follows that K = K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{r} and |K| = |K_{1}| ∪ |K_{2}| ∪ ⋯ ∪ |K_{r}|, as required.

The direct sum A_{1}⊕A_{2}⊕⋯⊕A_{r} of additive Abelian groups A_{1}, A_{2}, …, A_{r} is defined to be the additive group consisting of all r-tuples (a_{1}, a_{2}, …, a_{r}) with a_{i} ∈ A_{i} for i = 1, 2, …, r, where

(a_{1}, a_{2}, …, a_{r}) + (b_{1}, b_{2}, …, b_{r}) ≡ (a_{1} + b_{1}, a_{2} + b_{2}, …, a_{r} + b_{r}).

Lemma 5.6 Let K be a simplicial complex. Suppose that K = K_{1} ∪ K_{2} ∪ ⋯ ∪ K_{r}, where K_{1}, K_{2}, … K_{r} are pairwise disjoint. Then

H_{q}(K) ≅ H_{q}(K_{1}) ⊕ H_{q}(K_{2}) ⊕ ⋯ ⊕ H_{q}(K_{r})

for all integers q.

Proof We may restrict our attention to the case when 0 ≤ q ≤ dim K, since H_{q}(K) = {0} if q < 0 or q > dim K. Now any q-chain c of K can be expressed uniquely as a sum of the form c = c_{1} + c_{2} + ⋯ + c_{r}, where c_{j} is a q-chain of K_{j} for j = 1, 2, …, r. It follows that

C_{q}(K) ≅ C_{q}(K_{1}) ⊕ C_{q}(K_{2}) ⊕ ⋯ ⊕ C_{q}(K_{r}).

Now let z be a q-cycle of K (i.e., z ∈ C_{q}(K) satisfies ∂_{q}(z) = 0). We can express z uniquely in the form z = z_{1} + z_{2} + ⋯ + z_{r}, where z_{j} is a q-chain of K_{j} for j = 1, 2, …, r. Now

0 = ∂_{q}(z) = ∂_{q}(z_{1}) + ∂_{q}(z_{2}) + ⋯ + ∂_{q}(z_{r}),

and ∂_{q}(z_{j}) is a (q−1)-chain of K_{j} for j = 1, 2, …, r. It follows that ∂_{q}(z_{j}) = 0 for j = 1, 2, …, r. Hence each z_{j} is a q-cycle of K_{j}, and thus

Z_{q}(K) ≅ Z_{q}(K_{1}) ⊕ Z_{q}(K_{2}) ⊕ ⋯ ⊕ Z_{q}(K_{r}).

Now let b be a q-boundary of K. Then b = ∂_{q+1}(c) for some (q + 1)- chain c of K. Moreover c = c_{1} + c_{2} + ⋯ c_{r}, where c_{j} ∈ C_{q+1}(K_{j}). Thus b = b_{1} + b_{2} + ⋯ b_{r}, where b_{j} ∈ B_{q}(K_{j}) is given by b_{j} = ∂_{q+1}c_{j} for j = 1, 2, …, r. We deduce that

B_{q}(K) ≅ B_{q}(K_{1}) ⊕ B_{q}(K_{2}) ⊕ ⋯ ⊕ B_{q}(K_{r}).

It follows from these observations that there is a well-defined isomorphism

ν: H_{q}(K_{1}) ⊕ H_{q}(K_{2}) ⊕ ⋯ ⊕ H_{q}(K_{r}) → H_{q}(K)

which maps ([z_{1}], [z_{2}], …, [z_{r}]) to [z_{1} + z_{2} + ⋯ + z_{r}], where [z_{j}] denotes the homology class of a q-cycle z_{j} of K_{j} for j = 1, 2, …, r.

Let K be a simplicial complex, and let y and z be vertices of K. We say that y and z can be joined by an edge path if there exists a sequence v_{0}, v_{1}, …, v_{m} of vertices of K with v_{0} = y and v_{m} = z such that the line segment with endpoints v_{j−1} and v_{j} is an edge belonging to K for j = 1, 2, …, m.

Lemma 5.7 The polyhedron |K| of a simplicial complex K is a connected topological space if and only if any two vertices of K can be joined by an edge path.

Proof It is easy to verify that if any two vertices of K can be joined by an edge path then |K| is path-connected and is thus connected. (Indeed any two points of |K| can be joined by a path made up of a finite number of straight line segments.)

We must show that if |K| is connected then any two vertices of K can be joined by an edge path. Choose a vertex v_{0} of K. It suffices to verify that every vertex of K can be joined to v_{0} by an edge path.

Let K_{0} be the collection of all of the simplices of K having the property that one (and hence all) of the vertices of that simplex can be joined to v_{0} by an edge path. If σ is a simplex belonging to K_{0} then every vertex of σ can be joined to v_{0} by an edge path, and therefore every face of σ belongs to K_{0}. Thus K_{0} is a subcomplex of K. Clearly the collection K_{1} of all simplices of K which do not belong to K_{0} is also a subcomplex of K. Thus K = K_{0} ∪ K_{1}, where K_{0} ∩ K_{1} = ∅, and hence |K| = |K_{0}| ∪ |K_{1}|, where |K_{0}| ∩ |K_{1}| = ∅. But the polyhedra |K_{0}| and |K_{1}| of K_{0} and K_{1} are closed subsets of |K|. It follows from the connectedness of |K| that either |K_{0}| = ∅ or |K_{1}| = ∅. But v_{0} ∈ K_{0}. Thus K_{1} = ∅ and K_{0} = K, showing that every vertex of K can be joined to v_{0} by an edge path, as required.

Theorem 5.8 Let K be a simplicial complex. Suppose that the polyhedron |K| of K is connected. Then H_{0}(K) ≅ ℤ.

Proof Let u_{1}, u_{2}, …, u_{r} be the vertices of the simplicial complex K. Every 0-chain of K can be expressed uniquely as a formal sum of the form

n_{1}⟨u_{1}⟩ + n_{2}⟨u_{2}⟩ + ⋯ + n_{r}⟨u_{r}⟩

for some integers n_{1}, n_{2}, …, n_{r}. It follows that there is a well-defined homomorphism ε: C_{0}(K) → ℤ defined by

ε (n_{1}⟨u_{1}⟩ + n_{2}⟨u_{2}⟩ + ⋯ + n_{r}⟨u_{r}⟩) = n_{1} + n_{2} + ⋯ + n_{r}.

Now ε(∂_{1}(⟨y, z⟩)) = ε(⟨z⟩ − ⟨y⟩) = 0 whenever y and z are endpoints of an edge of K. It follows that ε ∘ ∂_{1} = 0, and hence B_{0}(K) ⊂ ker ε.

Let v_{0}, v_{1}, …, v_{m} be vertices of K determining an edge path. Then

⟨v_{m}⟩ − ⟨v_{0}⟩ = ∂_{1} (∑_{j=1}^{m}⟨v_{j−1}, v_{j}⟩) ∈ B_{0}(K).

Now |K| is connected, and therefore any pair of vertices of K can be joined by an edge path (Lemma 5.7). We deduce that ⟨z⟩ − ⟨y⟩ ∈ B_{0}(K) for all

vertices y and z of K. Thus if c ∈ ker ε, where c = ∑_{j=1}^{r} n_{j}⟨u_{j}⟩, then ∑_{j=1}^{r} n_{j} = 0,

and hence c = ∑_{j=2}^{r} n_{j}(⟨u_{j}⟩ − ⟨u_{1}⟩). But ⟨u_{j}⟩ − ⟨u_{1}⟩ ∈ B_{0}(K). It follows that

c ∈ B_{0}(K). We conclude that ker ε ⊂ B_{0}(K), and hence ker ε = B_{0}(K).

Now the homomorphism ε: C_{0}(K) → ℤ is surjective and its kernel is B_{0}(K). Therefore it induces an isomorphism from C_{0}(K)/B_{0}(K) to ℤ. However Z_{0}(K) = C_{0}(K) (since ∂_{0} = 0 by definition). Thus H_{0}(K) ≡ C_{0}(K)/B_{0}(K) ≅ ℤ, as required.

On combining Theorem 5.8 with Lemmas 5.5 and 5.6 we obtain immediately the following result.

Corollary 5.9 Let K be a simplicial complex. Then

H_{0}(K) ≅ ℤ ⊕ ℤ ⊕ ⋯ ⊕ ℤ (r times),

where r is the number of connected components of |K|.

6 Introduction to Homological Algebra

6.1 Exact Sequences

In homological algebra we consider sequences

⋯ ⟶F ⟶^{p}G⟶^{q}H⟶^{···}

where F, G, H etc. are Abelian groups and p, q etc. are homomorphisms. We denote the trivial group {0} by 0, and we denote by 0⟶G and G⟶0 the zero homomorphisms from 0 to G and from G to 0 respectively. (These zero homomorphisms are of course the only homomorphisms mapping out of and into the trivial group 0.)

Definition The sequence F ⟶^{p}G⟶^{q}H of Abelian groups and homomorphisms is said to be exact at G if and only if image(p: F → G) = ker(q: G → H). A sequence of Abelian groups and homomorphisms is said to be exact if it is exact at each Abelian group occurring in the sequence (so that the image of each homomorphism is the kernel of the succeeding homomorphism).

A monomorphism is an injective homomorphism. An epimorphism is a surjective homomorphism. An isomorphism is a bijective homomorphism.

The following result follows directly from the relevant definitions.

Lemma 6.1 Let h: G → H be a homomorphism of Abelian groups.

• h: G → H is a monomorphism if and only if 0⟶G⟶^{h}H is an exact sequence.

• h: G → H is an epimorphism if and only if G⟶^{h}H⟶0 is an exact sequence.

• h: G → H is an isomorphism if and only if 0⟶G⟶^{h}H⟶0 is an exact sequence.

Let F be a subgroup of an Abelian group G. Then the sequence

0⟶F ⟶^{i}G⟶^{q}G/F ⟶0,

is exact, where G/F is the quotient group, i: F ↪ G is the inclusion homomorphism, and q: G → G/F is the quotient homomorphism. Conversely, given any exact sequence of the form

0⟶F ⟶^{i}G⟶^{q}H⟶0,

we can regard F as a subgroup of G (on identifying F with i(F)), and then H is isomorphic to the quotient group G/F. Exact sequences of this type are referred to as short exact sequences.

We now introduce the concept of a commutative diagram. This is a diagram depicting a collection of homomorphisms between various Abelian groups occurring on the diagram. The diagram is said to commute if, whenever there are two routes through the diagram from an Abelian group G to an Abelian group H, the homomorphism from G to H obtained by forming the composition of the homomorphisms along one route in the diagram agrees with that obtained by composing the homomorphisms along the other route. Thus, for example, the diagram

A ⟶^{f} B ⟶^{g} C
↓^{p}  ↓^{q}  ↓^{r}
D ⟶^{h} E ⟶^{k} F

commutes if and only if q ∘ f = h ∘ p and r ∘ g = k ∘ q.

Proposition 6.2 Suppose that the following diagram of Abelian groups and homomorphisms

G_{1} ⟶^{θ_{1}} G_{2} ⟶^{θ_{2}} G_{3} ⟶^{θ_{3}} G_{4} ⟶^{θ_{4}} G_{5}
↓^{ψ_{1}}  ↓^{ψ_{2}}  ↓^{ψ_{3}}  ↓^{ψ_{4}}  ↓^{ψ_{5}}
H_{1} ⟶^{ϕ_{1}} H_{2} ⟶^{ϕ_{2}} H_{3} ⟶^{ϕ_{3}} H_{4} ⟶^{ϕ_{4}} H_{5}

commutes and that both rows are exact sequences. Then the following results follow:

(i) if ψ_{2} and ψ_{4} are monomorphisms and if ψ_{1} is a epimorphism then ψ_{3} is an monomorphism,

(ii) if ψ_{2} and ψ_{4} are epimorphisms and if ψ_{5} is a monomorphism then ψ_{3} is an epimorphism.

Proof First we prove (i). Suppose that ψ_{2} and ψ_{4} are monomorphisms and that ψ_{1} is an epimorphism. We wish to show that ψ_{3} is a monomorphism. Let x ∈ G_{3} be such that ψ_{3}(x) = 0. Then ψ_{4} (θ_{3}(x)) = ϕ_{3} (ψ_{3}(x)) = 0, and hence θ_{3}(x) = 0. But then x = θ_{2}(y) for some y ∈ G_{2}, by exactness. Moreover

ϕ_{2} (ψ_{2}(y)) = ψ_{3} (θ_{2}(y)) = ψ_{3}(x) = 0,

hence ψ_{2}(y) = ϕ_{1}(z) for some z ∈ H_{1}, by exactness. But z = ψ_{1}(w) for some w ∈ G_{1}, since ψ_{1} is an epimorphism. Then

ψ_{2} (θ_{1}(w)) = ϕ_{1} (ψ_{1}(w)) = ψ_{2}(y),

and hence θ_{1}(w) = y, since ψ_{2} is a monomorphism. But then

x = θ_{2}(y) = θ_{2} (θ_{1}(w)) = 0

by exactness. Thus ψ_{3} is a monomorphism.

Next we prove (ii). Thus suppose that ψ_{2} and ψ_{4} are epimorphisms and that ψ_{5} is a monomorphism. We wish to show that ψ_{3} is an epimorphism. Let a be an element of H_{3}. Then ϕ_{3}(a) = ψ_{4}(b) for some b ∈ G_{4}, since ψ_{4} is an epimorphism. Now

ψ_{5} (θ_{4}(b)) = ϕ_{4} (ψ_{4}(b)) = ϕ_{4} (ϕ_{3}(a)) = 0,

hence θ_{4}(b) = 0, since ψ_{5} is a monomorphism. Hence there exists c ∈ G_{3} such that θ_{3}(c) = b, by exactness. Then

ϕ_{3} (ψ_{3}(c)) = ψ_{4} (θ_{3}(c)) = ψ_{4}(b),

hence ϕ_{3} (a − ψ_{3}(c)) = 0, and thus a − ψ_{3}(c) = ϕ_{2}(d) for some d ∈ H_{2}, by exactness. But ψ_{2} is an epimorphism, hence there exists e ∈ G_{2} such that ψ_{2}(e) = d. But then

ψ_{3} (θ_{2}(e)) = ϕ_{2} (ψ_{2}(e)) = a − ψ_{3}(c).

Hence a = ψ_{3} (c + θ_{2}(e)), and thus a is in the image of ψ_{3}. This shows that ψ_{3} is an epimorphism, as required.

The following result is an immediate corollary of Proposition 6.2.

Lemma 6.3 (Five-Lemma) Suppose that the rows of the commutative diagram of Proposition 6.2 are exact sequences and that ψ_{1}, ψ_{2}, ψ_{4} and ψ_{5} are isomorphisms. Then ψ_{3} is also an isomorphism.

6.2 Chain Complexes

Definition A chain complex C_{∗} is a (doubly infinite) sequence (C_{i} : i ∈ ℤ) of Abelian groups, together with homomorphisms ∂_{i}: C_{i} → C_{i−1} for each i ∈ ℤ, such that ∂_{i} ∘ ∂_{i+1} = 0 for all integers i.

The ith homology group H_{i}(C_{∗}) of the complex C_{∗} is defined to be the quotient group Z_{i}(C_{∗})/B_{i}(C_{∗}), where Z_{i}(C_{∗}) is the kernel of ∂_{i}: C_{i} → C_{i−1} and B_{i}(C_{∗}) is the image of ∂_{i+1}: C_{i+1} → C_{i}.

Definition Let C_{∗} and D_{∗} be chain complexes. A chain map f: C_{∗} → D_{∗} is a sequence f_{i}: C_{i} → D_{i} of homomorphisms which satisfy the commutativity condition ∂_{i} ∘ f_{i} = f_{i−1} ∘ ∂_{i} for all i ∈ ℤ.

Note that a collection of homomorphisms f_{i}: C_{i} → D_{i} defines a chain map f_{∗}: C_{∗} → D_{∗} if and only if the diagram

⋯ ⟶ C_{i+1} ⟶^{∂_{i+1}} C_{i} ⟶^{∂_{i}} C_{i−1} ⟶ ⋯
↓^{f_{i+1}}  ↓^{f_{i}}  ↓^{f_{i−1}}
⋯ ⟶ D_{i+1} ⟶^{∂_{i+1}} D_{i} ⟶^{∂_{i}} D_{i−1} ⟶ ⋯

is commutative.

Let C_{∗} and D_{∗} be chain complexes, and let f_{∗}: C_{∗} → D_{∗} be a chain map. Then f_{i}(Z_{i}(C_{∗})) ⊂ Z_{i}(D_{∗}) and f_{i}(B_{i}(C_{∗})) ⊂ B_{i}(D_{∗}) for all i. It follows from this that f_{i}: C_{i} → D_{i} induces a homomorphism f_{∗}: H_{i}(C_{∗}) → H_{i}(D_{∗}) of homology groups sending [z] to [f_{i}(z)] for all z ∈ Z_{i}(C_{∗}), where [z] = z + B_{i}(C_{∗}), and [f_{i}(z)] = f_{i}(z) + B_{i}(D_{∗}).

Definition A short exact sequence 0⟶A_{∗}⟶^{p_{∗}}B_{∗}⟶^{q_{∗}}C_{∗}⟶0 of chain complexes consists of chain complexes A_{∗}, B_{∗} and C_{∗} and chain maps p_{∗}: A_{∗} → B_{∗} and q_{∗}: B_{∗} → C_{∗} such that the sequence

0⟶A_{i}⟶^{p_{i}}B_{i}⟶^{q_{i}}C_{i}⟶0

is exact for each integer i.

We see that 0⟶A_{∗}⟶^{p_{∗}}B_{∗}⟶^{q_{∗}}C_{∗}⟶0 is a short exact sequence of chain complexes if and only if the diagram

\begin{aligned}& & ⋮ & & ⋮ & & ⋮ \\ & & ↓^{∂_{i+2}} & & ↓^{∂_{i+2}} & & ↓^{∂_{i+2}} \\ 0 & ⟶ & A_{i+1} & ⟶^{p_{i+1}} & B_{i+1} & ⟶^{q_{i+1}} & C_{i+1} & ⟶ & 0 \\ & & ↓^{∂_{i+1}} & & ↓^{∂_{i+1}} & & ↓^{∂_{i+1}} \\ 0 & ⟶ & A_{i} & ⟶^{p_{i}} & B_{i} & ⟶^{q_{i}} & C_{i} & ⟶ & 0 \\ & & ↓^{∂_{i}} & & ↓^{∂_{i}} & & ↓^{∂_{i}} \\ 0 & ⟶ & A_{i−1} & ⟶^{p_{i−1}} & B_{i−1} & ⟶^{q_{i−1}} & C_{i−1} & ⟶ & 0 \\ & & ↓^{∂_{i−1}} & & ↓^{∂_{i−1}} & & ↓^{∂_{i−1}} \\ & & ⋮ & & ⋮ & & ⋮ \end{aligned} .

is a commutative diagram whose rows are exact sequences and whose columns are chain complexes.

Lemma 6.4 Given any short exact sequence 0⟶A_{∗}⟶^{p_{∗}}B_{∗}⟶^{q_{∗}}C_{∗}⟶0 of chain complexes, there is a well-defined homomorphism

α_{i}: H_{i}(C_{∗}) → H_{i−1}(A_{∗})

which sends the homology class [z] of z ∈ Z_{i}(C_{∗}) to the homology class [w] of any element w of Z_{i−1}(A_{∗}) with the property that p_{i−1}(w) = ∂_{i}(b) for some b ∈ B_{i} satisfying q_{i}(b) = z.

Proof Let z ∈ Z_{i}(C_{∗}). Then there exists b ∈ B_{i} satisfying q_{i}(b) = z, since q_{i}: B_{i} → C_{i} is surjective. Moreover

q_{i−1}(∂_{i}(b)) = ∂_{i}(q_{i}(b)) = ∂_{i}(z) = 0.

But p_{i−1}: A_{i−1} → B_{i−1} is injective and p_{i−1}(A_{i−1}) = ker q_{i−1}, since the sequence

0⟶A_{i−1}⟶^{p_{i−1}}B_{i−1}⟶^{q_{i−1}}C_{i−1}

is exact. Therefore there exists a unique element w of A_{i−1} such that ∂_{i}(b) = p_{i−1}(w). Moreover

p_{i−2}(∂_{i−1}(w)) = ∂_{i−1}(p_{i−1}(w)) = ∂_{i−1}(∂_{i}(b)) = 0

(since ∂_{i−1} ∘ ∂_{i} = 0), and therefore ∂_{i−1}(w) = 0 (since p_{i−2}: A_{i−2} → B_{i−2} is injective). Thus w ∈ Z_{i−1}(A_{∗}).

Now let b, b′ ∈ B_{i} satisfy q_{i}(b) = q_{i}(b′) = z, and let w, w′ ∈ Z_{i−1}(A_{∗}) satisfy p_{i−1}(w) = ∂_{i}(b) and p_{i−1}(w′) = ∂_{i}(b′). Then q_{i}(b − b′) = 0, and hence b′ − b = p_{i}(a) for some a ∈ A_{i}, by exactness. But then

p_{i−1}(w + ∂_{i}(a)) = p_{i−1}(w) + ∂_{i}(p_{i}(a)) = ∂_{i}(b) + ∂_{i}(b′ − b) = ∂_{i}(b′) = p_{i−1}(w′),

and p_{i−1}: A_{i−1} → B_{i−1} is injective. Therefore w + ∂_{i}(a) = w′, and hence [w] = [w′] in H_{i−1}(A_{∗}). Thus there is a well-defined function α̃_{i}: Z_{i}(C_{∗}) → H_{i−1}(A_{∗}) which sends z ∈ Z_{i}(C_{∗}) to [w] ∈ H_{i−1}(A_{∗}), where w ∈ Z_{i−1}(A_{∗}) is chosen such that p_{i−1}(w) = ∂_{i}(b) for some b ∈ B_{i} satisfying q_{i}(b) = z. This function α̃_{i} is clearly a homomorphism from Z_{i}(C_{∗}) to H_{i−1}(A_{∗}).

Suppose that elements z and z′ of Z_{i}(C_{∗}) represent the same homology class in H_{i}(C_{∗}). Then z′ = z + ∂_{i+1}c for some c ∈ C_{i+1}. Moreover c = q_{i+1}(d) for some d ∈ B_{i+1}, since q_{i+1}: B_{i+1} → C_{i+1} is surjective. Choose b ∈ B_{i} such that q_{i}(b) = z, and let b′ = b + ∂_{i+1}(d). Then

q_{i}(b′) = z + q_{i}(∂_{i+1}(d)) = z + ∂_{i+1}(q_{i+1}(d)) = z + ∂_{i+1}(c) = z′.

Moreover ∂_{i}(b′) = ∂_{i}(b + ∂_{i+1}(d)) = ∂_{i}(b) (since ∂_{i} ∘ ∂_{i+1} = 0). Therefore α̃_{i}(z) = α̃_{i}(z′). It follows that the homomorphism α̃_{i}: Z_{i}(C_{∗}) → H_{i−1}(A_{∗}) induces a well-defined homomorphism α_{i}: H_{i}(C_{∗}) → H_{i−1}(A_{∗}), as required.

Let 0⟶A_{∗}⟶^{p_{∗}}B_{∗}⟶^{q_{∗}}C_{∗}⟶0 and 0⟶A′_{∗}⟶^{p′_{∗}}B′_{∗}⟶^{q′_{∗}}C′_{∗}⟶0 be short ex-

act sequences of chain complexes, and let λ_{∗}: A_{∗} → A′_{∗}, μ_{∗}: B_{∗} → B′_{∗} and ν_{∗}: C_{∗} → C′_{∗} be chain maps. For each integer i, let α_{i}: H_{i}(C_{∗}) → H_{i−1}(A_{∗}) and α′_{i}: H_{i}(C′_{∗}) → H_{i−1}(A′_{∗}) be the homomorphisms defined as described in Lemma 6.4. Suppose that the diagram

0 ⟶ A_{∗} ⟶^{p_{∗}} B_{∗} ⟶^{q_{∗}} C_{∗} ⟶ 0
↓^{λ_{∗}}  ↓^{μ_{∗}}  ↓^{ν_{∗}}
0 ⟶ A′_{∗} ⟶^{p′_{∗}} B′_{∗} ⟶^{q′_{∗}} C′_{∗} ⟶ 0

commutes (i.e., p′_{i} ∘ λ_{i} = μ_{i} ∘ p_{i} and q′_{i} ∘ μ_{i} = ν_{i} ∘ q_{i} for all i). Then the square

H_{i}(C_{∗}) ⟶^{α_{i}} H_{i−1}(A_{∗})
↓^{ν_{∗}}  ↓^{λ_{∗}}
H_{i}(C′_{∗}) ⟶^{α′_{i}} H_{i−1}(A′_{∗})

commutes for all i ∈ ℤ (i.e., λ_{∗} ∘ α_{i} = α′_{i} ∘ ν_{∗}).

Proposition 6.5 Let 0⟶A_{∗}⟶^{p_{∗}}B_{∗}⟶^{q_{∗}}C_{∗}⟶0 be a short exact sequence of chain complexes. Then the (infinite) sequence

⋯ ⟶^{α_{i+1}}H_{i}(A_{∗})⟶^{p_{∗}}H_{i}(B_{∗})⟶^{q_{∗}}H_{i}(C_{∗})⟶^{α_{i}}H_{i−1}(A_{∗})⟶^{p_{∗}}H_{i−1}(B_{∗})⟶^{q_{∗}} ⋯

of homology groups is exact, where α_{i}: H_{i}(C_{∗}) → H_{i−1}(A_{∗}) is the well-defined homomorphism that sends the homology class [z] of z ∈ Z_{i}(C_{∗}) to the homology class [w] of any element w of Z_{i−1}(A_{∗}) with the property that p_{i−1}(w) = ∂_{i}(b) for some b ∈ B_{i} satisfying q_{i}(b) = z.

Proof First we prove exactness at H_{i}(B_{∗}). Now q_{i} ∘ p_{i} = 0, and hence q_{∗} ∘ p_{∗} = 0. Thus the image of p_{∗}: H_{i}(A_{∗}) → H_{i}(B_{∗}) is contained in the kernel of q_{∗}: H_{i}(B_{∗}) → H_{i}(C_{∗}). Let x be an element of Z_{i}(B_{∗}) for which [x] ∈ ker q_{∗}. Then q_{i}(x) = ∂_{i+1}(c) for some c ∈ C_{i+1}. But c = q_{i+1}(d) for some d ∈ B_{i+1}, since q_{i+1}: B_{i+1} → C_{i+1} is surjective. Then

q_{i}(x − ∂_{i+1}(d)) = q_{i}(x) − ∂_{i+1}(q_{i+1}(d)) = q_{i}(x) − ∂_{i+1}(c) = 0,

and hence x − ∂_{i+1}(d) = p_{i}(a) for some a ∈ A_{i}, by exactness. Moreover

p_{i−1}(∂_{i}(a)) = ∂_{i}(p_{i}(a)) = ∂_{i}(x − ∂_{i+1}(d)) = 0,

since ∂_{i}(x) = 0 and ∂_{i} ∘ ∂_{i+1} = 0. But p_{i−1}: A_{i−1} → B_{i−1} is injective. Therefore ∂_{i}(a) = 0, and thus a represents some element [a] of H_{i}(A_{∗}). We deduce that

[x] = [x − ∂_{i+1}(d)] = [p_{i}(a)] = p_{∗}([a]).

We conclude that the sequence of homology groups is exact at H_{i}(B_{∗}).

Next we prove exactness at H_{i}(C_{∗}). Let x ∈ Z_{i}(B_{∗}). Now α_{i}(q_{∗}[x]) = α_{i}([q_{i}(x)]) = [w], where w is the unique element of Z_{i}(A_{∗}) satisfying p_{i−1}(w) = ∂_{i}(x). But ∂_{i}(x) = 0, and hence w = 0. Thus α_{i} ∘ q_{∗} = 0. Now let z be an element of Z_{i}(C_{∗}) for which [z] ∈ ker α_{i}. Choose b ∈ B_{i} and w ∈ Z_{i−1}(A_{∗}) such that q_{i}(b) = z and p_{i−1}(w) = ∂_{i}(b). Then w = ∂_{i}(a) for some a ∈ A_{i}, since [w] = α_{i}([z]) = 0. But then q_{i}(b − p_{i}(a)) = z and ∂_{i}(b − p_{i}(a)) = 0. Thus b − p_{i}(a) ∈ Z_{i}(B_{∗}) and q_{∗}([b − p_{i}(a)]) = [z]. We conclude that the sequence of homology groups is exact at H_{i}(C_{∗}).

Finally we prove exactness at H_{i−1}(A_{∗}). Let z ∈ Z_{i}(C_{∗}). Then α_{i}([z]) = [w], where w ∈ Z_{i−1}(A_{∗}) satisfies p_{i−1}(w) = ∂_{i}(b) for some b ∈ B_{i} satisfying q_{i}(b) = z. But then p_{∗}(α_{i}([z])) = [p_{i−1}(w)] = [∂_{i}(b)] = 0. Thus p_{∗} ∘ α_{i} = 0. Now let w be an element of Z_{i−1}(A_{∗}) for which [w] ∈ ker p_{∗}. Then [p_{i−1}(w)] = 0 in H_{i−1}(B_{∗}), and hence p_{i−1}(w) = ∂_{i}(b) for some b ∈ B_{i}. But

∂_{i}(q_{i}(b)) = q_{i−1}(∂_{i}(b)) = q_{i−1}(p_{i−1}(w)) = 0.

Therefore [w] = α_{i}([z]), where z = q_{i}(b). We conclude that the sequence of homology groups is exact at H_{i−1}(A_{∗}), as required.

6.3 The Mayer-Vietoris Sequence

Let K be a simplicial complex and let L and M be subcomplexes of K such that K = L ∪ M. Let

i_{q}: C_{q}(L ∩ M) → C_{q}(L), j_{q}: C_{q}(L ∩ M) → C_{q}(M),
u_{q}: C_{q}(L) → C_{q}(K), v_{q}: C_{q}(M) → C_{q}(K)

be the inclusion homomorphisms induced by the inclusion maps i: L ∩ M ↪ L, j: L ∩ M ↪ M, u: L ↪ K and v: M ↪ K. Then

0⟶C_{∗}(L ∩ M)⟶^{k_{∗}}C_{∗}(L) ⊕ C_{∗}(M)⟶^{w_{∗}}C_{∗}(K)⟶0

is a short exact sequence of chain complexes, where

k_{q}(c) = (i_{q}(c), −j_{q}(c)),
w_{q}(c′, c′′) = u_{q}(c′) + v_{q}(c′′),
∂_{q}(c′, c′′) = (∂_{q}(c′), ∂_{q}(c′′))

for all c ∈ C_{q}(L∩M), c′ ∈ C_{q}(L) and c′′ ∈ C_{q}(M). It follows from Lemma 6.4 that there is a well-defined homomorphism α_{q}: H_{q}(K) → H_{q−1}(L ∩ M) such that α_{q}([z]) = [∂_{q}(c′)] = −[∂_{q}(c′′)] for any z ∈ Z_{q}(K), where c′ and c′′ are any q-chains of L and M respectively satisfying z = c′ + c′′. (Note that ∂_{q}(c′) ∈ Z_{q−1}(L ∩ M) since ∂_{q}(c′) ∈ Z_{q−1}(L), ∂_{q}(c′′) ∈ Z_{q−1}(M) and ∂_{q}(c′) = −∂_{q}(c′′).) It now follows immediately from Proposition 6.5 that the infinite sequence

⋯ ⟶^{α_{q+1}}H_{q}(L ∩ M)⟶^{k_{∗}}H_{q}(L) ⊕ H_{q}(M)⟶^{w_{∗}}H_{q}(K)⟶^{α_{q}}H_{q−1}(L ∩ M)⟶^{k_{∗}} ⋯,

of homology groups is exact. This long exact sequence of homology groups is referred to as the Mayer-Vietoris sequence associated with the decomposition of K as the union of the subcomplexes L and M.

7 The Topological Invariance of Simplicial Homology Groups

7.1 Contiguous Simplicial Maps

Definition Two simplicial maps s: K → L and t: K → L between simplicial complexes K and L are said to be contiguous if, given any simplex σ of K, there exists a simplex τ of L such that s(v) and t(v) are vertices of τ for each vertex v of σ.

Lemma 7.1 Let K and L be simplicial complexes, and let s: K → L and t: K → L be simplicial approximations to some continuous map f: |K| → |L|. Then the simplicial maps s and t are contiguous.

Proof Let x be a point in the interior of some simplex σ of K. Then f(x) belongs to the interior of a unique simplex τ of L, and moreover s(x) ∈ τ and t(x) ∈ τ, since s and t are simplicial approximations to the map f. But s(x) and t(x) are contained in the interior of the simplices s(σ) and t(σ) of L. It follows that s(σ) and t(σ) are faces of τ, and hence s(v) and t(v) are vertices of τ for each vertex v of σ, as required.

Proposition 7.2 Let s: K → L and t: K → L be simplicial maps between simplicial complexes K and L. Suppose that s and t are contiguous. Then the homomorphisms s_{∗}: H_{q}(K) → H_{q}(L) and t_{∗}: H_{q}(K) → H_{q}(L) coincide for all q.

Proof Choose an ordering of the vertices of K. Then there are well-defined homomorphisms D_{q}: C_{q}(K) → C_{q+1}(L) characterized by the property that

D_{q}(⟨v_{0}, v_{1}, …, v_{q}⟩) = ∑_{j=0}^{q}(−1)^{j}⟨s(v_{0}), …, s(v_{j}), t(v_{j}), …, t(v_{q})⟩.

whenever v_{0}, v_{1}, … v_{q} are the vertices of a q-simplex of K listed in increasing order (with respect to the chosen ordering of the vertices of K). Then

∂_{1}(D_{0}(⟨v⟩)) = ∂_{1}(⟨s(v), t(v)⟩) = ⟨t(v)⟩ − ⟨s(v)⟩,

and thus ∂_{1} ∘ D_{0} = t_{0} − s_{0}. Also

D_{q−1}(∂_{q}(⟨v_{0}, …, v_{q}⟩))

= ∑_{i=0}^{q}(−1)^{i}D_{q−1}(⟨v_{0}, …, v̂_{i}, …, v_{q}⟩)

= ∑_{i=0}^{q} ∑_{j=0}^{i−1}(−1)^{i+j}⟨s(v_{0}), …, s(v_{j}), t(v_{j}), …, { \atop t(v_{i})}, …, t(v_{q})⟩

+ ∑_{i=0}^{q} ∑_{j=i+1}^{q}(−1)^{i+j−1}⟨s(v_{0}), …, { \atop s(v_{i})}, …, s(v_{j}), t(v_{j}), …, t(v_{q})⟩

and

∂_{q+1}(D_{q}(⟨v_{0}, … v_{q}⟩))

= ∑_{j=0}^{q}(−1)^{j}∂_{q+1}(⟨s(v_{0}), …, s(v_{j}), t(v_{j}), …, t(v_{q})⟩)

= ∑_{j=0}^{q} ∑_{i=0}^{j−1}(−1)^{i+j}⟨s(v_{0}), …, { \atop s(v_{i})}, …, s(v_{j}), t(v_{j}), …, t(v_{q})⟩

+⟨t(v_{0}), …, t(v_{q})⟩ + ∑_{j=1}^{q}⟨s(v_{0}), …, s(v_{j−1}), t(v_{j}), …, t(v_{q})⟩

− ∑_{j=0}^{q−1}⟨s(v_{0}), …, s(v_{j}), t(v_{j+1}), …, t(v_{q})⟩ − ⟨s(v_{0}), …, s(v_{q})⟩

+ ∑_{j=0}^{q} ∑_{i=j+1}^{q}(−1)^{i+j+1}⟨s(v_{0}), …, s(v_{j}), t(v_{j}), …, { \atop t(v_{i})}, …, t(v_{q})⟩

= −D_{q−1}(∂_{q}(⟨v_{0}, …, v_{q}⟩)) + ⟨t(v_{0}), …, t(v_{q})⟩ − ⟨s(v_{0}), …, s(v_{q})⟩

and thus

∂_{q+1} ∘ D_{q} + D_{q−1} ∘ ∂_{q} = t_{q} − s_{q}

for all q > 0. It follows that t_{q}(z) − s_{q}(z) = ∂_{q+1} (D_{q}(z)) for any q-cycle z of K, and therefore s_{∗}([z]) = t_{∗}([z]). Thus s_{∗} = t_{∗} as homomorphisms from H_{q}(K) to H_{q}(L), as required.

7.2 The Homology of Barycentric Subdivisions

We shall show that the homology groups of a simplicial complex are isomorphic to those of its first barycentric subdivision.

We recall that the vertices of the first barycentric subdivision K′ of a simplicial complex K are the barycentres σ̂ of the simplices σ of K, and that K′ consists of the simplices spanned by σ̂_{0}, σ̂_{1}, …, σ̂_{q}, where σ_{0}, σ_{1}, …, σ_{q} ∈ K and σ_{j−1} is a proper face of σ_{j} for j = 1, 2, …, q.

Lemma 7.3 Let K′ be the first barycentric subdivision of a simplicial complex K. Then a function ζ: Vert K′ → Vert K from the vertices of K′ to those of K represents a simplicial approximation to the identity map of |K| if and only if it sends the barycentre of any simplex of K to some vertex of that simplex.

Proof If ζ represents a simplicial approximation to the identity map of |K| then ζ(σ̂) ∈ σ for any σ ∈ K, and hence ζ(σ̂) is a vertex of σ.

Conversely suppose that the function ζ sends the barycentre of any simplex of K to a vertex of that simplex. Let τ be a simplex of K′. Then it follows from the definition of K′ that the interior of τ is contained in the interior of some simplex σ of K, and the vertices of τ are barycentres of faces of σ. Then ζ must map the vertices of τ to vertices of σ, and hence ζ represents a simplicial map from K′ to K. Moreover this simplicial map is a simplicial approximation to the identity map, since the interior of τ is contained in σ and ζ maps the interior of τ into σ.

It follows from Lemma 7.3 that there exist simplicial approximations ζ: K′ → K to the identity map of |K|: such a simplicial approximation can be obtained by choosing, for each σ ∈ K, a vertex v_{σ} of σ, and defining ζ(σ̂) = v_{σ}.

Suppose that ζ: K′ → K and θ: K′ → K are both simplicial approximations to the identity map of |K|. Then ζ and θ are contiguous (Lemma 7.1), and therefore the homomorphisms ζ_{∗} and θ_{∗} of homology groups induced by ζ and θ must coincide. It follows that there is a well-defined natural homomorphism ν_{K}: H_{q}(K′) → H_{q}(K) of homology groups which coincides with ζ_{∗} for any simplicial approximation ζ: K′ → K to the identity map of |K|.

Theorem 7.4 The natural homomorphism ν_{K}: H_{q}(K′) → H_{q}(K) is an isomorphism for any simplicial complex K.

Proof Let M be the simplicial complex consisting of some simplex σ together with all of its faces. Then H_{0}(M) ≅ ℤ, H_{0}(M′) ≅ ℤ, and H_{q}(M) = 0 = H_{q}(M′) for all q > 0 (see Proposition 5.4 and the following example). Let v be a vertex of M. If θ: M′ → M is any simplicial approximation to the identity map of |M| then θ(v) = v. But the homology class of ⟨v⟩ generates both H_{0}(M) and H_{0}(M′). It follows that θ_{∗}: H_{0}(M′) → H_{0}(M) is an isomorphism, and thus ν_{M}: H_{q}(M′) → H_{q}(M) is an isomorphism for all q.

We now use induction on the number of simplices in K to prove the theorem in the general case. It therefore suffices to prove that the required result holds for a simplicial complex K under the additional assumption that the result is valid for all proper subcomplexes of K.

Let σ be a simplex of K whose dimension equals the dimension of K. Then σ is not a face of any other simplex of K, and therefore K \ {σ} is a subcomplex of K. Let M be the subcomplex of K consisting of the simplex σ, together with all of its faces. We have already proved the result in the special case when K = M. Thus we only need to verify the result in the case when M is a proper subcomplex of K. In that case K = L∪M, where L = K \{σ}.

Let ζ: K′ → K be a simplicial approximation to the identity map of |K|. Then the restrictions ζ|L′, ζ|M′ and ζ|L′ ∩ M′ of ζ to L′, M′ and L′ ∩ M′ are simplicial approximations to the identity maps of |L|, |M| and |L| ∩ |M| respectively. Therefore the diagram

0⟶C_{q}(L′∩M′)⟶C_{q}(L′)⊕C_{q}(M′)  ⟶  C_{q}(K′)⟶0
↓^{ζ|L′∩M′}  ↓^{(ζ|L′)⊕(ζ|M′)}  ↓^{ζ}
0⟶ C_{q}(L∩M) ⟶ C_{q}(L)⊕C_{q}(M)  ⟶  C_{q}(K) ⟶0

commutes, and its rows are short exact sequences. But the restrictions ζ|L′, ζ|M′ and ζ|L′ ∩ M′ of ζ to L′, M′ and L′ ∩ M′ are simplicial approximations to the identity maps of |L|, |M| and |L| ∩ |M| respectively, and therefore induce the natural homomorphisms ν_{L}, ν_{M} and ν_{L∩M}. We therefore obtain a commutative diagram

H_{q}(L′∩M′)⟶H_{q}(L′)⊕H_{q}(M′)⟶H_{q}(K′)⟶^{αq}H_{q−1}(L′∩M′)⟶H_{q−1}(L′)⊕H_{q−1}(M′)
↓^{ν_{L∩M}}  ↓^{ν_{L}⊕ν_{M}}  ↓^{ν_{K}}  ↓^{ν_{L∩M}}  ↓^{ν_{L}⊕ν_{M}}
H_{q}(L∩M) ⟶ H_{q}(L)⊕H_{q}(M) ⟶ H_{q}(K) ⟶^{αq} H_{q−1}(L∩M) ⟶ H_{q−1}(L)⊕H_{q−1}(M)

in which the rows are exact sequences, and are the Mayer-Vietoris sequences corresponding to the decompositions K = L ∪ M and K′ = L′ ∪ M′ of K and K′. But the induction hypothesis ensures that the homomorphisms ν_{L}, ν_{M} and ν_{L∩M} are isomorphisms, since L, M and L ∩ M are all proper subcomplexes of K. It now follows directly from the Five-Lemma (Lemma 6.3) that ν_{K}: H_{q}(K′) → H_{q}(K) is also an isomorphism, as required.

We refer to the isomorphism ν_{K}: H_{q}(K′) → H_{q}(K) as the canonical isomorphism from the qth homology group of K′ to that of K.

For each j > 0, we define the canonical isomorphism ν_{K,j}: H_{q}(K^{(j)}) → H_{q}(K) from the homology groups of the jth barycentric subdivision K^{(j)} of K to those of K itself to be the composition of the natural isomorphisms

H_{q}(K^{(j)}) → H_{q}(K^{(j−1)}) → ⋯ → H_{q}(K′) → H_{q}(K)

induced by appropriate simplicial approximations to the identity map of |K|. Note that if i ≤ j then ν_{K,i}^{−1} ∘ ν_{K,j} is induced by a composition of simplicial approximations to the identity map of |K|. But any composition of simplicial approximations to the identity map is itself a simplicial approximation to the identity map (Corollary 4.10). We deduce the following result.

Lemma 7.5 Let K be a simplicial complex, let i and j be positive integers satisfying i ≤ j. Then ν_{K,j} = ν_{K,i} ∘ ζ_{∗} for some simplicial approximation ζ: K^{(j)} → K^{(i)} to the identity map of |K|.

7.3 Continuous Maps and Induced Homomorphisms

Proposition 7.6 Any continuous map f: |K| → |L| between the polyhedra of simplicial complexes K and L induces a well-defined homomorphism f_{∗}: H_{q}(K) → H_{q}(L) of homology groups such that f_{∗} = s_{∗} ∘ ν_{K,i}^{−1} for any simplicial approximation s: K^{(i)} → L to the map f, where s_{∗}: H_{q}(K^{(i)}) → H_{q}(L) is the homomorphism induced by the simplicial map s and ν_{K,i}: H_{q}(K^{(i)}) → H_{q}(K) is the canonical isomorphism.

Proof The Simplicial Approximation Theorem (Theorem 4.11) guarantees the existence of a simplicial approximation s: K^{(i)} → L to the map f defined on the ith barycentric subdivision K^{(i)} of K for some sufficiently large i. Thus it only remains to verify that if s: K^{(i)} → L and t: K^{(j)} → L are both simplicial approximations to the map f then s_{∗} ∘ ν_{K,i}^{−1} = t_{∗} ∘ ν_{K,j}^{−1}.

Suppose that i ≤ j. Then ν_{K,i}^{−1}ν_{K,j} = ζ_{∗} for some simplicial approximation ζ: K^{(j)} → K^{(i)} to the identity map of |K| (Lemma 7.5). Thus s_{∗} ∘ ν_{K,i}^{−1} = s_{∗} ∘ ζ_{∗} ∘ ν_{K,j}^{−1} = (s ∘ ζ)_{∗} ∘ ν_{K,j}^{−1}. Moreover ζ: K^{(j)} → K^{(i)} and s: K^{(i)} → L are simplicial approximations to the identity map of |K| and to f: |K| → |L| respectively, and therefore s ∘ ζ: K^{(j)} → L is a simplicial approximation to f: |K| → |L| (Corollary 4.10). But then s∘ζ and t are simplicial approximations to the same continuous map, and thus are contiguous simplicial maps from K^{(j)} to L (Lemma 7.1). It follows that (s ∘ ζ)_{∗} and t_{∗} coincide as homomorphisms from H_{q}(K^{(j)}) to H_{q}(L) (Lemma 7.2), and therefore s_{∗} ∘ ν_{K,i}^{−1} = t_{∗} ∘ ν_{K,j}^{−1}, as required.

Proposition 7.7 Let K, L and M be simplicial complexes and let f: |K| → |L| and g: |L| → |M| be continuous maps. Then the homomorphisms f_{∗}, g_{∗} and (g ∘ f)_{∗} of homology groups induced by the maps f, g and g ∘ f satisfy (g ∘ f)_{∗} = g_{∗} ∘ f_{∗}.

Proof Let t: L^{(m)} → M be a simplicial approximation to g and let s: K^{(j)} → L^{(m)} be a simplicial approximation to f. Now the canonical isomorphism ν_{L,m} from H_{q}(L^{(m)}) to H_{q}(L) is induced by some simplicial approximation to the identity map of |L|. It follows that ν_{L,m} ∘s_{∗} is induced by some simplicial approximation to f (Corollary 4.10), and therefore f_{∗} = ν_{L,m} ∘ s_{∗} ∘ ν_{K,j}^{−1}. Also g_{∗} = t_{∗} ∘ ν_{L,m}^{−1}. It follows that g_{∗} ∘ f_{∗} = t_{∗} ∘ s_{∗} ∘ ν_{K,j}^{−1} = (t ∘ s)_{∗} ∘ ν_{K,j}^{−1}. But t ∘ s: K^{(j)} → M is a simplicial approximation to g ∘ f (Corollary 4.10). Thus (g ∘ f)_{∗} = g_{∗} ∘ f_{∗}, as required.

Corollary 7.8 If the polyhedra |K| and |L| of simplicial complexes K and L are homeomorphic then the homology groups of K and L are isomorphic.

Proof Let h: |K| → |L| be a homeomorphism. Then h_{∗}: H_{q}(K) → H_{q}(L) is an isomorphism whose inverse is (h^{−1})_{∗}: H_{q}(L) → H_{q}(K).

One can make use of induced homomorphisms in homology theory in order to prove the Brouwer Fixed Point Theorem (Theorem 4.14) in all dimensions. The Brouwer Fixed Point Theorem is a consequence of the fact that there is no continuous map r: Δ → ∂Δ from an n-simplex Δ to its boundary ∂Δ with the property that r(x) = x for all x ∈ ∂Δ (Proposition 4.13). Such a continuous map would induce homomorphisms r_{∗}: H_{q}(Δ) → H_{q}(∂Δ) of homology groups for all non-negative integers q, and r_{∗} ∘ i_{∗} would be the identity automorphism of H_{q}(∂Δ) for all q, where i_{∗}: H_{q}(∂Δ) → H_{q}(Δ) is induced by the inclusion map i: ∂Δ ↪ Δ. But this would imply that r_{∗}: H_{q}(Δ) → H_{q}(∂Δ) was surjective for all non-negative integers q, which is impossible, since H_{n−1}(Δ) = 0 and H_{n−1}(∂Δ) ≅ ℤ when n ≥ 2 (and H_{n−1}(Δ) ≅ ℤ and H_{n−1}(∂Δ) ≅ ℤ ⊕ ℤ when n = 1). We conclude therefore that there is no continuous map r: Δ → ∂Δ that fixes all points of ∂Δ, and therefore the Brouwer Fixed Point Theorem is satisfied in all dimensions.

We next show that homotopic maps between the polyhedra of simplicial complexes induce the same homomorphisms of homology groups. For this we require the following result.

Lemma 7.9 For any simplicial complex L there is some ε > 0 with the following property: given continuous maps f: |K| → |L| and g: |K| → |L| defined on the polyhedron of some simplicial complex K, where f(x) is within a distance ε of g(x) for all x ∈ |K|, there exists a simplicial map defined on K^{(i)} for some sufficiently large i which is a simplicial approximation to both f and g.

Proof An application of the Lebesgue Lemma shows that there exists ε > 0 such that the open ball of radius 2ε about any point of |L| is contained in st_{L}(b) for some vertex b of L. Let f: |K| → |L| and g: |K| → |L| be continuous maps. Suppose that f(x) is within a distance ε of g(x) for all x ∈ |K|. Another application of the Lebesgue Lemma (to the open cover of |K| by preimages of open balls of radius ε) shows that there exists δ > 0 such that any subset S of |K| whose diameter is less than δ is mapped by f into an open ball of radius ε about some point of |L|, and is therefore mapped by g into an open ball of of radius 2ε about that point. But then f(S) ⊂ st_{L}(b) and g(S) ⊂ st_{L}(b) for some vertex b of L. Now choose i such that μ(K^{(i)}) < \frac{1}{2}δ. As in the proof of the Simplicial Approximation Theorem (Theorem 4.11) we see that, for every vertex a of K^{(i)}, the diameter of st_{K^{(i)}}(a) is less than δ, and hence f(st_{K^{(i)}}(a)) ⊂ st_{L}(s(a)) and g(st_{K^{(i)}}(a)) ⊂ st_{L}(s(a)) for some vertex s(a) of L. It then follows from Proposition 4.9 that the function s: Vert K^{(i)} → Vert L constructed in this manner is the required simplicial approximation to f and g.

Theorem 7.10 Let K and L be simplicial complexes and let f: |K| → |L| and g: |K| → |L| be continuous maps from |K| to |L|. Suppose that f and g are homotopic. Then the induced homomorphisms f_{∗} and g_{∗} from H_{q}(K) to H_{q}(L) are equal for all q.

Proof Let F: |K| × [0, 1] → |L| be a homotopy with F (x, 0) = f(x) and F (x, 1) = g(x), and let ε > 0 be given. Using the well-known fact that continuous functions defined on compact metric spaces are uniformly continuous (which is easily proved using the Lebesgue Lemma), we see that there exists some δ > 0 such that if |s − t| < δ then the distance from F (x, s) to F (x, t) is less than ε. Let f_{i}(x) = F (x, t_{i}) for i = 0, 1, …, r, where t_{0}, t_{1}, …, t_{r} have been chosen such that 0 = t_{0} < t_{1} < ⋯ < t_{r} = 1 and t_{i} − t_{i−1} < δ for all i. Then f_{i−1}(x) is within a distance ε of f_{i}(x) for all x ∈ |K|. Using Lemma 7.9, we see that the maps f_{i−1} and f_{i} from |K| to |L| have a common simplicial approximation, and thus f_{i−1} and f_{i} induce the same homomorphisms of homology groups, provided that ε > 0 has been chosen sufficiently small. It follows that the maps f and g induce the same homomorphisms of homology groups, as required.

7.4 Homotopy Equivalence

Definition Let X and Y be topological spaces. A continuous map f: X → Y is said to be a homotopy equivalence if there exists a continuous map g: Y → X such that g ∘ f is homotopic to the identity map of X and f ∘ g is homotopic to the identity map of Y. The spaces X and Y are said to be homotopy equivalent if there exists a homotopy equivalence from X to Y.

Lemma 7.11 A composition of homotopy equivalences is itself a homotopy equivalence.

Proof Let X, Y and Z be topological spaces, and let f: X → Y and h: Y → Z be homotopy equivalences. Then there exist continuous maps g: Y → X and k: Z → Y such that g ∘ f ≃ i_{X}, f ∘ g ≃ i_{Y}, k ∘ h ≃ i_{Y} and h ∘ k ≃ i_{Z}, where i_{X}, i_{Y} and i_{Z} denote the identity maps of the spaces X, Y, Z. Then (g ∘ k) ∘ (h ∘ f) = g ∘ (k ∘ h) ∘ f ≃ g ∘ i_{Y} ∘ f = g ∘ f ≃ i_{X} and (h ∘ f) ∘ (g ∘ k) = h ∘ (f ∘ g) ∘ k ≃ h ∘ i_{Y} ∘ k = h ∘ k ≃ i_{Z}. Thus h ∘ f: X → Z is a homotopy equivalence from X to Z.

Lemma 7.12 Let f: |K| → |L| be a homotopy equivalence between the polyhedra of simplicial complexes K and L. Then, for each non-negative integer q, the induced homomorphism f_{∗}: H_{q}(K) → H_{q}(L) of homology groups is an isomorphism.

Proof There exists a continuous map g: |L| → |K| such that g ∘ f is homotopic to the identity map of |K| and f ∘ g is homotopic to the identity map of |L|. It follows that the induced homomorphisms (g ∘ f)_{∗}: H_{q}(K) → H_{q}(K) and (f ∘ g)_{∗}: H_{q}(L) → H_{q}(L) are the identity automorphisms of H_{q}(K) and H_{q}(L) for each q. But (g ∘ f)_{∗} = g_{∗} ∘ f_{∗} and (f ∘ g)_{∗} = f_{∗} ∘ g_{∗}. It follows that f_{∗}: H_{q}(K) → H_{q}(L) is an isomorphism with inverse g_{∗}: H_{q}(L) → H_{q}(K).

Definition A subset A of a topological space X is said to be a deformation retract of X if there exists a continuous map H: X × [0, 1] → X such that H(x, 0) = x and H(x, 1) ∈ A for all x ∈ X and H(a, 1) = a for all a ∈ A.

Thus a subset A of a topological space X is a deformation retract of X if and only if there exists a function r: X → A such that r(a) = a for all a ∈ A and r is homotopic in X to the identity map of X.

Example The unit sphere S^{n−1} in ℝ^{n} is a deformation retract of ℝ^{n} \ {0}. For if H(x, t) = (1 − t + t/|x|)x for all x ∈ ℝ^{n} \ {0} and t ∈ [0, 1] then H(x, 0) = x and H(x, 1) ∈ S^{n−1} for all x ∈ ℝ^{n} \ {0} and H(x, 1) = x when x ∈ S^{n−1}.

If A is a deformation retract of a topological space X then the inclusion map i: A ↪ X is a homotopy equivalence.

Theorem 7.13 The spaces ℝ^{m} and ℝ^{n} are not homeomorphic if m ≠ n.

Proof Let S^{m−1} and S^{n−1} denote the unit spheres in ℝ^{m} and ℝ^{n} respectively. Then S^{m−1} and S^{n−1} are homeomorphic to the polyhedra of simplicial complexes K and L respectively. Let i_{m}: S^{m−1} → ℝ^{m} \ {0} be the inclusion map and let r_{n}: ℝ^{n} \ {0} → S^{n−1} be the map that sends x ∈ ℝ^{n} \ {0} to (1/|x|)x. Then both i_{m}: S^{m−1} → ℝ^{m} \ {0} and r_{n}: ℝ^{n} \ {0} → S^{n−1} are homotopy equivalences.

Suppose that there were to exist a homeomorphism h: ℝ^{m} → ℝ^{n}. Let f(x) = h(x)−h(0) for all x ∈ ℝ^{m} \{0}. Then f: ℝ^{m} \{0} → ℝ^{n} \{0} would also be a homeomorphism, and therefore r_{n} ∘ f ∘ i_{m}: S^{m−1} → S^{n−1} would be a homotopy equivalence. Thus if ℝ^{m} and ℝ^{n} were homeomorphic then S^{m−1} and S^{n−1} would be homotopy equivalent, and therefore the homology groups of the simplicial complexes K and L would be isomorphic. But H_{q}(K) ≅ ℤ when q = 0 and q = m − 1 and H_{q}(K) = 0 for all other values of q, whereas H_{q}(L) ≅ ℤ when q = 0 and q = n−1 and H_{q}(L) = 0 for all other values of q. Thus if m ≠ n then the homology groups of the simplicial complexes K and L are not isomorphic, and therefore ℝ^{m} and ℝ^{n} are not homeomorphic.