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	~~In [[algebraic topology]]~~, the ~~'''pushforward''' of~~ a ~~[[continuous function]]~~ <~~math~~>f<~~/math~~> : <~~math~~>~~X \rightarrow Y~~<~~/math~~> ~~between two [[topological space]]s~~ is a [~~[homomorphism]] <math>f_{*}~~:~~H_n\left(X\right) \rightarrow H_n\left(Y\right)<~~/~~math> between the [[homology group~~]~~]s for <math>n \geq 0</math>~~.		Αre you awarе something about nutгients? Perhaps you have produced a nutrition prepare? Do ʏou want to ensure it is greater? How confident are you thаt you will be ingesting effectively? Should you be not ɑble to satisfactoгily solution these գuestions with total self confidence, then reading through the following tips will certainly give yoս a hand.<br><br>A beneficіal avenue to diet is the idea of not depriving oneself of the food items you love, but swap tҺem witҺ far more nourishing variations the exact same plate. 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	~~Homology is a~~ [~~[functor]] which converts a topological space <math>X<~~/~~math> into a sequence of homology groups <math>H_{n}\left(X\right)<~~/~~math>~~. ~~(Often,~~ the ~~collection~~ of ~~all such groups is referred~~ to ~~using the notation~~ <~~math~~>~~H_{*}\left(X\right)~~<~~/math~~>~~; this collection has the structure of~~ a ~~[[graded group]].) In any [[category (mathematics)\|category]]~~, a ~~functor must induce a corresponding [[morphism]]. The pushforward is the morphism corresponding~~ to ~~the homology functor~~.

	~~== Definition~~ for ~~singular~~ and ~~simplicial homology ==~~

	~~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]]~~ <~~math~~>~~C_n\left(X\right)~~<~~/math~~> and ~~<math>C_n\left(Y\right)</math> defined by composing~~ each ~~singular n-[[simplex]]~~ <~~math~~>~~\sigma_X~~<~~/math> : <math>\Delta^n\rightarrow X</math~~> with <~~math~~>f<~~/math~~> ~~to obtain~~ a ~~singular n-simplex~~ of <~~math~~>Y<~~/math~~>, <~~math~~>~~f_{\#}\left(\sigma_X\right) = f\sigma_X~~<~~/math~~> : ~~<math>\Delta^n\rightarrow Y<~~/~~math>~~. ~~Then we extend <math>f_{\#}<~~/~~math> linearly via <math>f_{\#}\left(\sum_tn_t\sigma_t\right) = \sum_tn_tf_{\#}\left(\sigma_t\right)<~~/~~math>~~.

	~~The maps <math>f_{\#}</math> : <math>C_n\left(X\right)\rightarrow C_n\left(Y\right)</math> satisfy <math>f_{\#}\partial~~ = ~~\partial f_{\#}</math> where <math>\partial</math> is~~ the ~~[[boundary operator]] between chain groups, so <math>\partial f_{\#}</math> defines a '''[[chain complex#Chain maps\|chain map]]'~~''.

	<~~!-- We have a diagram~~
	~~<math~~>
	\[
	~~\begin{array}{ccccccccc}~~
	~~\dots & \rightarrow & C_{n+1}\left(X\right) & \overset{\partial}{\rightarrow} & C_{n}\left(X\right) & \overset{\partial}{\rightarrow} & C_{n-1}\left(X\right) & \rightarrow & \dots\\~~
	~~& & \downarrow{\scriptstyle f_{\#}} & & \downarrow{\scriptstyle f_{\#}} & & \downarrow{\scriptstyle f_{\#}}\\~~
	~~\dots & \rightarrow & C_{n+1}\left(Y\right) & \overset{\partial}{\rightarrow} & C_{n}\left(Y\right) & \overset{\partial}{\rightarrow} & C_{n-1}\left(Y\right) & \rightarrow & \dots\end{array}\]~~
	<~~/math~~>
	~~such that~~ in each ~~square~~ <~~math~~>~~f_{\#}\partial = \partial f_{\#}~~<~~/math~~>, ~~so <math>\partial f_{\#}</math> defines a '''chain map'~~''.
	~~-->~~

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

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

	~~== Properties and homotopy invariance ==~~

	~~{{Main\|singular homology#Homotopy invariance}}~~

	~~Two basic properties~~ of the ~~push~~-~~forward are:~~

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

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

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

	~~The maps~~ <~~math~~>~~f_{*}:H_n\left(X\right) \rightarrow H_n\left(Y\right)~~<~~/math~~> ~~induced by a homotopy equivalence <math>f</math> : <math>X\rightarrow Y</math> are isomorphisms for all <math>n</math>.~~

	~~==References==~~
	* Allen Hatcher, ~~[http://www~~.~~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]]~~

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