For an oriented proper smooth map of oriented manifolds, the Gysin homomorphism is a degree-shifting pushforward , 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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The Gysin homomorphism is a concept from algebraic topology and algebraic geometry, particularly in the study of cohomology theories, intersection theory, and the topology of manifolds. It is most commonly associated with the theory of fiber bundles and the intersection products in cohomology.