Fixed point and tilting

Speaker: Vignesh Subramanian – Indian Statistical Institute, Kolkata

Abstract: Given an $\mathbb{E}_{\infty}$-algebra $A$ over $\mathbb{F}_p$, just as in classical algebra, there exists a homotopically coherent version of Frobenius on $A$, called the Tate valued Frobenius $A \rightarrow A^{tC_p}$. In this talk, we recall the notion Frobenius perfect $\mathbb{F}_p$-algebra and construct a certain version of perfection $A^{\flat}$, we call this construction tilting and give an explicit formula for the computation of homotopy groups of the tilt via power operations. As an application, given $X$ a finite $G$-CW complex where $G$ is an elementary abelian group, we offer a recipe to recover the $p$-local homotopy type of the genuine fixed point $X^{G}$ from the Borel equivariant cohomology of $X$. This application can be considered a categorification of Smith theory, which plays a significant role in the ideas surrounding proof of the Sullivan conjecture. This is joint work with Robert Burklund.