Singular Homology and Mayer–Vietoris

7A.9 Homology in Brief introduced homology for spaces with a triangulation, computed it by linear algebra, and stated the facts needed for the Poincaré conjecture. This chapter builds the theory properly, for all spaces, and proves those facts. The construction is singular homology: chains are formal sums of continuous maps of simplices, so no triangulation is needed and functoriality is automatic. Its power comes from three properties, proved here:

• Homotopy invariance: homotopic maps induce the same maps on homology.
• Exactness: a pair $(X, A)$ gives a long exact sequence linking the homology of $A$, of $X$, and of $X$ relative to $A$.
• Excision: relative homology does not see what happens deep inside $A$. Equivalently, the Mayer–Vietoris sequence computes the homology of a union from the pieces and their intersection.

With these three, the homology of spheres follows by induction, and with it the classical theorems of topology in every dimension: Brouwer's fixed point theorem, the Jordan–Brouwer separation theorem, invariance of domain and of dimension. Cellular homology then computes everything from a CW structure (7A.6 CW Complexes and Cellular Approximation) with almost no work, and proves that the Euler characteristic is a topological invariant. The homology of lens spaces and of the Poincaré homology sphere, which 7A.12 The Poincaré Conjecture, Precisely uses to explain what the conjecture is about, comes out in a few lines.

By the end of this chapter you will be able to:

• define singular homology, prove its functoriality and homotopy invariance (with the prism operator), and compute $H_0$;
• prove the zig-zag lemma and use the long exact sequence of a pair; state excision and use it for good pairs, $H_*(X, A) \cong \tilde H_*(X/A)$;
• derive and apply the Mayer–Vietoris sequence: spheres, wedges, connected sums;
• define the degree of a map $S^n \to S^n$, compute it for reflections and the antipodal map, and prove the hairy ball and Brouwer theorems with it;
• prove the Jordan–Brouwer separation theorem and invariance of domain;
• compute cellular homology of $\mathbb{R}P^n$, lens spaces and surfaces, prove that $\chi$ is a topological invariant, and state the Lefschetz fixed point theorem.

10.1Inside or outside

10.2Singular homology

Let $\Delta^n = \{(t_0, \dots, t_n) \in \mathbb{R}^{n+1} : t_i \geq 0, \sum t_i = 1\}$ be the standard $n$-simplex, with vertices $e_0, \dots, e_n$ and faces $\Delta^{n-1} \to \Delta^n$ given by inserting a zero in position $i$.

The groups $C_n(X)$ are enormous (one generator for each continuous map), but homology is computable, and for simplicial complexes agrees with the simplicial homology of 7A.9 Homology in Brief (Hatcher, Theorem 2.27). The advantage is functoriality: a continuous map $f : X \to Y$ sends $\sigma$ to $f\circ\sigma$, commuting with $\partial$, so it induces $f_* : H_n(X) \to H_n(Y)$, with $(g\circ f)_* = g_*\circ f_*$ and $\mathrm{id}_* = \mathrm{id}$. Homeomorphic spaces have isomorphic homology.

$H_0$ counts path components. A 0-simplex is a point, a 1-simplex a path, and $\partial\sigma = \sigma(e_1) - \sigma(e_0)$. So two points are homologous iff they are joined by a path, and $H_0(X)$ is free abelian on the path components. The reduced homology $\tilde H_*$ replaces $H_0$ by the kernel of the augmentation $\sum n_ix_i \mapsto \sum n_i$; for nonempty $X$, $H_0 = \tilde H_0\oplus\mathbb{Z}$ and $\tilde H_n = H_n$ for $n > 0$, and a point has $\tilde H_* = 0$.

10.3Exact sequences

For $A \subseteq X$, the relative chains $C_n(X, A) = C_n(X)/C_n(A)$ inherit $\partial$, and their homology $H_n(X, A)$ consists of chains whose boundary lies in $A$, modulo chains in $A$ and boundaries. Intuitively, it is the homology of $X$ with $A$ "collapsed".

Applied to $0 \to C(A) \to C(X) \to C(X, A) \to 0$, the lemma gives the long exact sequence of the pair

where $\partial$ takes a relative cycle to its boundary, a cycle in $A$.

Excision. The deep property of singular homology is that relative homology is local.

The proof (Hatcher, §2.1) shows that every chain is homologous to one made of small simplices, each lying in $A$ or in $B$, by repeated barycentric subdivision: subdividing a simplex into smaller ones is chain homotopic to the identity, and after enough subdivisions every simplex is small (the Lebesgue number lemma, 2B.3 Compactness). The idea is the same as in van Kampen's theorem (7A.5 Computing π₁): break loops, or chains, into small pieces that each lie in one open set.

For a good pair (a closed $A$ that is a deformation retract of a neighbourhood, for instance a CW pair), excision gives

For instance $(D^n, S^{n-1})$ is good with $D^n/S^{n-1} \cong S^n$.

Mayer–Vietoris. If $X = \operatorname{int}A\cup\operatorname{int}B$, the chains in $X$ made of simplices lying in $A$ or in $B$ compute $H_*(X)$ (the subdivision argument again), and $0 \to C(A\cap B) \to C(A)\oplus C(B) \to C(A) + C(B) \to 0$, with maps $x \mapsto (x, -x)$ and $(y, z) \mapsto y + z$, is exact. The zig-zag lemma gives the Mayer–Vietoris sequence

the homological van Kampen theorem. The connecting map takes a cycle $z = a + b$ ($a$ in $A$, $b$ in $B$) to $\partial a = -\partial b$, a cycle in $A\cap B$: cut the cycle along the intersection and take the cut. The same holds with reduced homology when $A\cap B \neq \emptyset$.

10.4The homology of spheres

Immediate consequences:

• $S^n$ is not contractible, and $S^n \not\simeq S^m$ for $n \neq m$.
• Invariance of dimension: $\mathbb{R}^n \not\cong \mathbb{R}^m$ for $n \neq m$, because $\mathbb{R}^n\setminus\{0\} \simeq S^{n-1}$.
• No retraction $D^n \to S^{n-1}$: it would give $\tilde H_{n-1}(S^{n-1}) = \mathbb{Z}$ factoring through $\tilde H_{n-1}(D^n) = 0$. Hence Brouwer's fixed point theorem in every dimension, by the argument of 7A.4 The Fundamental Group (a fixed-point-free map gives a retraction), without the smooth approximation used in 7A.8 Smooth Topology.

Degree. A map $f : S^n \to S^n$ induces multiplication by an integer on $H_n(S^n) = \mathbb{Z}$, its degree $\deg f$. It is a homotopy invariant, multiplicative, and $\deg\mathrm{id} = 1$. It agrees with the smooth degree of 7A.8 Smooth Topology (counting preimages of a regular value with signs; Hatcher, Proposition 2.30).

The hairy ball theorem. If $v$ is a nowhere vanishing tangent vector field on $S^n$, then $f_t(x) = \cos(\pi t)x + \sin(\pi t)\frac{v(x)}{|v(x)|}$ is a homotopy from the identity to the antipodal map. So $1 = (-1)^{n+1}$, and $n$ is odd. On even-dimensional spheres every vector field vanishes somewhere, as 7A.8 Smooth Topology proved with Poincaré–Hopf.

10.5Jordan–Brouwer and invariance of domain

In dimension $2$ more is true, the Schoenflies theorem: each component is homeomorphic to an open disc. In dimension $3$ this fails: Alexander's horned sphere (1924) is an embedded $S^2$ in $S^3$ whose outside is not simply connected. Wild embeddings like this are one reason three-dimensional topology needs care, and why the smooth or piecewise-linear categories, where Alexander's theorem says every embedded 2-sphere bounds a ball on each side, are used (10A.5 The Prime Decomposition).

Consequences: a manifold has a well-defined dimension; a manifold's boundary points (those with half-space neighbourhoods) are distinct from its interior points, so the boundary of a manifold is well defined (7A.3 Manifolds and Surfaces); and a continuous injection of a closed $n$-manifold into another closed connected $n$-manifold is onto (Exercise 10.19).

10.6Cellular homology

For a CW complex $X$, the long exact sequences of the pairs $(X^n, X^{n-1})$ and the fact that $X^n/X^{n-1}$ is a wedge of $n$-spheres (one for each $n$-cell) give:

The proof is a diagram chase with the long exact sequences, using $H_k(X^n, X^{n-1}) = 0$ for $k \neq n$ (Hatcher, Theorem 2.35). In practice the degrees are easy to read off.

This is the invariance promised in 7A.3 Manifolds and Surfaces and 7A.6 CW Complexes and Cellular Approximation: Euler's $V - E + F = 2$ for every polyhedral sphere, and $\chi = 2 - 2g$ for every cell structure on $\Sigma_g$.

Connected sums. For closed connected $n$-manifolds $M$, $N$, $n \geq 3$: removing a ball does not change $H_k$ for $k < n - 1$, and Mayer–Vietoris over the gluing sphere gives $H_k(M\#N) \cong H_k(M)\oplus H_k(N)$ for $0 < k < n$ (for orientable $M$, $N$; Exercise 10.20). In dimension three, $H_1(M\#N) = H_1(M)\oplus H_1(N)$, the abelianisation of the free product $\pi_1(M)*\pi_1(N)$ (7A.5 Computing π₁, 10A.5 The Prime Decomposition).

10.7The Lefschetz fixed point theorem

For a map $f : X \to X$ of a finite CW complex, the Lefschetz number is

For $f = \mathrm{id}$, $L = \chi(X)$; for a contractible $X$, $L(f) = 1$ for every $f$, recovering Brouwer. Every map $\mathbb{R}P^{2k} \to \mathbb{R}P^{2k}$ has a fixed point (its rational homology is that of a point), and a map $S^n \to S^n$ without fixed points has degree $(-1)^{n+1}$, the antipodal map's. The proof approximates $f$ by a cellular map that moves every cell off itself if there are no fixed points, so that the trace on cellular chains, which equals $L(f)$ by the same telescoping as for $\chi$, is zero. For smooth maps on closed manifolds, $L(f)$ is the signed count of fixed points, the analogue of Poincaré–Hopf (7A.8 Smooth Topology), which is the case of the flow of a vector field for a short time.

10.8History

Poincaré introduced homology in Analysis Situs (1895) and its supplements, with Betti numbers and torsion; Emmy Noether, around 1925, recast it in terms of groups. Singular homology was developed by Lefschetz and by Eilenberg (1944), who gave its modern definition; Eilenberg and Steenrod axiomatised homology in 1945 by homotopy invariance, exactness and excision. Walther Mayer (1929) and Leopold Vietoris (1930) found their sequence. Camille Jordan stated his curve theorem in 1887; Oswald Veblen gave the first rigorous proof in 1905, and Brouwer proved the higher-dimensional separation theorem and invariance of domain in 1911–12. Lefschetz proved his fixed point theorem in 1926–37, and Alexander's horned sphere appeared in 1924.

10.9Exercises

Solution

Removing $e_j$ then $e_i$ ($i < j$, where $e_i$ is still in position $i$) gives sign $(-1)^j(-1)^i$. Removing $e_i$ first, $e_j$ moves to position $j - 1$, so removing it gives $(-1)^i(-1)^{j-1}$. The two terms cancel.

Solution

$\partial(\text{path}) = \text{end} - \text{start} = F(x, 1) - F(x, 0) = g(x) - f(x)$, and $P\partial x = 0$. For $\sigma$: $P\sigma = F(\sigma\times\mathrm{id})|_{[v_0, w_0, w_1]} - F(\sigma\times\mathrm{id})|_{[v_0, v_1, w_1]}$. Its boundary consists of the top edge $[w_0, w_1]$ ($= g\sigma$), the bottom edge $[v_0, v_1]$ with a minus sign ($-f\sigma$), the vertical edges $[v_0, w_0]$ and $[v_1, w_1]$, which are $P$ of the endpoints with signs giving $-P\partial\sigma$, and the diagonal $[v_0, w_1]$ twice with opposite signs. So $\partial P\sigma = g\sigma - f\sigma - P\partial\sigma$.

Solution

At $H_n(B)$: $j_*i_* = (ji)_* = 0$. If $j_*[b] = 0$, then $j(b) = \partial c'$; lift $c' = j(b')$, then $j(b - \partial b') = 0$, so $b - \partial b' = i(a)$, and $a$ is a cycle since $i$ is injective and $i(\partial a) = \partial b = 0$; $[b] = i_*[a]$. At $H_{n-1}(A)$: $i_*\delta[c] = [i(a)] = [\partial b] = 0$. If $i_*[a] = 0$, $i(a) = \partial b$, and $c = j(b)$ is a cycle with $\delta[c] = [a]$.

Solution

Orient both hemispheres by projecting them to the same disc in $\mathbb{R}^n$; then both have boundary the equator with the same orientation, $d(D_1) = d(D_2) = [S^{n-1}]$, and $d(D_1 - D_2) = 0$. The $n$-cycles are multiples of $D_1 - D_2$ (the kernel of $\mathbb{Z}^2 \to \mathbb{Z}$, $(a, b) \mapsto a + b$), and there are no $(n + 1)$-cells, so $H_n = \mathbb{Z}(D_1 - D_2)$. The reflection maps $D_1$ to $D_2$ and vice versa, preserving the orientations induced from the equator, so $D_1 - D_2 \mapsto D_2 - D_1$.

Solution

$S^2\setminus A$ is connected by part (a) of Proposition 10.7. It is the disjoint union of $U$, $V$ and $\operatorname{int}D$. $U$ is open in $S^2\setminus A$, and closed in it: its closure in $S^2$ is $U\cup\partial U$, and $\partial U \subseteq \Sigma$ with $\partial U\cap D = \emptyset$ by the choice of $D$, so $\partial U \subseteq A$; and the closure of $U$ in $S^2\setminus A$ is $U$. A nonempty proper clopen subset of a connected space is impossible, since $V \neq \emptyset$ lies outside $U$. So every point of $\Sigma$ is in $\partial U$.

Solution

By invariance of domain in charts, $f$ is open, so $f(M)$ is open; it is compact, hence closed. $N$ is connected, so $f(M) = N$, and an open continuous bijection is a homeomorphism.

Solution

(a) $H_k(M, M\setminus B) = H_k(D^n, S^{n-1})$ is $\mathbb{Z}$ for $k = n$ and $0$ otherwise. The sequence $H_n(M) \to H_n(M, M\setminus B) \to H_{n-1}(M\setminus B) \to H_{n-1}(M) \to 0$ with the first map onto gives the isomorphism in degree $n - 1$; lower degrees are isomorphisms since the relative groups vanish. (b) $M\#N = M'\cup N'$ with intersection a collar $\simeq S^{n-1}$, contributing only in degrees $0$ and $n - 1$; the map $H_{n-1}(S^{n-1}) \to H_{n-1}(M')\oplus H_{n-1}(N')$ is zero, because the gluing sphere is the boundary of $M'$ (it is the boundary of a chain representing the fundamental class of $M'$ relative to its boundary) and likewise of $N'$; this gives the splitting. (c) $\mathbb{Z}, (\mathbb{Z}/2)^2, 0, \mathbb{Z}$; and $\mathbb{Z}, \mathbb{Z}^2, \mathbb{Z}^2, \mathbb{Z}$.

Solution

(a) With rational coefficients the torsion disappears: $H_0 = \mathbb{Q}$ and all others vanish (the top group $H_{2k} = 0$ since $2k$ is even). So $L(f) = 1 \neq 0$. (b) A fixed point would need $(-x_1, x_0, -x_3, x_2) = \lambda(x_0, x_1, x_2, x_3)$, so $x_1 = -\lambda x_0$ and $x_0 = \lambda x_1 = -\lambda^2x_0$, and similarly for $x_2$, $x_3$; this forces $\lambda^2 = -1$, impossible over $\mathbb{R}$ unless all coordinates vanish. $H_3(\mathbb{R}P^3; \mathbb{Q}) = \mathbb{Q}$, and the map acts by $1$ there (it comes from a rotation of $S^3$), so $L = 1 - 1 = 0$, consistent.