Definition:Homology Group

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

$$H_p(X) = Z_p(X) / B_p(X) = (\text{ker}\partial_p)/(\text{im}\partial_{p+1}$$

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

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