Equivariant Homotopy: M & N Manifolds, G-Action & Cohomology Rings

  • Thread starter Kreizhn
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In summary, the question is whether two manifolds M and N with G-actions, which are homotopy equivalent, will also be G-equivariantly homotopy equivalent. The conversation discusses the possibility of this being true, but concludes that it is unlikely without some additional compatibility between the actions. The person asking the question is reading a paper that mentions equivariant cohomology, but only uses the fact that the spaces are homotopy equivalent, and is curious if they are overlooking something. The conversation suggests working out simple examples, such as the circle with two different Z2 actions, to see if equivariant cohomology can differ even when the spaces are homotopy equivalent.
  • #1
Kreizhn
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This might be a really silly question, but suppose that you have two (possibly Frechet) manifolds M and N both endowed with a G-action. If M and N are homotopy equivalent, is it necessary that they will be G-equivariantly homotopy equivalent?

Edit: That is, should I expect them to have the same equivariant cohomology rings?
 
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  • #2
I don't know what any of those words mean but I will say no, because why on Earth would this be true when the hypothesis (homotopy equivalent) does not contain the key word "G". I mean if this were true why would G actions be interesting? Maybe if you give us some of the relevant definitions, or at least a few examples, we can see more what is going on.
 
  • #3
I am leaning towards agreeing with your assessment: it seems unreasonable that one should imply the other, especially since I could put really weird actions on each manifold.

Perhaps there needs to be some sort of added "compatibility" requirement between the actions. Upon thinking about it, that compatibility is probably exactly the fact that a homotopy equivalence be equivariant.

Anyway, thanks for the response. I ask because I'm reading a paper which makes some assertions on equivariant cohomology, but only utilizes the fact that two spaces are homotopy equivalent. I was wondering if perhaps I was missing something obvious.
 
  • #4
work out some very simple examples of equivariant cohomology, and see if they can differ. e.g. try to cook up a non trivial action that gives a non trivial equivariant cohomology on R^n, which of course is homotopic to a point.
 
  • #5
The circle is homotopy equivalent to itself. Let Z2 act on it in two ways: first trivially by the identity map and second by reflection around the y axis. These maps are not homotopically equivalent.
 

Related to Equivariant Homotopy: M & N Manifolds, G-Action & Cohomology Rings

1. What is "equivariant homotopy"?

Equivariant homotopy is a branch of mathematics that studies continuous transformations and symmetries of spaces, such as manifolds, while taking into account a group action. It combines ideas from homotopy theory, algebraic topology, and group theory.

2. What are M & N manifolds?

M & N manifolds are two types of smooth manifolds that are used in equivariant homotopy. M manifolds are spaces that have a group action by a compact Lie group, while N manifolds have a group action by a finite group. These two types of manifolds play important roles in the study of equivariant homotopy.

3. What is a G-action?

A G-action refers to a group action, where G is a group, on a space. In equivariant homotopy, the group G is usually a compact Lie group or a finite group and the space is a manifold. This action is important because it allows us to study the symmetries and invariants of a space under these transformations.

4. What are cohomology rings?

Cohomology rings are algebraic structures that are used to study the topological properties of spaces. They are derived from the cohomology groups, which are a set of algebraic invariants that can be associated with a space. Cohomology rings are useful in equivariant homotopy because they can be used to study the symmetries and invariants of spaces under group actions.

5. What are some applications of equivariant homotopy?

Equivariant homotopy has various applications in mathematics and physics. In mathematics, it is used to study the symmetries and invariants of spaces, as well as to prove theorems about the topology of manifolds. In physics, it has been used to study gauge theories and string theory, as well as to understand the symmetries and invariants of physical systems.

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