# Homology and Homotopy groups from properties

1. Dec 25, 2009

### heras1985

I am looking for results which provides the homology and homotopy groups from some property of the space.
For instance, if a space $$X$$ is contractible then $$H_0(X)=\mathbb{Z}$$ and $$H_n(X)=0$$ if $$n\neq 0$$. Another example is the Eilenberg MacLane spaces $$K(\pi,n)$$ where $$\pi_n(K(\pi,n))=\pi$$ and $$\pi_r(K(\pi,n))=0$$ if $$n\neq r$$. 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?

Last edited: Dec 25, 2009
2. Dec 26, 2009

### quasar987

3. Dec 26, 2009

### heras1985

Yeah, I'm sure that there are hundreds of such rules, but it is difficult to find these rules explicitely.

4. Jan 1, 2010

### owlpride

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.

5. Jan 6, 2010

### wofsy

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.