Title: Strict units in the sphere spectrum
Speaker: Shachar Carmeli
Speaker Info: University of Copenhagen
Brief Description: Part of the Chicagoland Topology Seminar Series. *Note the unusual time*. Contact the organizer for the Zoom link.
For a commutative ring spectrum R, there are two natural candidates for the "multiplicative group" of R. One is the spectrum of invertible elements gl_1(R), and the other is the spectrum of "strict units" in R, denoted G_m(R). The latter is obtained from the former by taking the mapping spectrum out of the Eilenberg-McLane spectrum Z.Date: Tuesday , January 18, 2022
The spectrum gl_1(R) is closely related to R itself. For example, the homotopy groups of R and gl_1(R) agree in all degrees above 0. On the other hand, the spectrum G_m(R) is a more subtle object and the subject of active research.
The initial example of a commutative ring spectrum is the sphere spectrum S. In my talk, I will describe a work in progress dedicated to the computation of G_m(S) using Tate's construction for the finite group C_p. I will also explain how to compute the connective spectrum of maps from Z to the Picard spectrum of S, which gives a (non-trivial) delooping of G_m(S).