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

Homology and Homotopy groups from properties

  1. Dec 25, 2009 #1
    I am looking for results which provides the homology and homotopy groups from some property of the space.
    For instance, if a space [tex]X[/tex] is contractible then [tex]H_0(X)=\mathbb{Z}[/tex] and [tex]H_n(X)=0[/tex] if [tex]n\neq 0[/tex]. Another example is the Eilenberg MacLane spaces [tex]K(\pi,n)[/tex] where [tex]\pi_n(K(\pi,n))=\pi[/tex] and [tex]\pi_r(K(\pi,n))=0[/tex] if [tex]n\neq r[/tex]. It is also known the result for the homology groups of the spheres.
    Do you know some similar result or some book where I can find them?
    Thank you in advance.
     
    Last edited: Dec 25, 2009
  2. jcsd
  3. Dec 26, 2009 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

  4. Dec 26, 2009 #3
    Yeah, I'm sure that there are hundreds of such rules, but it is difficult to find these rules explicitely.
     
  5. Jan 1, 2010 #4
    Have you heard of cellular homology? Readily lets you compute the homology groups of any CW complex.

    Other than that, Seifert-van-Kampen and Mayer-Vietoris are your friend.
     
  6. Jan 6, 2010 #5
    In a good first book on algebraic topology you will find many homology computations.
    Homotopy is much harder. Rational homotopy of simply connected spaces can be computed from minimal models of rational cohomology. This is a powerful technique.

    A good exercise is to compute the homology of an arbitrary closed surface.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Homology and Homotopy groups from properties
Loading...