pushforward (homology)

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

First, the map $f\backslash colon\; X\backslash to\; Y$ induces a homomorphism between the singular or simplicial chain complex $C\_n\backslash left(X\backslash right)$ and $C\_n\backslash left(Y\backslash right)$ defined by composing each singular n-simplex $\backslash sigma\_X\backslash colon\backslash Delta^n\backslash rightarrow\; X$ with $f$ to obtain a singular n-simplex of $Y$, $f\_\{\backslash \#\}\backslash left(\backslash sigma\_X\backslash right)\; =\; f\backslash sigma\_X\backslash colon\backslash Delta^n\backslash rightarrow\; Y$, and extending this linearly via $f\_\{\backslash \#\}\backslash left(\backslash sum\_tn\_t\backslash sigma\_t\backslash right)\; =\; \backslash sum\_tn\_tf\_\{\backslash \#\}\backslash left(\backslash sigma\_t\backslash right)$.

The maps $f\_\{\backslash \#\}\backslash colon\; C\_n\backslash left(X\backslash right)\backslash rightarrow\; C\_n\backslash left(Y\backslash right)$ satisfy $f\_\{\backslash \#\}\backslash partial\; =\; \backslash partial\; f\_\{\backslash \#\}$ where $\backslash partial$ is the boundary operator between chain groups, so $\backslash partial\; f\_\{\backslash \#\}$ defines a chain map.

Therefore, $f\_\{\backslash \#\}$ takes cycles to cycles, since $\backslash partial\; \backslash alpha\; =\; 0$ implies $\backslash partial\; f\_\{\backslash \#\}\backslash left(\; \backslash alpha\; \backslash right)\; =\; f\_\{\backslash \#\}\backslash left(\backslash partial\; \backslash alpha\; \backslash right)\; =\; 0$. Also $f\_\{\backslash \#\}$ takes boundaries to boundaries since $f\_\{\backslash \#\}\backslash left(\backslash partial\; \backslash beta\; \backslash right)\; =\; \backslash partial\; f\_\{\backslash \#\}\backslash left(\backslash beta\; \backslash right)$.

Hence $f\_\{\backslash \#\}$ induces a homomorphism between the homology groups $f\_\{*\}\; :\; H\_n\backslash left(X\backslash right)\; \backslash rightarrow\; H\_n\backslash left(Y\backslash right)$ for $n\backslash geq0$.

{{Main|Singular homology#Homotopy invariance}}

Two basic properties of the push-forward are:

1. $\backslash left(\; f\backslash circ\; g\backslash right)\_\{*\}\; =\; f\_\{*\}\backslash circ\; g\_\{*\}$ for the composition of maps $X\backslash overset\{g\}\{\backslash rightarrow\}Y\backslash overset\{f\}\{\backslash rightarrow\}Z$.
2. $\backslash left(\; \backslash text\{id\}\_X\; \backslash right)\_\{*\}\; =\; \backslash text\{id\}$ where $\backslash text\{id\}\_X$ : $X\backslash rightarrow\; X$ refers to identity function of $X$ and $\backslash text\{id\}\backslash colon\; H\_n\backslash left(X\backslash right)\; \backslash rightarrow\; H\_n\backslash left(X\backslash right)$ 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 $f,g\backslash colon\; X\backslash rightarrow\; Y$ are homotopic, then they induce the same homomorphism $f\_\{*\}\; =\; g\_\{*\}\backslash colon\; H\_n\backslash left(X\backslash right)\; \backslash rightarrow\; H\_n\backslash left(Y\backslash right)$.

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

• Pullback (cohomology)

• Allen Hatcher, [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic topology.] Cambridge University Press, {{ISBN|0-521-79160-X}} and {{ISBN|0-521-79540-0}}

Category:Topology

Category:Homology theory