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homotopy invariance (Definition)

Let $ \cal F$ be a functor from the category of topological spaces to some category $ \cal C$. Then $ \cal F$ is called homotopy invariant if for any two homotopic maps $ f,g\colon X\to Y$ between topological spaces $ X$ and $ Y$ the morphisms $ {\cal F}f$ and $ {\cal F}g$ in $ \cal C$ induced by $ \cal F$ are identical.

Suppose $ \cal F$ is a homotopy invariant functor, and $ X$ and $ Y$ are homotopy equivalent topological spaces. Then there are continuous maps $ f\colon X\to Y$ and $ g\colon Y\to X$ such that $ g\circ f\simeq{\rm id}_X$ and $ f\circ g\simeq{\rm id}_Y$ (i.e. $ g\circ f$ and $ f\circ g$ are homotopic to the identity maps on $ X$ and $ Y$, respectively). Assume that $ \cal F$ is a covariant functor. Then the homotopy invariance of $ \cal F$ implies

$\displaystyle {\cal F}g\circ{\cal F}f={\cal F}(g\circ f)={\rm id}_{{\cal F}X} $
and
$\displaystyle {\cal F}f\circ{\cal F}g={\cal F}(f\circ g)={\rm id}_{{\cal F}Y}. $
From this we see that $ {\cal F}X$ and $ {\cal F}Y$ are isomorphic in $ \cal C$. (The same argument clearly holds if $ \cal F$ is contravariant instead of covariant.)

An important example of a homotopy invariant functor is the fundamental group $ \pi_1$; here $ \cal C$ is the category of groups.



"homotopy invariance" is owned by pbruin.
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See Also: homotopy equivalence

Also defines:  homotopy invariant
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Cross-references: groups, fundamental group, argument, isomorphic, implies, identity maps, homotopic, continuous maps, homotopy equivalent, induced, morphisms, homotopic maps, topological spaces, category, functor
There are 7 references to this entry.

This is version 1 of homotopy invariance, born on 2004-06-15.
Object id is 5919, canonical name is HomotopyInvariance.
Accessed 2932 times total.

Classification:
AMS MSC55Pxx (Homotopy theory)

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