Abstract: Equivariant homotopy and homology theories are invariants of spaces or spectra with group actions. In this talk, we focus on cyclic groups of order 2 and 3, $C_2$ and $C_3$. I will start by describing the algebraic structures that equivariant theories carry and in particular the equivariant homology of a point, which is a non-trivial bi-graded ring. Then, I will explain a $C_3$-equivariant computation of the Borelification of $tmf(2)$. We used a relative Adams spectral sequence, with input from the Hopf algebroid structure of the $C_3$-equivariant dual Steenrod algebra. This yields an entirely algebraic computation of the 3-local homotopy groups of $tmf$, the topological modular form spectrum.
This is joint work with Jeremy Hahn, Andrew Senger, and Adela Zhang.
