Topology/Homology Groups

A homology group is a group derived from a space's chain complex.

Definition

Given a chain complex

\cdots {\xrightarrow {\partial _{2}}}C_{2}{\xrightarrow {\partial _{1}}}C_{1}{\xrightarrow {\partial _{0}}}C_{0}

the n-th homology group is

H_{n}=Ker(\partial _{n})/Im(\partial _{n+1})

. We have a similar situation to the fundamental group.

Theorem

A continuous function on topological spaces $f:X\to Y$ always induces homomorphisms $f_{*}:H_{n}(X)\to H_{n}(Y)$ . If $f$ is a homeomorphism, $f_{*}$ is an isomorphism.

Examples

(under construction)

Exercises

(under construction)