Definition:Homology Group

Definition
The $p^{th}$ singular homology group of a space $X$ is:


 * $\displaystyle H_p(X) = \frac {Z_p \left({X}\right)} {B_p \left({X}\right)} = \frac {\operatorname{ker} \left({\partial_p}\right)} {\operatorname{Im} \left({\partial_{p+1}}\right)}$

where:
 * $\partial_p: \Delta_p \left({X}\right) \to \Delta_{p-1} \left({X}\right)$ is a homomorphism of the singular p-chain groups.


 * $\Delta_p(X)=$ the free abelian group generated by the singular p-simplices of $X$.