I am looking for results which provides the homology and homotopy groups from some property of the space.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homology and Homotopy groups from properties

**Physics Forums | Science Articles, Homework Help, Discussion**