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For an oriented proper smooth map $f:M^m\to N^n$ of oriented manifolds, the Gysin homomorphism is a degree-shifting pushforward $f_{!}:H^k(M)\to H^{k+n-m}(N)$, defined using Poincare duality and the appropriate pushforward on homology. For an oriented sphere bundle it is integration along the fiber and participates in the Gysin sequence of a sphere bundle. Oriented embeddings give the corresponding Thom-class construction.

A Gysin sequence of a sphere bundle is the long exact sequence induced by the Gysin homomorphism and the Euler class of a vector bundle.

For the unit sphere bundle $S(E)\to B$ of an oriented rank-$d$ vector bundle, the Gysin sequence contains