# Problems in homotopy theory Equivariant homotopy theory

• (D. Ravenel) Does there exist a ${C}_{p}$-equivariant analogue of Real bordism theory? Such an object would have an underlying homotopy type given by a $\left(p-1\right)$-fold smash product of copies of $\mathrm{MU}$, with action analogous to the reduced regular representation of ${C}_{p}$. The geometric fixed point object should be a wedge of Eilenberg-Mac Lane objects. Such an object would be relevant to odd-primary analogues of the Hill-Hopkins-Ravenel constructions.

• (D. Ravenel) Revisit $K\left(n{\right)}^{*}\mathrm{BG}$ for a finite group G. It is known to have finite rank, and HKR gives a formula for its Euler characteristic. We tried but failed to prove that it is concentrated in even dimensions. This is known to be true when $n=1$, and when G is abelian or a symmetric group and a few other cases, and it would suffice to prove it for G a finite p-group. Later Kriz and his student (Lee?) found a counter example, the p-Sylow subgroup of ${\mathrm{GL}}_{4}\left({F}_{p}\right)$, which has order ${p}^{6}$. It would be interesting to have general statement about the ranks of the even and odd dimensional parts of $K\left(n{\right)}^{*}\mathrm{BG}$.

• (J.Greenlees)Is MU_G^* in even degrees? The answer is yes for abelian compact Lie groups, though the proof is computational (Loeffler, Comezana).

• Stefan Schwede and Anna-Marie Bohmann have both been doing work on global equivariant homotopy theory that has open avenues to pursue.

• (M. Mandell) What is the relation between this global perspective and the Greenlees-May paper on completion?

• (S. Schwede) In what sense is (periodic, complex, topological) global K-theory obtained from the global classifying space of the circle Lie group by inverting a Bott element?

• (S. Schwede) A theory of global orientations, global ${\mathrm{gl}}_{1}\left(R\right)$, global Thom spectra

• (P. Goerss) Equivariant formal group laws and equivariant complex cobordism have been around for a while, but it may be time to take it up a notch. Recent work of Abrams-Kriz gives a place to start.

• First think about the Abrams-Kriz work; they give a calculation, but it might be fruitful to think about that calculation in terms of the functors represented by these calculated rings—a sort of Lazard ring interpretation. There is some unpublished work of Greenlees (on his home page) worth reading.

• Abrams-Kriz works only for finite abelian groups, mostly because they have good duals. Is there a good notion of global equivariant formal group laws, in the fashion on Bohmann and Schwede? Does it good interpretation algebraically (Is there a Lazard object?) or homotopically?

• (S. Schwede) In global equivariant homotopy theory, the role of complex bordism is the universal “globally complex oriented theory”; the coefficients of global complex bordism have a very rich algebraic structure (global power functor with Euler classes). What is the universal property that the coefficients enjoy?