Pushforward (homology)

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Let $X$ and $Y$ be two topological spaces and $f:X\rightarrow Y$ a continuous function. Then $f$ induces a homomorphism between the homology groups $f_{*}:H_n\left(X\right) \rightarrow H_n\left(Y\right)$ for $n\geq0$ . We say that $f *$ is the pushforward induced by $f$ .

[edit] Definition for singular and simplicial homology

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

First we have a induced homomorphism between the singular or simplicial chain complex $C_n\left(X\right)$ and $C_n\left(Y\right)$ defined by composing each singular n-simplex $\sigma_X:\Delta^n\rightarrow X$ with $f$ to obtain a singular n-simplex of $Y$ , $f_{\#}\left(\sigma_X\right)=f\sigma_X:\Delta^n\rightarrow Y$ . Then we extend $f_{\#}$ linearly via $f_{\#}\left(\sum_tn_t\sigma_t\right) = \sum_tn_tf_{\#}\left(\sigma_t\right)$ .

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

We have that $f_{\#}$ takes cycles to cycles, since $\partial \alpha = 0$ implies $\partial f_{\#}\left( \alpha \right) = f_{\#}\left(\partial \alpha \right) = 0$ . Also $f_{\#}$ takes boundaries to boundaries since $f_{\#}\left(\partial \beta \right) = \partial f_{\#}\left(\beta \right)$ .

Hence $f_{\#}$ induces a homomorphism between the homology groups $f_{*}:H_n\left(X\right) \rightarrow H_n\left(Y\right)$ for $n\geq0$ .

[edit] Properties and homotopy invariance

Main article: singular homology#Homotopy invariance

Two basic properties of the push-forward are:

$\left( f\circ g\right)_{*} = f_{*}\circ g_{*}$ for the composition of maps $X\overset{f}{\rightarrow}Y\overset{g}{\rightarrow}Z$ .
$\left( id_X \right)_{*} = id$ where $id_X:X\rightarrow X$ refers to identity function of $X$ and $id:H_n\left(X\right) \rightarrow H_n\left(X\right)$ refers to the identity isomorphism of homology groups.

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