I'm in the process of learning equivariant homotopy theory, so I was wondering: what is the importance of equivariant homotopy theory, and what has it been applied to so far? I know of HHR's solution to the Kervaire invariant problem, but what are some other applications?
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2$\begingroup$ More a comment than an answer: when studying one "generalised homotopy theory" (e.g. A^1-homotopy theory) it can be useful to study others (e.g. equivariant homotopy theory), to see the formalism played out. Actually you can relate G-equivariant homotopy theory to A^1-homotopy theory over $k$, when $G$ is related to the Galois group of $k$. $\endgroup$– Tom BachmannNov 2, 2015 at 10:33
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3 Answers
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Let me give an example of the use of Smith's theory in real algebraic geometry. Smith theory gives a way to compute fixed point sets in terms of equivariant cohomology, for a "modern" treatment see: Dwyer, William G.; Wilkerson, Clarence W. "Smith theory revisited." Ann. of Math. (2) 127 (1988), no. 1, 191–198.
Let $M^n_{\mathbb{R}}$ be a smooth manifold that is the set of real points of a smooth complex manifold $M^n_{\mathbb{C}}$ then we have: $$\sum_{j=0}^n b_j(M^n_{\mathbb{R}})\leq \sum_{j=0}^{2n}b_j(M^n_{\mathbb{C}}).$$ Where $b_j$ is the $j$-th Betti number with $\mathbb{Z}/2$-coefficients. We consider $M^n_{\mathbb{R}}$ as the set of fixed points of the involution given by by complex conjugation on $M^n_{\mathbb{C}}$.
A modern proof of this inegality can be found in: A. Borel. "Seminar on Transformation groups, (with contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais)" Ann. of M. Studies n. 46, Princeton UniversityPress, 1960.
It uses spectral sequences for the $\mathbb{Z}/2$-equivariant cohomology.
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One of the applications that motivated the development of the theory is the solution of the Segal conjecture which states that the zeroth cohomotopy of a classifying space of a finite group is isomorphic to the completion of the Burnside ring. You can read about this in
- G. Carlsson. A survey of equivariant stable homotopy theory. Topology 31 (1992), 1-27.
There is also a book by J.P. May "Equivariant homotopy and cohomology theory" which discusses such applications.
Another application of equivariant cohomology is to the computation of ordinary cohomology of spaces with group actions. Equivariant cohomology has a nice property referred to as "localization", which allows to get cohomology information from just looking at fixed points. This can make computations of ring structures on equivariant cohomology easier than in ordinary cohomology (but one can then recover the ordinary cohomology ring from the equivariant one). You can read about this in the applications section of Tu's "What is ... equivariant cohomology?", the paper by Goresky-Kottwitz-MacPherson on equivariantly formal spaces, or the paper of Brion-Vergne on localization in equivariant cohomology
Finally, equivariant homotopy theory can also be used for classification of group actions on e.g. spheres. One can use equivariant versions of obstruction theory to classify maps between representation spheres of finite groups, cf. e.g.
- S.J. Willson, Equivariant maps between representation spheres. Pacific J. Math. 56, (1975), 291-296.
See also the works of tom Dieck, Petrie,... This application of equivariant homotopy theory is sort of a non-abelian version of representation theory.
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There are many beautiful applications of equivariant topology to combinatorics, via the test map method. I recommend the diploma thesis of Benjamin Matschke as a good place to start reading about this (the test map method is nicely illustrated in Chapter 1).
There are also applications of equivariant cohomology to embedding problems, via the Haefliger-Shapiro-Wu deleted product construction.