Integer Cohomology of Real Infinite-Dimensional Grassmann Manifold

In summary, the conversation discusses the difficulty in finding information on the web about the Z cohomology of the infinite dimensional Grassmann manifold of real unoriented k planes in Euclidean space. The speaker also mentions their interest in computing the Bockstein exact sequence and using classifying spaces for this computation, but notes that they can only find information on the Z2 cohomology of the Grassmann manifold.
  • #1
lavinia
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I can't seem to find on the web a site that gives the Z cohomology of the infinite dimensional Grassmann manifold of real unoriented k planes in Euclidean space.

I am interested in computing the Bockstein exact sequence for the coefficient sequence,

0 -> Z ->Z ->Z/2Z -> 0

to see which products of the Stiefel-Whitney classes are mod 2 reductions of integer classes.
 
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  • #2


Isn't this done in Milnor Stacheff ?!?
 
  • #3


quasar987 said:
Isn't this done in Milnor Stacheff ?!?

No. I think just the Z2 cohomology. I will check again.
 
  • #4


Don't you use classifying spaces for this?
 
  • #5


Bacle2 said:
Don't you use classifying spaces for this?

yes but for the Grassmann of unoriented planes I can only find the Z2 cohomology.
 

1. What is the purpose of studying integer cohomology of real infinite-dimensional Grassmann manifold?

The purpose of studying integer cohomology of real infinite-dimensional Grassmann manifold is to understand the topological properties of this space. This can provide insights into the geometric structure and how it relates to other mathematical concepts.

2. What is the definition of integer cohomology?

Integer cohomology is a mathematical concept that assigns a group to a topological space, which measures the number of holes in the space. It is a way of quantifying the topological properties of a given space.

3. How is integer cohomology calculated for real infinite-dimensional Grassmann manifold?

The integer cohomology of real infinite-dimensional Grassmann manifold can be calculated using the Serre spectral sequence. This involves breaking down the manifold into smaller, simpler spaces and then using algebraic techniques to determine the cohomology groups.

4. What are the applications of studying integer cohomology of real infinite-dimensional Grassmann manifold?

The applications of studying integer cohomology of real infinite-dimensional Grassmann manifold can be found in various areas of mathematics and physics. It can be used to understand the topology of other spaces, such as flag manifolds, and also has applications in quantum field theory and string theory.

5. Are there any open problems or unresolved questions related to integer cohomology of real infinite-dimensional Grassmann manifold?

Yes, there are still many open problems and unresolved questions related to integer cohomology of real infinite-dimensional Grassmann manifold. Some of these include determining the cohomology ring structure, computing the cohomology classes, and understanding the relationship between the cohomology and the geometry of the manifold.

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