 Gysin homomorphism (source code)

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= Gysin homomorphism
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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.

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