# Topology/Homology Groups

 Topology ← Exact Sequences Homology Groups Relative Homology →

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

## Definition

Given a chain complex

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

the n-th homology group is

${\displaystyle H_{n}=Ker(\partial _{n})/Im(\partial _{n+1})}$. We have a similar situation to the fundamental group.
Theorem

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

## Examples

(under construction)

## Exercises

(under construction)

 Topology ← Exact Sequences Homology Groups Relative Homology →