Chain map

A chain map $f:C\to C^{\prime }$ is a family $f_i:C_i\to C_i^{\prime }$ satisfying $d_i^{\prime }{f}_i=f_{i-1}{d}_i$.

A chain map inducing isomorphisms on all homology groups. The same definition applies to a map of cochain complexes using cohomology groups. A quasi-isomorphism need not be an isomorphism in each degree.

A chain map sends cycles to cycles and boundaries to boundaries, and therefore induces $f_{\ast }:H_i(C)\to H_i(C^{\prime })$ by $[x]\mapsto [f_i(x)]$.

Chain maps $f,g:C\to C^{\prime }$ are chain homotopic when there are maps $h_i:C_i\to C_{i+1}^{\prime }$ such thatChain-homotopic maps induce the same map on homology.

A contracting homotopy on a chain complex is a degree-one map $h$ whose displayed identity makes the identity chain-homotopic to zero. It implies that all homology vanishes. The converse is not valid for arbitrary complexes of modules.

For cochain maps $f,g:C^{\bullet }\to D^{\bullet }$, a cochain homotopy is a family $K:C^q\to D^{q-1}$ satisfyingCochain-homotopic maps induce the same map on cohomology.

For an open cover $\mathcal{U}$ of $X$, singular chains generated by simplices lying inside one member of $\mathcal{U}$ form a subcomplex chain-homotopy equivalent to the full singular chain complex. Repeated barycentric subdivision makes chains small.