## UBC Topology Seminars: Krishanu Sankar

- Date: 10/25/2017
- Time: 15:15

University of British Columbia

Symmetric Powers and the Dual Steenrod Algebra - Part 1

The mod $p$ Steenrod algebra is the (Hopf) algebra of stable operations on mod $p$ cohomology, and in part measures the subtle behavior of $p$-local homotopy theory (as opposed rational homotopy theory, which is much simpler). A classical theorem of Dold-Thom tells us that the infinite symmetric power of the $n$-dimensional sphere is the Eilenberg-Maclane space K(Z, n),and one can use an appropriate modification of this construction to compute the dual Steenrod algebra. The infinite symmetric power of the sphere spectrum has a filtration whose $k$-th cofiber miraculously turns out to be the Steinberg summand (from modular representation theory of GL_k(F_p)) of the classifying space of (Z/p)^k. This opens the door for slick computations - for example, the Milnor indecomposables can be picked out as explicit cells.

In this talk, I will introduce the concepts and results chronologically. I will also include hands-on homotopy theory computations as time permits.

Location: ESB 4133