Definition. A sequence of Group homomorphism between Chain groups of two Chain complexes, which satisfies $f\partial = \partial f$. This ensures that the chain map induces a Group isomorphism $f_*$ between the Homology groups of the two Chain complexes.

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A chain map induces a map in homology

Used to show Homotopy invariance of homology groups: Video1/Video2 Homotopy equivalent spaces have isomorphic Homology groups. See also Hatcher. We define a Functor by finding an induced function $f_\#$ between Chain groups which is a Chain map, which itself induces a function $f_*$ between Homology groups by lifting $f_\#$ to the Cosets defining the homology groups, and showing it's well defined because it maps cycles to cycles and boundaries to boundaries. Finally, there is a theorem that shows that homotopic maps lead to equal $f_*$, from which one can conclude (after showing Functoriality) that homotopy equivalences induce group isomorphisms between the homology groups.