Definition:Homology Group
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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 $X$.
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$Z_p \left({X}\right)$ and $B_p \left({X}\right)$: presumably $Z_p \left({X}\right)$ is the center of $X$, but clarification needed.
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