Mayer-Vietoris Theorem for Singular Homology

Subjects: Algebraic Topology
Links: Singular Homology, The Seifert-Van Kampen Theorem, The Zigzag Lemma, Simplicial Complexes, Abstract Simplicial Complexes

Def: Suppose $\mathcal{U}$ is an open cover of $X$. A singular chain $c$ is said to be $\mathcal{U}$-small if every singular simplex that appears in $c$ has image lying entirely in one of the open subsets in $\mathcal{U}$. Let $C_p^{\mathcal{U}}(X)$ denote the subgroup of $C_p(X)$ consisting of $\mathcal{U}$-small chains, and let $H_p^{\mathcal{U}}(X)$ denote the homology of the complex $C_{\bullet }^{\mathcal{U}}(X)$.

Def: If $\alpha =A(v_0,\dots ,v_p)$ is an affine singular $p$-simplex in some convex set $K\subseteq {\mathbb{R}}^m$ and $w$ is any point in $K$, we define an affine singular $(p+1)$-simplex $w\ast \alpha$ called the cone on $\alpha$ from $w$ by $$w* \alpha := w* A(v_0,\dots, v_p) = A(w, v_0,\dots, v_p). $$In other words, $w\ast \alpha :{\mathrm{\Delta }}_{p+1}\to K$ is the unique affine simplex that sends $e_0$ to $w$ and whose $0$th dace map is equal $\alpha$. We extend this operator to affine chains by linearity $$w* \left(\sum_i n_i\alpha_i\right) := \sum_i n_i (w *\alpha_i).$$It is not defined for arbitrary singular chains.

Lemma: If $c$ is an affine chain. then $$\partial(wc) + w\partial c= c,$$or equivalently, $$\partial(wc) = c-w\partial c.$$
Def: For any $k$-simplex $\sigma =[v_0,\dots ,v_k]\in {\mathbb{R}}^n$, we define the barycentre of $\sigma$ to be the points $b_{\sigma }\in \text{Int }\sigma$ whose barycentric coordinates are all equal: $$b_\sigma := \sum_{i = 0}^k \frac1{k+1} v_i. $$
Now we define an operator $s$ taking affine $p$-chains to affine $p$-chains, called the singular subdivision operator. For $p=0$, simply set $s=\text{id}$. For $p>0$, assuming that $s$ has been defined for chains of dimensions less than $p$, for any affine $p$-simplex $\alpha :{\mathrm{\Delta }}_p\to {\mathbb{R}}^n$ we set $$s\alpha := \alpha(b_p)* s (\partial \alpha), $$where $b_p$ is the barycentre of ${\mathrm{\Delta }}_p$, and extend linearly to affine chains.

Lemma: Suppose $\alpha :{\mathrm{\Delta }}_p\to {\mathbb{R}}^n$ is an affine simplex that is a homeomorphism onto a $p$-simplex $\sigma \subseteq {\mathbb{R}}^n$. Let $\beta :{\mathrm{\Delta }}_p\to {\mathbb{R}}^n$ be any one of the affine singular $p$-simplices that appear in the chains $s\alpha$.

• $\beta$ is an affine homeomorphism onto a $p$-simplex of the form $[b_p,\dots ,b_0]$, where each $b_i$ is the barycentre of an $i$-dimensional face of $\sigma$.
• The diameter of any such simplex $[b_p,\dots ,b_0]$ is at most $p/(p+1)$ times the diameter of $\sigma$.

Now we need to extend the singular subdivision operator to arbitrary, not necessarily affine, singular chains. For a singular $p$-simples $\sigma$ in any space $X$, note that $\sigma ={\sigma }_{\mathrm{\#}}{i}_p$, where $i_p:{\mathrm{\Delta }}_p\to {\mathrm{\Delta }}_p$ is the the identity map considered as an affine singular $p$-simplex in ${\mathrm{\Delta }}_p$, and ${\sigma }_{\mathrm{\#}}:C_p({\mathrm{\Delta }}_p)\to C_p(X)$ is the chain map obtained from the continuous map $\sigma :{\mathrm{\Delta }}_p\to X$. we define $$s\sigma := \sigma_# (si_p),$$and extend by linearity to all of $C_p(X)$. We can iterate $s$ to obtain operators $s^2:=s\circ s$ and more generally $s^{k+1}:=s^k\circ s$.

Lemma: The singular subdivision operators $s:C_p(X)\to C_p(X)$ have the following properties.

• $s\circ f_{\mathrm{\#}}=f_{\mathrm{\#}}\circ s$ for any continuous map $f$.
• $\partial \circ s=s\circ \partial$.
• Given an open cover $\mathcal{U}$ of $X$ and any $c\in C_p(X)$, there exists $m$ such that $s^{m}c\in C_p^{\mathcal{U}}(X)$.

The proof of the third fact, actually needs the Lebesgue Number Lemma

Prop: Suppose $\mathcal{U}$ is any open cover of $X$. The inclusion map $C_{\bullet }^{\mathcal{U}}(X)\to C_{\bullet }(X)$ induces a homology isomorphism $H_p^{\mathcal{U}}(X)\cong H_p(X)$ for all $p$.

We are given a space $X$ and two open subsets $U,V\subseteq X$ whose union is $X$. There are four inclusion maps

\usepackage{tikz-cd}
\usepackage{amsfonts, amsmath, amssymb}

\begin{document}
\begin{tikzcd}
& U\arrow[dr, "k"] & \\
U \cap V\arrow[ur, "i"]\arrow[dr, "j"] && X \\
&V\arrow[ur,"\ell"']&
\end{tikzcd}
\end{document}

Mayer-Vietoris Theorem: Let $X$ be a topological space, and let $U,V$ be open subsets of whose union is $X$. Then for each $p$ there is a homomorphism ${\partial }_{\ast }:H_p(X)\to H_{p-1}(U\cap V)$ such that the following sequence is exact: $$\cdots \stackrel{\partial_}{\longrightarrow}H_p(U \cap V) \stackrel{i_ \oplus j_}{\longrightarrow} H_p(U) \oplus H_p(V)\stackrel{k_- l_}{\longrightarrow}H_p(X)\stackrel{\partial_}{\longrightarrow} H_{p-1}(U\cap V)\stackrel{i_* \oplus j_*}{\longrightarrow}\cdots$$
Prop: Let $X_1,\dots ,X_k$ be spaces with nondegenerate base points. For every $p>0$, then $$H_p(X_1\vee \dots \vee X_k) = H_p(X_1) \oplus \dots \oplus H_p(X_k)$$