Mayer–Vietoris sequences

Overview

For a pair of subspaces $A,B\subseteq X$ such that $X$ is the union of the interiors of $A,B$, the Mayer–Vietoris sequence is a long exact sequence of the form

$$\cdots \to H_p(A\cap B)\to H_p(A)\oplus H_p(B)\to H_p(X){\to }_{\partial }{H}_{p-1}(A\cap B)\to \cdots \to H_0(X)\to 0.$$

Like the long exact sequence in relative homology, the Mayer–Vietoris sequence is useful for calculations and sometimes more convenient to use.

For $A\cap B$ path-connected, we have the abelianized statement of (Theorem) van Kampen: the $H_1$ terms of the reduced Mayer–Vietoris sequence give an isomorphism

$$H_1(X)\cong \frac{H_1(A)\oplus H_1(B)}{\text{Im}((j_0{)}_{\ast },-(j_1{)}_{\ast }):H_1(A\cap B)\to H_1(A)\oplus H_p(B)},$$

where $(j_0{)}_{\ast },(j_1{)}_{\ast }$ are the maps induced on relative homology by inclusions $j_0:A\cap B\to A$ and $j_1:A\cap B\to B$.

---

Preliminaries: Small chains

Recall the definition of a $\mathcal{U}$-small chain from the proof of the excision theorem.

Small chain

Let $X$ be any space and $\mathcal{U}$ be a collection of subsets of $X$. We say that a singular simplex $\sigma :{\Delta }^p\to X$ is $\mathcal{U}$-small if there exists $U\in \mathcal{U}$ such that $\text{Im}(\sigma )\subseteq U$. The subgroup spanned by $\mathcal{U}$-small simplices is denoted

$$C_p^{\mathcal{U}}(X)\subseteq C_p(X).$$

Note that if $\sigma :{\Delta }^p\to X$ is $\mathcal{U}$-small, then so is $\sigma \circ {\delta }^i:{\Delta }^{p-1}\to X$ and hence $\partial \sigma \in C_p^{\mathcal{U}}$. Therefore we have a chain complex $(C_{\ast }^{\mathcal{U}}(X),\delta )$ given by restricting the boundary on $C_{\ast }(X)$, and the inclusion

$$C_{\ast }^{\mathcal{U}}(X)\to C_{\ast }(X)$$

is a chain map.

---

Derivation of the Mayer–Vietoris sequence

Let $U_0,U_1$ be subsets of a topological space $X$ with corresponding inclusions $i_0:U_0\to X$ and $i_1:U_1\to X$, respectively, and set $\mathcal{U}=\{U_0,U_1\}$.

• The abelian group $C_p^{\mathcal{U}}(X)$ is precisely the free abelian group on continuous maps $\sigma :{\Delta }^p\to X$ with image contained in either $U_0$ or $U_1$, so it is equal to the image of the homomorphism induced by inclusions

$$C_p(U_0)\oplus C_p(U_1)\to C_p(X),(\alpha ,\beta )\mapsto (i_0{)}_{\ast }\alpha +(i_1{)}_{\ast }\beta .$$

• The kernel of this homomorphism has the formal linear combinations $({\sum }_{\sigma }{n}_{\sigma }\sigma ,{\sum }_{\sigma ’}{n}_{{\sigma }^{\prime }}\sigma )$ of simplices $\sigma \in U_0$ and $\sigma \in U_1$ such that:
  • (i) When $\sigma ({\Delta }^p)\in U_0\cap U_1$, we have $n_{\sigma }+n_{\sigma ’}=0$;
  • (ii) When $\sigma ({\Delta }^p)\notin U_0\cap U_1$, the coefficients are 0.
• The kernel is equal to the image of the map

$$C_p(U_{01})\to C_p(U_0)\oplus C_p(U_1),c\mapsto ((j_0{)}_{\ast }(c),-(j_1{)}_{\ast }(c)),$$

where $U_{01}=U_0\cap U_1$ and $j_0:U_{01}\to U_0$ and $j_1:U_{01}\to U_1$ are the inclusions.

• Each coordinate of the homomorphism above is injective, so we may define a short exact sequence

$$0\to C_p(U_{01})\to C_p(U_0)\oplus C_p(U_1)\to C_p^{\mathcal{U}}(X)\to 0.$$

In particular, we can view this as an SES of chain complexes by viewing the middle term as a chain complex with boundary map $(\alpha ,\beta )\mapsto (\partial \alpha ,\partial \beta )$. Therefore this SES induces a long exact sequence in homology.

• Assuming that the interiors of $U_0,U_1$ cover $X$, i.e., $X=\text{int}(U_0)\cup \text{int}(U_1)$, the homology of $C_{\ast }^{\mathcal{U}}(X)$ agrees with the homology of $X$ and we obtain the unreduced Meyer–Vietoris sequence

$$\cdots \to H_p(U_{01})\to H_p(U_0)\oplus H_p(U_1)\to H_p(X){\to }_{\delta }{H}_{p-1}(U_{01})\to \cdots ,$$

where the two mappings are

$$(i_0{)}_{\ast }+(i_1{)}_{\ast }:H_p(U_0)\oplus H_p(U_1)\to H_p(X)$$ $$((j_0{)}_{\ast },-(j_1{)}_{\ast }):H_p(U_{01})\to H_p(U_0)\oplus H_p(U_1).$$

---

The reduced Mayer–Vietoris sequence

The Mayer–Vietoris sequence for reduced singular homology is obtained by augmenting the short exact sequence

$$0\to C_0(U_{01})\to C_0(U_0)\oplus C_0(U_1)\to C_0^{\mathcal{U}}(X)\to 0$$

using the mapping ${\tilde{\partial }}_0:C_0\to \mathbb{Z}$ defined by $\sigma \mapsto 1\in \mathbb{Z}$ for the first and last chain groups, and ${\tilde{\partial }}_0\oplus {\tilde{\partial }}_0$ in the middle group, which gives

$$0\to \mathbb{Z}\to \mathbb{Z}\oplus \mathbb{Z}\to \mathbb{Z}\to 0.$$

---

Examples

wip

---

Exercises

• Check that the SES that induces the Mayer–Vietoris sequence is indeed exact (see Hatcher, p. 150).

---

Code snippets

C^{\mathcal U}_*(X)