Chain Homotopy and Chain Equivalence

Core Idea

A chain homotopy between two chain maps f, g : C_* -> D_* is a sequence of homomorphisms P_n : C_n -> D_{n+1} satisfying f_n - g_n = d_{n+1} compose P_n + P_{n-1} compose d_n. Chain homotopic maps induce identical homomorphisms on homology: f_* = g_*. This algebraic notion mirrors topological homotopy — when two continuous maps are homotopic, the induced chain maps are chain homotopic. Chain homotopy equivalence is the correct notion of "sameness" for chain complexes, just as homotopy equivalence is for topological spaces.

Explainer

Chain homotopy is the algebraic analog of topological homotopy. In topology, two maps f, g : X -> Y are homotopic if there is a continuous deformation between them. In homological algebra, two chain maps f, g : C_* -> D_* are chain homotopic if there exists a sequence of homomorphisms P_n : C_n -> D_{n+1} (called a chain homotopy or homotopy operator) satisfying f_n - g_n = d_{n+1} compose P_n + P_{n-1} compose d_n for all n. The chain homotopy P "connects" f and g at the chain level, just as a homotopy H connects two maps at the space level.

The crucial consequence of chain homotopy is that chain homotopic maps induce identical maps on homology. The proof is direct: for a cycle z in ker(d_n), we have f(z) - g(z) = dP(z) + Pd(z) = dP(z) (since d(z) = 0). So f(z) - g(z) is a boundary, meaning [f(z)] = [g(z)] in homology. This is the algebraic mechanism underlying the topological theorem that homotopic maps induce the same homomorphism on homology. The proof of homotopy invariance of singular homology constructs the chain homotopy P (the "prism operator") from the given topological homotopy H : X x [0,1] -> Y, and the rest follows from this algebraic lemma.

Two chain complexes C_* and D_* are chain homotopy equivalent if there exist chain maps f : C_* -> D_* and g : D_* -> C_* with g compose f chain homotopic to id_{C_*} and f compose g chain homotopic to id_{D_*}. Chain homotopy equivalent complexes have isomorphic homology in every degree. This is the chain-level analog of homotopy equivalence: just as homotopy equivalent spaces have the same homology, chain homotopy equivalent complexes have the same homology. The theory of derived categories formalizes this by treating chain homotopy equivalent complexes as "the same object."

The prism operator construction is worth understanding in detail. Given a homotopy H : X x [0,1] -> Y between f = H(-, 0) and g = H(-, 1), define P_n : C_n(X) -> C_{n+1}(Y) as follows. For a singular n-simplex sigma : Deltaⁿ -> X, consider the prism Deltaⁿ x [0,1]. Triangulate this prism into (n+1)-simplices (there is a standard triangulation using n+1 simplices, corresponding to the n+1 orderings of "when each vertex moves from the bottom to the top"). Compose with sigma x id : Deltaⁿ x [0,1] -> X x [0,1] and then with H to get (n+1)-chains in Y. The alternating sum of these gives P_n(sigma). The boundary of the prism equals the top face minus the bottom face minus the lateral faces, which translates algebraically to dP + Pd = f_# - g_#.

Chain homotopy equivalence is strictly weaker than isomorphism of chain complexes but strictly stronger than isomorphism of homology. Two non-isomorphic chain complexes can be chain homotopy equivalent (just as non-homeomorphic spaces can be homotopy equivalent), and two chain complexes with isomorphic homology need not be chain homotopy equivalent (just as spaces with the same homology need not be homotopy equivalent — e.g., the lens spaces L(7,1) and L(7,2) have isomorphic homology but are not homotopy equivalent). Chain homotopy equivalence occupies the "Goldilocks zone" for algebraic topology: it is fine enough to capture meaningful topological distinctions but coarse enough to identify spaces and complexes that are "the same" from the perspective of homological invariants.