Thom Isomorphism

A statement in algebraic topology that for a (suitably oriented) rank‑k vector bundle E → B there exists a Thom class in the cohomology of the Thom space Th(E) that induces an isomorphism H^*(B; R) → H^{*+k}(Th(E); R) (equivalently H^*(Th(E)) ≅ H^{*-k}(B) after degree shift), enabling transfer and Gysin maps.