Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group of the stable module category of representations for certain finite groups. I was wondering if there are more results of a similar flavor, providing applications of equivariant homotopy theory to representation theory (or I suppose group theory in general).
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1$\begingroup$ There are some connections with equivariant homotopy theory, e.g., to define “level”, in work on the Verlinde formula. $\endgroup$– Jason StarrCommented Nov 9, 2022 at 23:07
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There are decades and decades of algebraic results that use techniques from equivariant homotopy theory. Some examples ...
(1) Quillen's work on ring theoretic aspects of the cohomology of finite groups [The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602]. He uses `descent' arguments that leave the world of finite groups: to learn about $BG$, he studies $EG \times_G X$ for suitable $G$--spaces $X$.
(2) Quillen's calculation of $H^*(GL_{\infty}(\mathbb F_q); k)$ and his use of this to compute the algebraic $K$--theory of finite fields involves serious excursions into homotopy theory.
(3) Quillen's paper (hmm ... do we see a trend here?) [Homotopy properties of the poset of nontrivial p-subgroups of a group. Adv. in Math. 28 (1978), no. 2, 101–128] amounts to a gorgeous use of elementary applications of categorical/homotopical methods to prove things about groups.
And Quillen is not the only person to use homotopy to prove results about groups. Here is an example:
(4) Peter Symonds, in [On the Castelnuovo-Mumford regularity of the cohomology ring of a group. J. Amer. Math. Soc. 23 (2010), no. 4, 1159–1173], proves a conjecture of Benson's about regularity of the group cohomology of a finite group by first using a descent argument, similar to Quillen, and then using a rather delicate results of Jeanne Duflot about actions of elementary abelian $p$-groups on smooth manifolds.
One could go on with many other examples, including results that are just being proved now.
