Equivariant topology
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In
The notion of symmetry is usually captured by considering a
Induced G-bundles
An important construction used in equivariant cohomology and other applications includes a naturally occurring group bundle (see principal bundle for details).
Let us first consider the case where acts freely on . Then, given a -equivariant map , we obtain sections given by , where gets the diagonal action , and the bundle is , with fiber and projection given by . Often, the total space is written .
More generally, the assignment actually does not map to generally. Since is equivariant, if (the isotropy subgroup), then by equivariance, we have that , so in fact will map to the collection of . In this case, one can replace the bundle by a homotopy quotient where acts freely and is bundle homotopic to the induced bundle on by .
Applications to discrete geometry
In the same way that one can deduce the ham sandwich theorem from the Borsuk-Ulam Theorem, one can find many applications of equivariant topology to problems of discrete geometry.[1][2] This is accomplished by using the configuration-space test-map paradigm:
Given a geometric problem , we define the configuration space, , which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) Additionally, we consider a test space and a map where is a solution to a problem if and only if . Finally, it is usual to consider natural symmetries in a discrete problem by some group that acts on and so that is equivariant under these actions. The problem is solved if we can show the nonexistence of an equivariant map .
Obstructions to the existence of such maps are often formulated algebraically from the topological data of and .[3] An archetypal example of such an obstruction can be derived having a vector space and . In this case, a nonvanishing map would also induce a nonvanishing section from the discussion above, so , the top Stiefel–Whitney class would need to vanish.
Examples
- The identity map will always be equivariant.
- If we let act antipodally on the unit circle, then is equivariant, since it is an odd function.
- Any map is equivariant when acts trivially on the quotient, since for all .
See also
References
- ^ Matoušek, Jiří (2003). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Universitext. Springer.
- ISBN 9781584883012.
- ^ Matschke, Benjamin. "Equivariant topology methods In discrete geometry" (PDF).