Homology theory

Definition

Let $C$ be the category of compact polyhedral pairs. A homology theory on $C$ is defined as follows.

Data

• For every nonnegative integer $n$, a functor $H_{n}:C\to Ab$ where $Ab$ denotes the category of Abelian groups
• For every positive integer, a natural transformation $\partial _{n}:H_{n}\to H_{n-1}\circ R$ where $R$ is the functor that sends $(X,A)$ to $(A,\emptyset )$.

Axioms

• Homotopy axiom: If $f_{0},f_{1}:(X,A)\to (Y,B)$ are homotopic, then $H_{n}(f_{0})=H_{n}(f_{1})$
• Exactness axiom: For every pair $(X,A)$ with inclusions $(A,\emptyset )\to (X,\emptyset )\to (X,A)$, there is a long exact sequence:

$\ldots \to H_{n}(A,\emptyset )\to H_{n}(X,\emptyset )\to H_{n}(X,A)\to H_{n-1}(A,\emptyset )\to \ldots$

• Excision axiom: For every open subset $U$ whose closure lies in the interior of $A$, the map of homotopy groups induced by the inclusion $(X-U,A-U)\to (X,A)$ is an isomorphism
• Dimension axiom: If $X$ is a one-point space, then $H_{n}(X,\emptyset )$ is trivial for all $n>0$. One calls $H_{0}(X,\emptyset )$ the coefficient group of the homology theory.

For a homology theory, the homology of a topological space $X$ is defined as the homology of the pair $(X,\emptyset )$.