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Pushforward (homology)

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In algebraic topology, the pushforward of a continuous function  : between two topological spaces is a homomorphism between the homology groups for .

Homology is a functor which converts a topological space into a sequence of homology groups . (Often, the collection of all such groups is referred to using the notation ; 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.

Definition for singular and simplicial homology

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We build the pushforward homomorphism as follows (for singular or simplicial homology):

First we have an induced homomorphism between the singular or simplicial chain complex and defined by composing each singular n-simplex  : with to obtain a singular n-simplex of ,  : . Then we extend linearly via .

The maps  : satisfy where is the boundary operator between chain groups, so defines a chain map.


We have that takes cycles to cycles, since implies . Also takes boundaries to boundaries since .

Hence induces a homomorphism between the homology groups for .

Properties and homotopy invariance

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Two basic properties of the push-forward are:

  1. for the composition of maps .
  2. where  : refers to identity function of and refers to the identity isomorphism of homology groups.


A main result about the push-forward is the homotopy invariance: if two maps are homotopic, then they induce the same homomorphism .

This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:

The maps induced by a homotopy equivalence are isomorphisms for all .

References

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  • Allen Hatcher, Algebraic topology. Cambridge University Press, ISBN 0-521-79160-X and ISBN 0-521-79540-0