excision theorem

Overview

The Excision Theorem is a result from singular homology where “deleting” a “small enough” subspace (of a subspace) doesn’t affect the relative homology.

Statement

Let $(X,A)$ be a pair and $U\subset X$ such that $\overline{U}\subset \text{int}(A)$. Then there is an isomorphism

$H_{\bullet }(X\backslash U,A\backslash U)\simeq H_{\bullet }(X,A)$

induced by the inclusion of pairs

$(X\backslash U,A\backslash U)\hookrightarrow (X,A)$

Proof

todo - This is pretty hard and involved. Need argument about baricentric subdivision.