In algebraic topology, the pushforward of a continuous function f {\displaystyle f} : X → Y {\displaystyle X\rightarrow Y} between two topological spaces is a homomorphism f ∗ : H n ( X ) → H n ( Y ) {\displaystyle f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)} between the homology groups for n ≥ 0 {\displaystyle n\geq 0} . Homology is a functor which converts a topological space X {\displaystyle X} into a sequence of homology groups H n ( X ) {\displaystyle H_{n}\left(X\right)} . (Often, the collection of all such groups is referred to using the notation H ∗ ( X ) {\displaystyle H_{*}\left(X\right)} ; this collection has the structure of a graded ring.) In any category, a functor must induce a corresponding morphism. The pushforward is the morphism corresponding to the homology functor.
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