a chain homotopy

Two continuous maps that can be continuously deformed into one another should not be distinguishable by homology — that is the homotopy invariance we want. Chain homotopy is the precise algebraic shadow of a geometric homotopy: it is the condition on two chain maps that guarantees they induce the same map on homology, and it is the tool by which homotopy invariance gets proved.

Let f and g be two chain maps from a complex (C, d) to a complex (C', d'). A chain homotopy from f to g is a family of homomorphisms P_n: C_n -> C'_{n+1} (note: raising degree by one) satisfying the identity f_n - g_n = d'_{n+1} P_n + P_{n-1} d_n. Read the right side: P slides a chain up one dimension, applies a boundary on either side, and the two terms combine to account exactly for the difference f - g. The geometric picture behind the degree-raising P is the prism: a homotopy of spaces sweeps each n-simplex into an (n+1)-dimensional prism, and P is the algebraic operation 'fill in the prism'.

The single payoff worth memorizing: if f and g are chain homotopic, then they induce the same homomorphism on homology, f_* = g_* : H_n(C) -> H_n(C'). The proof is one line — on a cycle z (where d z = 0) we get f(z) - g(z) = d'(P z) + P(d z) = d'(P z), a boundary, hence zero in homology. From this single lemma flow the cornerstones: homotopic continuous maps induce equal maps on H_n, homotopy-equivalent spaces have isomorphic homology, and contractible spaces have the homology of a point.

A caution: chain homotopy is an equivalence relation on chain maps, but it is strictly coarser than equality. Two genuinely different chain maps can be chain homotopic; what they share is only their effect on homology. So 'chain homotopic' never means 'equal as maps of chains' — it means 'equal after passing to homology', which is exactly the looseness that makes the invariant robust.