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Pushforward (homology) - Wikipedia

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In algebraic topology, the pushforward of a continuous function {\displaystyle f} : {\displaystyle X\rightarrow Y} between two topological spaces is a homomorphism {\displaystyle f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)} between the homology groups for {\displaystyle n\geq 0}.

Homology is a functor which converts a topological space {\displaystyle X} into a sequence of homology groups {\displaystyle H_{n}\left(X\right)}. (Often, the collection of all such groups is referred to using the notation {\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.

Definition for singular and simplicial homology

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

First, the map {\displaystyle f\colon X\to Y} induces a homomorphism between the singular or simplicial chain complex {\displaystyle C_{n}\left(X\right)} and {\displaystyle C_{n}\left(Y\right)} defined by composing each singular n-simplex {\displaystyle \sigma _{X}\colon \Delta ^{n}\rightarrow X} with {\displaystyle f} to obtain a singular n-simplex of {\displaystyle Y}, {\displaystyle f_{\#}\left(\sigma _{X}\right)=f\sigma _{X}\colon \Delta ^{n}\rightarrow Y}, and extending this linearly via {\displaystyle f_{\#}\left(\sum _{t}n_{t}\sigma _{t}\right)=\sum _{t}n_{t}f_{\#}\left(\sigma _{t}\right)}.

The maps {\displaystyle f_{\#}\colon C_{n}\left(X\right)\rightarrow C_{n}\left(Y\right)} satisfy {\displaystyle f_{\#}\partial =\partial f_{\#}} where {\displaystyle \partial } is the boundary operator between chain groups, so {\displaystyle \partial f_{\#}} defines a chain map.

Therefore, {\displaystyle f_{\#}} takes cycles to cycles, since {\displaystyle \partial \alpha =0} implies {\displaystyle \partial f_{\#}\left(\alpha \right)=f_{\#}\left(\partial \alpha \right)=0}. Also {\displaystyle f_{\#}} takes boundaries to boundaries since {\displaystyle f_{\#}\left(\partial \beta \right)=\partial f_{\#}\left(\beta \right)}.

Hence {\displaystyle f_{\#}} induces a homomorphism between the homology groups {\displaystyle f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)} for {\displaystyle n\geq 0}.

Properties and homotopy invariance

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Two basic properties of the push-forward are:

  1. {\displaystyle \left(f\circ g\right)_{*}=f_{*}\circ g_{*}} for the composition of maps {\displaystyle X{\overset {g}{\rightarrow }}Y{\overset {f}{\rightarrow }}Z}.
  2. {\displaystyle \left({\text{id}}_{X}\right)_{*}={\text{id}}} where {\displaystyle {\text{id}}_{X}} : {\displaystyle X\rightarrow X} refers to identity function of {\displaystyle X} and {\displaystyle {\text{id}}\colon H_{n}\left(X\right)\rightarrow H_{n}\left(X\right)} refers to the identity isomorphism of homology groups.

(This shows the functoriality of the pushforward.)

A main result about the push-forward is the homotopy invariance: if two maps {\displaystyle f,g\colon X\rightarrow Y} are homotopic, then they induce the same homomorphism {\displaystyle f_{*}=g_{*}\colon H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)}.

This immediately implies (by the above properties) that the homology groups of homotopy equivalent spaces are isomorphic: The maps {\displaystyle f_{*}\colon H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)} induced by a homotopy equivalence {\displaystyle f\colon X\rightarrow Y} are isomorphisms for all {\displaystyle n}.