Mahowald (and later many others) proved that HF_2 is a Thom spectrum for a bundle over the double loop space on the 3-sphere. This has had lots of applications, including (more recently) some nilpotence results for E_2-algebras and a calculation of THH of F_2. In this talk, we explain how to extend the result to the case of G-spectra where G is a cyclic group of prime power order. The case of HF_2 over C_2 is joint with Mark Behrens, and the general case is joint with Jeremy Hahn. If there’s time, we’ll discuss how this fits into a program of Hill-Hopkins-Ravenel for solving the 3-primary Kervaire invariant problem.

There will also be a pretalk in the same room at 2.