Eilenberg Mac Lane spectra as p-cyclonic Thom spectra

Abstract

Hopkins and Mahowald gave a simple description of the mod p Eilenberg Mac Lane spectrum HF p as the free E 2 -algebra with an equivalence of p and 0. We show for each faithful 2-dimensional representation λ of a p-group G that the G-equivariant Eilenberg Mac Lane spectrum HF p is the free E λ -algebra with an equivalence of p and 0. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral 2-subgroups of O(2). The main new idea is that HF p has a simple description as a p-cyclonic module over THH(HF p ). We show our result is the best possible one in that it gives all groups G and representations V such that HF p is the free E V -algebra with an equivalence of p and 0.