IsomH2

IsomH2 is a term used in algebraic topology to denote the study of isomorphisms between second homology groups H2(X) and H2(Y) of topological spaces, and how such isomorphisms can be realized explicitly. The focus is on determining when H2(X) and H2(Y) are structurally the same and on constructing concrete maps that induce those isomorphisms.

Formally, given spaces X and Y with chain complexes C_*(X) and C_*(Y), a chain map f: C_*(X) → C_*(Y) induces a homomorphism H2(f): H2(X) → H2(Y). If H2(f) is an isomorphism, this provides an instance of IsomH2, linking topological data to an algebraic equivalence. The program of IsomH2 emphasizes functoriality, naturality, and the explicit realization of the isomorphism, often via maps between spaces or algebraic constructions attached to them.

Tools commonly employed include the universal coefficient theorem, the Mayer–Vietoris sequence, and the Kunneth formula, which help identify when isomorphisms on H2 can be derived from simpler data. Orientation considerations of maps and spaces also play a role, since they influence the sign of induced maps on H2 in many cases.

Examples illustrate the concept: H2(S^2) ≅ Z, and any orientation-preserving homeomorphism of S^2 induces the identity on H2, while orientation-reversing maps induce multiplication by −1. The 2-torus T^2 also satisfies H2(T^2) ≅ Z, and various maps between spaces that preserve homological structure yield isomorphisms on H2.

Applications include classifying spaces up to homology, comparing spaces via H2 invariants, and aiding computational topology where explicit isomorphisms are required. See also homology, H2, isomorphism, and algebraic topology.