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Homology groups from Homotopy groups 
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#1
Oct709, 07:18 PM

P: 6

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 


#2
Oct709, 07:51 PM

P: 707

get more specific



#3
Oct809, 04:08 AM

P: 6

OK,
I have the first 3 Homotopy groups which are generically nontrivial and I need to compute H_3 


#4
Oct1409, 12:17 AM

P: 491

Homology groups from Homotopy groups
There's no way to do it in general. Looking at the twosphere, you get an idea how complicated this relationship can be.



#5
Dec2209, 05:04 PM

P: 429

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.



#6
Dec2209, 09:10 PM

P: 707

Most theorems are known for simply connected spaces. For a simply connected space the first nonzero homotopy group is the first nonzero 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. 


#7
Jan610, 06:23 AM

P: 707

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 nontorsion 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 4n1. (The differentials are dx = 0 and dy = x^2) Thus the 2n sphere has rational homotopy with one generator in dimensions 2n and 4n1 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. 


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