Vignesh Subramanian: Smith Theory Revisited

Abstract: The Sullivan conjecture is one of the landmark results in unstable homotopy theory, relating genuine fixed points of a finite \(p\)-group action to homotopy fixed points (after \(p\)-completion). In this talk, I will explain a new approach to the Sullivan conjecture for \(C_p\)-actions using perfect commutative \(\mathbb{F}_p\)-algebras.

The key idea is that the tilting functor provides an inverse to the Tate construction in the perfect setting, allowing us to recover the cochains on the genuine fixed-point space from Borel-equivariant cochains. Combining this with Mandell's theorem and the Segal conjecture gives a short proof of the Sullivan conjecture.

This is joint work with Robert Burklund.