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?
[Math] Motivation for equivariant homotopy theory
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I think a good reference for the first paragraph is "Equivariant Homotopy and Cohomology Theory" by Peter May and a bunch of other people. Chapter 5 includes "Elmendorf's theorem" that this homotopy theory of G-spaces is equivalent to the homotopy theory of diagrams of spaces on the orbit category O(G) of G. In the latter homotopy theory, the weak equivalences are "levelwise" as is usual in the homotopy theory of diagrams.
I'm less sure about the (∞,1)-categorical versions, but I would expect that the (∞,1)-category associated to a levelwise model structure on O(G)-diagrams will be essentially the (∞,1)-category of functors from O(G) to the (∞,1)-category of spaces. That ought to imply that it is locally presentable as well.
One might guess that the equivariant stable homotopy category would be the "stabilization" of this (∞,1)-category, but that's not entirely obvious to me. The point at issue is that there are two kinds of G-spectra: "naive" G-spectra, which are indexed on integers, and "true" G-spectra, which are indexed on G-representations. It seems possible to me that the standard "stabilization" process of an (∞,1)-category will only stabilize with respect to integers.
If $G$ acts linearly and isometrically on $V$ then the unit disk $D(V)$ and its boundary $S(V)$ are an appealing kind of equivariant disk and sphere. You can try taking the pairs $(D(V),S(V))$ as the building blocks for equivariant complexes, but you won't get all the equivariant homotopy types that you want, not even the manifolds with smooth $G$-action.
You will get them all if you consider also representations of subgroups. If $H$ acts on $V$ then look at $(G\times_HD(V),G\times_HS(V))$, where $G\times_HX$ means the quotient of $G\times X$ with $(g,hx)$ identified with $(gh,x)$. (This converts an $H$-action into a $G$-action.)
On the other hand, you also get everything you want if you only use trivial representations of arbitrary subgroups rather than arbitrary representations of arbitrary subgroups.
Best Answer
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.