Homology groups from Homotopy groups

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In summary, the conversation discusses the difficulty of computing homotopy groups while some homotopy groups are known, and the best way to do so. It is mentioned that there is no general way to solve this problem, and that most theorems are known for simply connected spaces. The relationship between homotopy and homology is also discussed, with the recipe of using a minimal model to compute rational homotopy groups for simply connected spaces. Examples of even and odd dimensional spheres are provided to illustrate this concept.
  • #1
wodhas
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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
 
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  • #2
get more specific
 
  • #3
OK,

I have the first 3 Homotopy groups which are generically non-trivial and I need to compute H_3
 
  • #4
There's no way to do it in general. Looking at the two-sphere, you get an idea how complicated this relationship can be.
 
  • #5
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
wodhas said:
OK,

I have the first 3 Homotopy groups which are generically non-trivial and I need to compute H_3

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.
 
  • #7
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.
 

1. What is the difference between homology groups and homotopy groups?

Homology groups and homotopy groups are both mathematical tools used in algebraic topology to study the properties of topological spaces. Homology groups are algebraic invariants that measure the number of holes in a topological space, while homotopy groups are algebraic invariants that measure the connectedness of a topological space.

2. How are homology groups and homotopy groups related?

Homology groups and homotopy groups are related through the Hurewicz theorem, which states that for a nice topological space X, the n-th homotopy group πn(X) is isomorphic to the n-th homology group Hn(X) for n ≥ 2. This means that these two tools provide complementary information about the structure of a topological space.

3. What is the significance of studying homology groups from homotopy groups?

Studying homology groups from homotopy groups allows us to understand the topological properties of a space in a more detailed and nuanced way. By using both of these tools together, we can gain a deeper understanding of the structure of a topological space and its underlying geometric properties.

4. How are homology groups from homotopy groups calculated?

The process of calculating homology groups from homotopy groups involves using the Hurewicz theorem, as well as other tools and techniques from algebraic topology. This usually involves constructing a long exact sequence using algebraic tools such as the Mayer-Vietoris sequence, and then using this sequence to calculate the homology groups of a given topological space.

5. What are some applications of homology groups from homotopy groups?

Homology groups from homotopy groups have a wide range of applications in various fields, including physics, engineering, and computer science. They are used to study and classify topological spaces, as well as to solve problems in data analysis, machine learning, and image processing. They also have important applications in the study of dynamical systems and differential equations.

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