Mayer-Vietoris Theorem

Overview

If you can break up a topological space into 2 subspaces in a nice way, then the Mayer-Vietoris Theorem gives a useful tool in computing singular homology groups.

This comes from the idea of using the (hopefully known) homology of the smaller pieces that you can use together with some algebraic reasoning about the larger space.

The Mayer-Vietoris theorem is the homology analogue to the Seifert-Van Kampen Theorem.

Statement

Let $X$ be a topological space and $U,V\subset X$ be subspaces such that

$\text{int}U\cup \text{int}V=X.$

Then there is a long exact sequence

where

$\partial :H_n(X)⟶H_n(X,V)⟶H_n(U,U\cap V)⟶H_{n-1}(U\cap V)$

Proof

todo Lots of diagrams… need some better explanations of what the maps above are.