homology

Overview

Definition

Recall the definitions of cycles and boundaries:

Definition of cycles

Let $A_{\bullet }$ be a chain complex. Then there is a graded abelian group $Z_{\bullet }(A)$ with

$Z_n:=\ker (d:A_n\to A_{n-1}).$

These are called cycles.

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Definition of boundaries

Let $A_{\bullet }$ be a chain complex. There is a graded abelian group $B_{\bullet }(A)$ with

$B_n:=\text{im}(d:A_{n+1}\to A_n).$

These are called boundaries.

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There is a graded abelian group $H_{\bullet }(A)$ with

$H_n(A):=Z_n(A)/B_n(A)$

$H_n(A)$ is called the n-th homology group of A.

Examples

• singular homology
• todo Add De Rham cohomology here…