Definition:Homology Group

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

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

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 $$X \ $$.