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# Algebra of chain complexes II

$\blacktriangleright$ Chain equivalence: $$f: (T,\partial ^T) \rightarrow (Q,\partial ^Q),$$ provided there is $$g: (Q,\partial ^Q) \rightarrow (T,\partial ^T),$$ such that $$fg \sim Id_{Q}, gf \sim Id_{T} .$$

$\blacktriangleright$ Dual cochain complex of chain complex: $$(R,\partial _R) = (S,\partial ^S)^*,$$ defined by $$R_k=(S_{-k})^*, \forall k,$$ $$\partial _R^k=(\partial _{-k}^S)^*, \forall k.$$

$\blacktriangleright$ Flip cochain complex of chain complex: $$(R,\partial _R) = (S,\partial ^S)^-,$$ defined by $$R_k=S_{-k}, \forall k,$$ $$\partial _R^k=\partial _{-k}^S, \forall k.$$

$\blacktriangleright$ $m$-shift chain complex of chain complex: $$(R,\partial ^R) = (S,\partial ^S)[m],$$ defined by $$R_k=S_{k+m}, \forall k,$$ $$\partial _R^k=\partial ^{k+m}_S, \forall k.$$

$\blacktriangleright$ Homology of chain complex: $$H(C, \partial),$$ defined by $$H_k(C, \partial)=\frac{ \ker \partial _k}{\text{im } \partial _{k-1}}.$$

$\blacktriangleright$ Cohomology of chain complex: $$H^*(C, \partial),$$ defined by: $$H^k(C, \partial)=H_{-k}(C^*, \partial ^*)=\frac{ \ker \partial ^{k-1}}{\text{im } \partial ^{k}}.$$

$\blacktriangleright$ Cap product on chain complex: $$\frown:C^{\ast} \otimes C \rightarrow C,$$ defined by: $$f\frown c=(1\times f)\Delta c,$$ where $\Delta : C → C⊗C$ is a "chain-diagonal": $$(\epsilon⊗1)\Delta = (1⊗\epsilon)\Delta,$$ and $\epsilon : C_0 → R$ is the augmentation map.

$\blacktriangleright$ Poincare duality with respect to $\mu \in R$: $$D_{\mu}:(R,\partial _R)^* \rightarrow (R,\partial _R),$$ defined by $$D_{\mu}(f)= f \frown \mu.$$