Course 421: Academic Year 1998-9 Part III: Simplicial Homology Theory

D. R. Wilkins

Contents

9 Simplicial Complexes 2 9.1 Geometrical Independence . . . . . . . . . . . . . . . . . . . . 2 9.2 Simplicial Complexes in Euclidean Spaces . . . . . . . . . . . 3 9.3 Simplicial Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.4 Barycentric Subdivision of a Simplicial Complex . . . . . . . . 6 9.5 The Simplicial Approximation Theorem . . . . . . . . . . . . . 9 9.6 The Brouwer Fixed Point Theorem . . . . . . . . . . . . . . . 11

10 Simplicial Homology Groups 14 10.1 The Chain Groups of a Simplicial Complex . . . . . . . . . . . 14 10.2 Boundary Homomorphisms . . . . . . . . . . . . . . . . . . . . 17 10.3 The Homology Groups of a Simplicial Complex . . . . . . . . 18 10.4 Simplicial Maps and Induced Homomorphisms . . . . . . . . . 20 10.5 Connectedness and H_{0}(K) . . . . . . . . . . . . . . . . . . . . 20

11 Introduction to Homological Algebra 24 11.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 24 11.2 Chain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . 26 11.3 The Mayer-Vietoris Sequence . . . . . . . . . . . . . . . . . . 30

12 The Topological Invariance of Simplicial Homology Groups 31 12.1 Contiguous Simplicial Maps . . . . . . . . . . . . . . . . . . . 31 12.2 The Homology of Barycentric Subdivisions . . . . . . . . . . . 33 12.3 Continuous Maps and Induced Homomorphisms . . . . . . . . 35 12.4 Homotopy Equivalence . . . . . . . . . . . . . . . . . . . . . . 38

9 Simplicial Complexes

9.1 Geometrical Independence

Definition Points v_{0}, v_{1}, …, v_{q} in some Euclidean space ℝ^{k} are said to be geometrically independent (or affine independent) if the only solution of the linear system

\{ \begin{matrix}∑_{j=0}^{q} λ_{j}v_{j} & = & 0, \\ ∑_{j=0}^{q} λ_{j} & = & 0 \end{matrix}

is the trivial solution λ_{0} = λ_{1} = ⋯ = λ_{q} = 0.

It is straightforward to verify that v_{0}, v_{1}, …, v_{q} are geometrically independent if and only if the vectors v_{1} − v_{0}, v_{2} − v_{0}, …, v_{q} − v_{0} are linearly independent. It follows from this that any set of geometrically independent points in ℝ^{k} has at most k + 1 elements. Note also that if a set consists of geometrically independent points in ℝ^{k}, then so does every subset of that set.

Definition A q-simplex in ℝ^{k} is defined to be a set of the form

{∑_{j=0}^{q} t_{j}v_{j} : 0 ≤ t_{j} ≤ 1 for j = 0, 1, …, q and ∑_{j=0}^{q} t_{j} = 1} ,

where v_{0}, v_{1}, …, v_{q} are geometrically independent points of ℝ^{k}. The points v_{0}, v_{1}, …, v_{q} are referred to as the vertices of the simplex. The non-negative integer q is referred to as the dimension of the simplex.

Note that a 0-simplex in ℝ^{k} is a single point of ℝ^{k}, a 1-simplex in ℝ^{k} is a line segment in ℝ^{k}, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron.

Let σ be a q-simplex in ℝ^{k} with vertices v_{0}, v_{1}, …, v_{q}. If x is a point of σ then there exist real numbers t_{0}, t_{1}, …, t_{q} such that

∑_{j=0}^{q} t_{j}v_{j} = x, ∑_{j=0}^{q} t_{j} = 1 and 0 ≤ t_{j} ≤ 1 for j = 0, 1, …, q.

Moreover t_{0}, t_{1}, …, t_{q} are uniquely determined: if ∑_{j=0}^{q} s_{j}v_{j} = ∑_{j=0}^{q} t_{j}v_{j} and

∑_{j=0}^{q} s_{j} = 1 = ∑_{j=0}^{q} t_{j}, then ∑_{j=0}^{q}(t_{j} − s_{j})v_{j} = 0 and ∑_{j=0}^{q}(t_{j} − s_{j}) = 0, hence

t_{j} − s_{j} = 0 for all j, since v_{0}, v_{1}, …, v_{q} are geometrically independent. We refer to t_{0}, t_{1}, …, t_{q} as the barycentric coordinates of the point x of σ.

Lemma 9.1 Let q be a non-negative integer, let σ be a q-simplex in ℝ^{m}, and let τ be a q-simplex in ℝ^{n}, where m ≥ q and n ≥ q. Then σ and τ are homeomorphic.

Proof Let v_{0}, v_{1}, …, v_{q} be the vertices of σ, and let w_{0}, w_{1}, …, w_{q} be the vertices of τ. The required homeomorphism h: σ → τ is given by

h (∑_{j=0}^{q} t_{j}v_{j}) = ∑_{j=0}^{q} t_{j}w_{j}

for all t_{0}, t_{1}, …, t_{q} satisfying 0 ≤ t_{j} ≤ 1 for j = 0, 1, …, q and ∑_{j=0}^{q} t_{j} = 1.

A homeomorphism between two q-simplices defined as in the above proof is referred to as a simplicial homeomorphism.

9.2 Simplicial Complexes in Euclidean Spaces

Definition Let σ and τ be simplices in ℝ^{k}. We say that τ is a face of σ if the set of vertices of τ is a subset of the set of vertices of σ. A face of σ is said to be a proper face if it is not equal to σ itself. An r-dimensional face of σ is referred to as an r-face of σ. A 1-dimensional face of σ is referred to as an edge of σ.

Note that any simplex is a face of itself. Also the vertices and edges of any simplex are by definition faces of the simplex.

Definition A finite collection K of simplices in ℝ^{k} is said to be a simplicial complex if the following two conditions are satisfied:—

• if σ is a simplex belonging to K then every face of σ also belongs to K,

• if σ_{1} and σ_{2} are simplices belonging to K then either σ_{1} ∩ σ_{2} = ∅ or else σ_{1} ∩ σ_{2} is a common face of both σ_{1} and σ_{2}.

The dimension of a simplicial complex K is the greatest non-negative integer n with the property that K contains an n-simplex. The union of all the simplices of K is a compact subset |K| of ℝ^{k} referred to as the polyhedron of K. (The polyhedron is compact since it is both closed and bounded in ℝ^{k}.)

Example Let K_{σ} consist of some n-simplex σ together with all of its faces. Then K_{σ} is a simplicial complex of dimension n, and |K_{σ}| = σ.

Lemma 9.2 Let K be a simplicial complex, and let X be a topological space. A function f: |K| → X is continuous on the polyhedron |K| of K if and only if the restriction of f to each simplex of K is continuous on that simplex.

Proof If a topological space can be expressed as a finite union of closed subsets, then a function is continuous on the whole space if and only if its restriction to each of the closed subsets is continuous on that closed set. The required result is a direct application of this general principle.

We shall denote by Vert K the set of vertices of a simplicial complex K (i.e., the set consisting of all vertices of all simplices belonging to K). A collection of vertices of K is said to span a simplex of K if these vertices are the vertices of some simplex belonging to K.

Definition Let K be a simplicial complex in ℝ^{k}. A subcomplex of K is a collection L of simplices belonging to K with the following property:—

• if σ is a simplex belonging to L then every face of σ also belongs to L.

Note that every subcomplex of a simplicial complex K is itself a simplicial complex.

Definition Let v_{0}, v_{1}, …, v_{q} be the vertices of a q-simplex σ in some Euclidean space ℝ^{k}. We define the interior of the simplex σ to be the set of all

points of σ that are of the form ∑_{j=0}^{q} t_{j}v_{j}, where t_{j} > 0 for j = 0, 1, …, q and

∑_{j=0}^{q} t_{j} = 1. One can readily verify that the interior of the simplex σ consists

of all points of σ that do not belong to any proper face of σ. (Note that, if σ ∈ ℝ^{k}, then the interior of a simplex defined in this fashion will not coincide with the topological interior of σ unless dim σ = k.)

Note that any point of a simplex σ belongs to the interior of a unique face of σ. Indeed let v_{0}, v_{1}, …, v_{q} be the vertices of σ, and let x ∈ σ. Then

∑_{j=0}^{q} t_{j}v_{j}, where 0 ≤ t_{j} ≤ 1 for j = 0, 1, …, q and ∑_{j=0}^{q} t_{j} = 1. The unique face

of σ containing x in its interior is then the face spanned by those vertices v_{j} for which t_{j} > 0.

Lemma 9.3 Let K be a finite collection of simplices in some Euclidean space ℝ^{k}, and let |K| be the union of all the simplices in K. Then K is a simplicial complex (with polyhedron |K|) if and only if the following two conditions are satisfied:— • K contains the faces of its simplices,

• every point of |K| belongs to the interior of a unique simplex of K.

Proof Suppose that K is a simplicial complex. Then K contains the faces of its simplices. We must show that every point of |K| belongs to the interior of a unique simplex of K. Let x ∈ |K|. Then x belongs to the interior of a face σ of some simplex of K (since every point of a simplex belongs to the interior of some face). But then σ ∈ K, since K contains the faces of all its simplices. Thus x belongs to the interior of at least one simplex of K.

Suppose that x were to belong to the interior of two distinct simplices σ and τ of K. Then x would belong to some common face σ ∩ τ of σ and τ (since K is a simplicial complex). But this common face would be a proper face of one or other of the simplices σ and τ (since σ ≠ τ), contradicting the fact that x belongs to the interior of both σ and τ. We conclude that the simplex σ of K containing x in its interior is uniquely determined, as required.

Conversely, we must show that any collection of simplices satisfying the given conditions is a simplicial complex. Since K contains the faces of all its simplices, it only remains to verify that if σ and τ are any two simplices of K with non-empty intersection then σ ∩ τ is a common face of σ and τ.

Let x ∈ σ ∩ τ. Then x belongs to the interior of a unique simplex ω of K. However any point of σ or τ belongs to the interior of a unique face of that simplex, and all faces of σ and τ belong to K. It follows that ω is a common face of σ and τ, and thus the vertices of ω are vertices of both σ and τ. We deduce that the simplices σ and τ have vertices in common, and that every point of σ ∩ τ belongs to the common face ρ of σ and τ spanned by these common vertices. But this implies that σ ∩ τ = ρ, and thus σ ∩ τ is a common face of both σ and τ, as required.

Definition A triangulation (K, h) of a topological space X consists of a simplicial complex K in some Euclidean space, together with a homeomorphism h: |K| → X mapping the polyhedron |K| of K onto X.

The polyhedron of a simplicial complex is a compact Hausdorff space. Thus if a topological space admits a triangulation then it must itself be a compact Hausdorff space.

Lemma 9.4 Let X be a Hausdorff topological space, let K be a simplicial complex, and let h: |K| → X be a bijection mapping |K| onto X. Suppose that the restriction of h to each simplex of K is continuous on that simplex. Then the map h: |K| → X is a homeomorphism, and thus (K, h) is a triangulation of X.

Proof Each simplex of K is a closed subset of |K|, and the number of simplices of K is finite. It follows from Lemma 9.2 that h: |K| → X is continuous. Also the polyhedron |K| of K is a compact topological space. But every continuous bijection from a compact topological space to a Hausdorff space is a homeomorphism. Thus (K, h) is a triangulation of X.

9.3 Simplicial Maps

Definition A simplicial map φ: K → L between simplicial complexes K and L is a function φ: Vert K → Vert L from the vertex set of K to that of L such that φ(v_{0}), φ(v_{1}), …, φ(v_{q}) span a simplex belonging to L whenever v_{0}, v_{1}, …, v_{q} span a simplex of K.

Note that a simplicial map φ: K → L between simplicial complexes K and L can be regarded as a function from K to L: this function sends a simplex σ of K with vertices v_{0}, v_{1}, …, v_{q} to the simplex φ(σ) of L spanned by the vertices φ(v_{0}), φ(v_{1}), …, φ(v_{q}).

A simplicial map φ: K → L also induces in a natural fashion a continuous map φ: |K| → |L| between the polyhedra of K and L, where

φ (∑_{j=0}^{q} t_{j}v_{j}) = ∑_{j=0}^{q} t_{j}φ(v_{j})

whenever 0 ≤ t_{j} ≤ 1 for j = 0, 1, …, q, ∑_{j=0}^{q} t_{j} = 1, and v_{0}, v_{1}, …, v_{q} span a

simplex of K. The continuity of this map follows immediately from a straightforward application of Lemma 9.2. Note that the interior of a simplex σ of K is mapped into the interior of the simplex φ(σ) of L.

There are thus three equivalent ways of describing a simplicial map: as a function between the vertex sets of two simplicial complexes, as a function from one simplicial complex to another, and as a continuous map between the polyhedra of two simplicial complexes. In what follows, we shall describe a simplicial map using the representation that is most appropriate in the given context.

9.4 Barycentric Subdivision of a Simplicial Complex

Let σ be a q-simplex in ℝ^{k} with vertices v_{0}, v_{1}, …, v_{q}. The barycentre of σ is defined to be the point

σ̂ = \frac{1}{q + 1}(v_{0} + v_{1} + ⋯ + v_{q}).

Let σ and τ be simplices in some Euclidean space. If σ is a proper face of τ then we denote this fact by writing σ < τ.

A simplicial complex K_{1} is said to be a subdivision of a simplicial complex K if |K_{1}| = |K| and each simplex of K_{1} is contained in a simplex of K.

Definition Let K be a simplicial complex in some Euclidean space ℝ^{k}. The first barycentric subdivision K′ of K is defined to be the collection of simplices in ℝ^{k} whose vertices are σ̂_{0}, σ̂_{1}, …, σ̂_{r} for some sequence σ_{0}, σ_{1}, …, σ_{r} of simplices of K with σ_{0} < σ_{1} < ⋯ < σ_{r}. Thus the set of vertices of K′ is the set of all the barycentres of all the simplices of K.

Note that every simplex of K′ is contained in a simplex of K. Indeed if σ_{0}, σ_{1}, …, σ_{r} ∈ K satisfy σ_{0} < σ_{1} < ⋯ < σ_{r} then the simplex of K′ spanned by σ̂_{0}, σ̂_{1}, …, σ̂_{r}, is contained in the simplex σ_{r} of K.

Proposition 9.5 Let K be a simplicial complex in some Euclidean space, and let K′ be the first barycentric subdivision of K. Then K′ is itself a simplicial complex, and |K′| = |K|.

Proof We prove the result by induction on the number of simplices in K. The result is clear when K consists of a single simplex, since that simplex must then be a point and therefore K′ = K. We prove the result for a simplicial complex K, assuming that it holds for all complexes with fewer simplices.

It is clear from the definition of the barycentric subdivision K′ that any face of a simplex of K′ must itself belong to K′. We must verify that any two simplices of K′ are disjoint or else intersect in a common face.

Choose a simplex σ of K for which dim σ = dim K, and let L = K \ {σ}. Then L is a subcomplex of K, since σ is not a proper face of any simplex of K. Now L has fewer simplices than K. It follows from the induction hypothesis that L′ is a simplicial complex and |L′| = |L|. Also it follows from the definition of K′ that K′ consists of the following simplices:—

• the simplices of L′,

• the barycentre σ̂ of σ,

• simplices σ̂ρ whose vertex set is obtained by adjoining σ̂ to the vertex set of some simplex ρ of L′, where the vertices of ρ are barycentres of proper faces of σ.

By checking all possible intersections of simplices of the above types, it is easy to verify that any two simplices of K′ intersect in a common face. Indeed any two simplices of L′ intersect in a common face, since L′ is a simplicial complex. If ρ_{1} and ρ_{2} are simplices of L′ whose vertices are barycentres of proper faces of σ, then ρ_{1} ∩ ρ_{2} is a common face of ρ_{1} and ρ_{2} which is of this type, and σ̂ρ_{1} ∩ σ̂ρ_{2} = σ̂(ρ_{1} ∩ ρ_{2}). Thus σ̂ρ_{1} ∩ σ̂ρ_{2} is a common face of σ̂ρ_{1} and σ̂ρ_{2}. Also any simplex τ of L′ is disjoint from the barycentre σ̂ of σ, and σ̂ρ ∩ τ = ρ ∩ τ. We conclude that K′ is indeed a simplicial complex.

It remains to verify that |K′| = |K|. Now |K′| ⊂ |K|, since every simplex of K′ is contained in a simplex of K. Let x be a point of the chosen simplex σ. Then there exists a point y belonging to a proper face of σ and some t ∈ [0, 1] such that x = (1−t)σ̂+ty. But then y ∈ |L|, and |L| = |L′| by the induction hypothesis. It follows that y ∈ ρ for some simplex ρ of L′ whose vertices are barycentres of proper faces of σ. But then x ∈ σ̂ρ, and therefore x ∈ |K′|. Thus |K| ⊂ |K′|, and hence |K′| = |K|, as required.

We define (by induction on j) the jth barycentric subdivision K^{(j)} of K to be the first barycentric subdivision of K^{(j−1)} for each j > 1.

Lemma 9.6 Let σ be a q-simplex and let τ be a face of σ. Let σ̂ and τ̂ be the barycentres of σ and τ respectively. If all the 1-simplices (edges) of σ have length not exceeding d for some d > 0 then

|σ̂ − τ̂| ≤ \frac{qd}{q + 1}.

Proof Let v_{0}, v_{1}, …, v_{q} be the vertices of σ. Let x and y be points of σ.

We can write y = ∑_{j=0}^{q} t_{j}v_{j}, where 0 ≤ t_{i} ≤ 1 for i = 0, 1, …, q and ∑_{j=0}^{q} t_{j} = 1.

Now

|x − y| = |∑_{i=0}^{q} t_{i}(x − v_{i})| ≤ ∑_{i=0}^{q} t_{i}|x − v_{i}|
≤ maximum (|x − v_{0}|, |x − v_{1}|, …, |x − v_{q}|) .

Applying this result with x = σ̂ and y = τ̂, we find that

|σ̂ − τ̂| ≤ maximum (|σ̂ − v_{0}|, |σ̂ − v_{1}|, …, |σ̂ − v_{q}|) .

But

σ̂ = \frac{1}{q + 1}v_{i} + \frac{q}{q + 1}z_{i}

for i = 0, 1, …, q, where z_{i} is the barycentre of the (q − 1)-face of σ opposite to v_{i}, given by

z_{i} = \frac{1}{q} ∑_{j≠i} v_{j}.

Moreover z_{i} ∈ σ. It follows that

|σ̂ − v_{i}| = \frac{q}{q + 1}|z_{i} − v_{i}| ≤ \frac{qd}{q + 1}

for i = 1, 2, …, q, and thus

|σ̂ − τ̂| ≤ maximum (|σ̂ − v_{0}|, |σ̂ − v_{1}|, …, |σ̂ − v_{q}|) ≤ \frac{qd}{q + 1},

as required.

The mesh μ(K) of a simplicial complex K is the length of the longest edge of K.

Lemma 9.7 Let K be a simplicial complex in ℝ^{k} for some k, and let n be the dimension of K. Let K′ be the first barycentric subdivision of K. Then

μ(K′) ≤ \frac{n}{n + 1}μ(K).

Proof A 1-simplex of K′ is of the form (τ̂, σ̂), where σ is a q-simplex of K for some q ≤ n and τ is a proper face of σ. Then

|τ̂ − σ̂| ≤ \frac{q}{q + 1}μ(K) ≤ \frac{n}{n + 1}μ(K)

by Lemma 9.6, as required.

It follows directly from the above lemma that lim_{j→+∞} μ(K^{(j)}) = 0, where

K^{(j)} is the jth barycentric subdivision of K.

9.5 The Simplicial Approximation Theorem

Definition Let f: |K| → |L| be a continuous map between the polyhedra of simplicial complexes K and L. A simplicial map s: K → L is said to be a simplicial approximation to f if, for each x ∈ |K|, s(x) is an element of the unique simplex of L which contains f(x) in its interior.

Note that if s: K → L is a simplicial approximation to f: |K| → |L| then s and f are homotopic. Indeed the map from |K| × [0, 1] to |L| sending (x, t) to (1 − t)f(x) + ts(x) is a well-defined homotopy between f and s.

Definition Let K be a simplicial complex, and let x ∈ |K|. The star st_{K}(x) of x in K is the union of the interiors of all simplices of K that contain the point x.

Lemma 9.8 Let K be a simplicial complex and let x ∈ |K|. Then the star st_{K}(x) of x is open in |K|, and x ∈ st_{K}(x).

Proof Every point of |K| belongs to the interior of a unique simplex of K (Lemma 9.3). It follows that the complement |K| \ st_{K}(x) of st_{K}(x) in |K| is the union of the interiors of those simplices of K that do not contain the point x. But if a simplex of K does not contain the point x, then the same is true of its faces. Moreover the union of the interiors of all the faces of some simplex is the simplex itself. It follows that |K| \ st_{K}(x) is the union of all simplices of K that do not contain the point x. But each simplex of K is closed in |K|. It follows that |K| \ st_{K}(x) is a finite union of closed sets, and is thus itself closed in |K|. We deduce that st_{K}(x) is open in |K|. Also x ∈ st_{K}(x), since x belongs to the interior of at least one simplex of K.

Proposition 9.9 A function s: Vert K → Vert L between the vertex sets of simplicial complexes K and L is a simplicial map, and a simplicial approximation to some continuous map f: |K| → |L|, if and only if f (st_{K}(v)) ⊂ st_{L} (s(v)) for all vertices v of K.

Proof Let s: K → L be a simplicial approximation to f: |K| → |L|, let v be a vertex of K, and let x ∈ st_{K}(v). Then x and f(x) belong to the interiors of unique simplices σ ∈ K and τ ∈ L. Moreover v must be a vertex of σ, by definition of st_{K}(v). Now s(x) must belong to τ (since s is a simplicial approximation to the map f), and therefore s(x) must belong to the interior of some face of τ. But s(x) must belong to the interior of s(σ), since x is in the interior of σ. It follows that s(σ) must be a face of τ, and therefore s(v) must be a vertex of τ. Thus f(x) ∈ st_{L}(s(v)). We conclude that if s: K → L is a simplicial approximation to f: |K| → |L|, then f (st_{K}(v)) ⊂ st_{L} (s(v)).

Conversely let s: Vert K → Vert L be a function with the property that f (st_{K}(v)) ⊂ st_{L} (s(v)) for all vertices v of K. Let x be a point in the interior of some simplex of K with vertices v_{0}, v_{1}, …, v_{q}. Then x ∈ st_{K}(v_{j}) and hence f(x) ∈ st_{L} (s(v_{j})) for j = 0, 1, …, q. It follows that each vertex s(v_{j}) must be a vertex of the unique simplex τ ∈ L that contains f(x) in its interior. In particular, s(v_{0}), s(v_{1}), …, s(v_{q}) span a face of τ, and s(x) ∈ τ.

We conclude that the function s: Vert K → Vert L represents a simplicial map which is a simplicial approximation to f: |K| → |L|, as required.

Corollary 9.10 If s: K → L and t: L → M are simplicial approximations to continuous maps f: |K| → |L| and g: |L| → |M|, where K, L and M are simplicial complexes, then t ∘ s: K → M is a simplicial approximation to g ∘ f: |K| → |M|.

Theorem 9.11 (Simplicial Approximation Theorem) Let K and L be simplicial complexes, and let f: |K| → |L| be a continuous map. Then, for some sufficiently large integer j, there exists a simplicial approximation s: K^{(j)} → L to f defined on the jth barycentric subdivision K^{(j)} of K.

Proof The collection consisting of the stars st_{L}(w) of all vertices w of L is an open cover of |L|, since each star st_{L}(w) is open in |L| (Lemma 9.8) and the interior of any simplex of L is contained in st_{L}(w) whenever w is a vertex of that simplex. It follows from the continuity of the map f: |K| → |L| that the collection consisting of the preimages f^{−1}(st_{L}(w)) of the stars of all vertices w of L is an open cover of |K|. It then follows from the Lebesgue Lemma that there exists some δ > 0 with the property that every subset of |K| whose diameter is less than δ is mapped by f into st_{L}(w) for some vertex w of L.

Now the mesh μ(K^{(j)}) of the jth barycentric subdivision of K tends to zero as j → +∞, since

μ(K^{(j)}) ≤ (\frac{dim K}{dim K + 1})^{j} μ(K)

for all j (Lemma 9.7). Thus we can choose j such that μ(K^{(j)}) < \frac{1}{2}δ. If v is a vertex of K^{(j)} then each point of st_{K^{(j)}}(v) is within a distance \frac{1}{2}δ of v, and hence the diameter of st_{K^{(j)}}(v) is at most δ. We can therefore choose, for each vertex v of K^{(j)} a vertex s(v) of L such that f(st_{K^{(j)}}(v)) ⊂ st_{L}(s(v)). In this way we obtain a function s: Vert K^{(j)} → Vert L from the vertices of K^{(j)} to the vertices of L. It follows directly from Proposition 9.9 that this is the desired simplicial approximation to f.

9.6 The Brouwer Fixed Point Theorem

Definition Let K be a simplicial complex which is a subdivision of some n-dimensional simplex Δ. We define a Sperner labelling of the vertices of K to be a function, labelling each vertex of K with an integer between 0 and n, with the following properties:— • for each j ∈ {0, 1, …, n}, there is exactly one vertex of Δ labelled by

j,

• if a vertex v of K belongs to some face of Δ, then some vertex of that face has the same label as v.

Lemma 9.12 (Sperner’s Lemma) Let K be a simplicial complex which is a subdivision of an n-simplex Δ. Then, for any Sperner labelling of the vertices of K, the number of n-simplices of K whose vertices are labelled by 0, 1, …, n is odd.

Proof Given integers i_{0}, i_{1}, …, i_{q} between 0 and n, let N(i_{0}, i_{1}, …, i_{q}) denote the number of q-simplices of K whose vertices are labelled by i_{0}, i_{1}, …, i_{q} (where an integer occurring k times in the list labels exactly k vertices of the simplex). We must show that N(0, 1, …, n) is odd.

We prove the result by induction on the dimension n of the simplex Δ; it is clearly true when n = 0. Suppose that the result holds in dimensions less than n. For each simplex σ of K of dimension n, let p(σ) denote the number of (n − 1)-faces of σ labelled by 0, 1, …, n − 1. If σ is labelled by 0, 1, …, n then p(σ) = 1; if σ is labelled by 0, 1, …, n − 1, j, where j < n, then p(σ) = 2; in all other cases p(σ) = 0. Therefore

∑_{{σ∈K \atop dim σ=n}} p(σ) = N(0, 1, …, n) + 2 ∑_{j=0}^{n−1} N(0, 1, …, n − 1, j).

Now the definition of Sperner labellings ensures that the only (n − 1)-face of Δ containing simplices of K labelled by 0, 1, …, n − 1 is that with vertices labelled by 0, 1, …, n − 1. Thus if M is the number of (n − 1)-simplices of K labelled by 0, 1, …, n−1 that are contained in this face, then N(0, 1, …, n− 1) − M is the number of (n − 1)-simplices labelled by 0, 1, …, n − 1 that intersect the interior of Δ. It follows that

∑_{{σ∈K \atop dim σ=n}} p(σ) = M + 2(N(0, 1, …, n − 1) − M),

since any (n−1)-simplex of K that is contained in a proper face of Δ must be a face of exactly one n-simplex of K, and any (n−1)-simplex that intersects the interior of Δ must be a face of exactly two n-simplices of K. On combining these equalities, we see that N(0, 1, …, n) − M is an even integer. But the induction hypothesis ensures that Sperner’s Lemma holds in dimension n−1, and thus M is odd. It follows that N(0, 1, …, n) is odd, as required.

Proposition 9.13 Let Δ be an n-simplex with boundary ∂Δ. Then there does not exist any continuous map r: Δ → ∂Δ with the property that r(x) = x for all x ∈ ∂Δ.

Proof Suppose that such a map r: Δ → ∂Δ were to exist. It would then follow from the Simplicial Approximation Theorem (Theorem 9.11) that there would exist a simplicial approximation s: K → L to the map r, where L is the simplicial complex consisting of all of the proper faces of Δ, and K is the jth barycentric subdivision, for some sufficiently large j, of the simplicial complex consisting of the simplex Δ together with all of its faces.

If v is a vertex of K belonging to some proper face Σ of Δ then r(v) = v, and hence s(v) must be a vertex of Σ, since s: K → L is a simplicial approximation to r: Δ → ∂Δ. In particular s(v) = v for all vertices v of Δ. Thus if v ↦ m(v) is a labelling of the vertices of Δ by the integers 0, 1, …, n, then v ↦ m(s(v)) is a Sperner labelling of the vertices of K. Thus Sperner’s Lemma (Lemma 9.12) guarantees the existence of at least one n-simplex σ of K labelled by 0, 1, …, n. But then s(σ) = Δ, which is impossible, since Δ is not a simplex of L. We conclude therefore that there cannot exist any continuous map r: Δ → ∂Δ satisfying r(x) = x for all x ∈ ∂Δ.

Theorem 9.14 (Brouwer Fixed Point Theorem) A continuous map f: E^{n} → E^{n} sending a closed n-dimensional ball E^{n} into itself has at least one fixed point (i.e., there exists x ∈ E^{n} for which f(x) = x).

Proof Suppose that the map f: E^{n} → E^{n} had no fixed point. For each x ∈ E, let q(x) be the point at which the half line starting at f(x) and passing through x intersects the boundary sphere S^{n−1} of E^{n}. Then q: E^{n} → S^{n−1} would be continuous, and q(x) = x for all x ∈ S^{n−1}. But the closed ndimensional ball E^{n} is homeomorphic to an n-simplex Δ. Therefore the map q: E^{n} → S^{n−1} would correspond under some homeomorphism h: Δ → E^{n} to a continuous map r: Δ → ∂Δ mapping Δ onto its boundary ∂Δ, where h(r(y)) = q(h(y)) for all y ∈ Δ. Moreover r(y) = y for all y ∈ ∂Δ. However Proposition 9.13 shows that there does not exist any continuous map r: Δ → ∂Δ with this property. Therefore the map f must have at least one fixed point, as required.

10 Simplicial Homology Groups

10.1 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 10.1 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 10.2 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}).

10.2 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 10.2 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 10.3 ∂_{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.

10.3 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 10.3. 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 10.4 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 10.2, 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.

10.4 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 10.2.) 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_{∗}.

10.5 Connectedness and H_{0}(K)

Lemma 10.5 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 10.6 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 10.7 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 10.8 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}. There is therefore 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 10.7). 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). Thus We conclude that 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 10.8 with Lemmas 10.5 and 10.6 we obtain immediately the following result.

Corollary 10.9 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|.

11 Introduction to Homological Algebra

11.1 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 11.1 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 11.2 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 11.2.

Lemma 11.3 (Five-Lemma) Suppose that the rows of the commutative diagram of Proposition 11.2 are exact sequences and that ψ_{1},ψ_{2},ψ_{4} and ψ_{5} are isomorphisms. Then ψ_{3} is also an isomorphism.

11.2 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 11.4 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−1}, 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 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 in described in Lemma 11.4. 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 11.5 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_{∗}) satisfing 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.

11.3 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 11.4 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 11.5 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.

12 The Topological Invariance of Simplicial Homology Groups

12.1 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 12.1 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 12.2 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.

12.2 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 12.3 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 12.3 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 12.1), 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 12.4 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 10.4 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 11.3) 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 9.10). We deduce the following result.

Lemma 12.5 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|.

12.3 Continuous Maps and Induced Homomorphisms

Proposition 12.6 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 9.11) 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 12.5). 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 9.10). 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 12.1). It follows that (s ∘ ζ)_{∗} and t_{∗} coincide as homomorphisms from H_{q}(K^{(j)}) to H_{q}(L) (Lemma 12.2), and therefore s_{∗} ∘ ν_{K,i}^{−1} = t_{∗} ∘ ν_{K,j}^{−1}, as required.

Proposition 12.7 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 9.10), 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 9.10). Thus (g ∘ f)_{∗} = g_{∗} ∘ f_{∗}, as required.

Corollary 12.8 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 9.14) 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 9.13). 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 12.9 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 9.11) 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 9.9 that the function s: Vert K^{(i)} → Vert L constructed in this manner is the required simplicial approximation to f and g.

Theorem 12.10 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 12.9, 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.

12.4 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 12.11 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 12.12 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_{∗} ∘ h_{∗} and (f ∘ g)_{∗} = g_{∗} ∘ h_{∗}. 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 12.13 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.