Stefan Schwede: Equivariant degrees and symmetric products
Tid: On 2023-04-05 kl 15.15 - 17.00
Medverkande: Stefan Schwede (University of Bonn)
FR4 (Oskar Klein), Albanova (directions available here ).
14:15–15:00 Pre-colloquium by Lucas Piessevaux in FB54.
15:15–16:15 Colloquium lecture by Stefan Schwede.
16:15–17:00 SMC social get together with refreshments.
The degree is a classical invariant -- an integer that completely determines a continuous map between spheres of the same dimension, up to homotopy. In the presence of additional symmetries, i.e., for equivariant maps between spheres of linear representations, this integer refines to a more subtle and powerful invariant. This equivariant degree can be viewed as an integer-valued function on the collection of subgroups, or as an element in the so-called `Burnside ring' of the group.
In this talk I will review the notion of degree of a map between spheres, and of its equivariant refinement. The answer is best organized as an isomorphism, due to Graeme Segal, between the Burnside ring of a finite group and the equivariant stable homotopy groups of spheres. Towards the end, I want to explain a more recent calculation of equivariant homotopy groups of symmetric products of spheres. This calculation is best phrased in terms of 'globally-equivariant' language, i.e., by considering the values at all groups at once, along with the operations of restriction and transfer that connect them.