Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fundamental groups and homotopy type

  1. Feb 15, 2007 #1


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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?
  2. jcsd
  3. Feb 15, 2007 #2


    User Avatar
    Homework Helper

    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.
  4. Feb 15, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    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.
    Last edited: Feb 15, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Fundamental groups and homotopy type