Hi, I am trying to compute homology groups of a space while some of the homotopy groups are known..what is the best way to do that. I hope that you can help. Thanks, Sandra
There's no way to do it in general. Looking at the two-sphere, you get an idea how complicated this relationship can be.
By abelianizing the first homotopy group, you obtain the first homology group (if the space is connected. If not, you can still use this to find the first homology group easily). This may help, although probably not.
I think you can not solve this problem in general. The relation of homotopy to homology is difficult. Why not send us the example you are working on? Most theorems are known for simply connected spaces. For a simply connected space the first non-zero homotopy group is the first non-zero homology group. For a simply connected manifold there is an incredible theorem that says that the rational homotopy groups have the same dimension as vector spaces in each dimension as the number of generators in a minimal model of the de Rham complex.
For simply connected spaces, it seems easier to compute homotopy from homology - up to torsion. Here is the recipe: From the rational homology construct a minimal model. The non-torsion part of the homotopy group in any dimension has the same number of generators as the minimal model in that dimension. Example: An even dimensional sphere. Its rational homology is zero except in dimension 2n where it is Q. A minimal model has two generators x and y with x of degree 2n, y of degree 4n-1. (The differentials are dx = 0 and dy = x^2) Thus the 2n sphere has rational homotopy with one generator in dimensions 2n and 4n-1 and zero in all other dimensions. Example: The odd dimensional sphere. The minimal model has one generator in dimension 2n+1 and no others. So the rational homotopy is Q in dimension 2n+1 and zero in all other dimensions.