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 $(A,\mathrm{\varnothing })$.

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 $(A,\mathrm{\varnothing })\to (X,\mathrm{\varnothing })\to (X,A)$, there is a long exact sequence:

$\dots \to H_n(A,\mathrm{\varnothing })\to H_n(X,\mathrm{\varnothing })\to H_n(X,A)\to H_{n-1}(A,\mathrm{\varnothing })\to \dots$

• 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 $H_n(X,\mathrm{\varnothing })$ is trivial for all $n>0$. One calls $H_0(X,\mathrm{\varnothing })$ 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 $(X,\mathrm{\varnothing })$.