Fundamental groups and homotopy type

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  • #1
quasar987
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I know of the "result" that if two pointed spaces are homeomorphic, then the group homomorphism induced by such an homeomorphism if actually an isomorphism between the fundamental groups of these pointed spaces.

But is there a link between the fundamental groups of homotopy equivalent spaces?
 

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  • #2
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Homotopy equivalences also induce isomorphisms. This is easy to show if they are homotopies relative to the basepoint, but not too much harder even if they're not.
 
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mathwonk
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a functor is something that atkes objkects to objects, takes maps betwen pairs of objects to similar maps, takes identities to identities, and takes compositions to compositions. hence it also takes inverses to inverses.

i.e. if ∏ is a functor from spaces to groups such as the fundamental group, and if f:X-->Y is a homeomorphism, that means there is a map g:Y-->X such that fg = idY and gf = idX are the identities on Y and X respectively.

Hence, since ∏ is a functor from top spaces to groups, then ∏(f) and ∏(g) are homomorphisms from ∏(f):∏(X)-->∏(Y), and ∏(g):∏(Y)-->∏(X), such that
∏(f)o∏(g) = ∏(fog) = ∏(idY) = id(∏(Y)), and similarly the other way.

Hence ∏(f) and ∏(g) are inverse homomorphisms of the groups ∏(Y) AND ∏(X), so those groups are isomorphic.

Now the fundamental group is a functor, so itab tkes homeomorphisms to isomorphisms of groups. But also the fundamental group is by tis very definition constant on homotopy classes of maps, hence also takes homotopy equivalences to isomorphisms.

so all this is ":trivial" from the category theoretic point of view. i.e., learn to think in terms of maps, not just objects, and these questions will become automatic to you.
 
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