- #1

heras1985

- 8

- 0

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.

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: