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 →