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Topology Seminar

Tue., Jan. 31, 2023 1:00 p.m.

Location: Zoom

Speaker: Igor Sikora, Bilkent University

Title: RO(C2)-graded coefficients of C2-Eilenberg-MacLane spectra (113 kB) PDF file

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In non-equivariant topology, the ordinary homology of a point is described by the dimension axiom and is quite simple - namely, it is concentrated in degree zero. The situation in G-equivariant topology is different. This is because Bredon homology - the equivariant counterpart of ordinary homology - is naturally graded over RO(G), the ring of G-representations. Whereas the equivariant dimension axiom describes the part of the Bredon homology of a point graded over trivial representations, it does not put any requirements on the rest of the grading - in which the homology may be quite complicated.

The RO(G)-graded Bredon homology theories are represented by G-Eilenberg-MacLane spectra, and thus the Bredon homology of a point is the same as coefficients of these spectra. During the talk, I will present the method of computing the RO(C2)-graded coefficients of C2-Eilenberg-MacLane spectra based on the Tate square. As demonstrated by Greenlees, the Tate square gives an algorithmic approach to computing the coefficients of equivariant spectra. In the talk, we will discuss how to use this method to obtain the RO(C2)-graded coefficients of a C2-Eilenberg-MacLane spectrum as a RO(C2)-graded abelian group. We will also present the multiplicative structure of the C2-Eilenberg-MacLane spectrum associated to the Burnside Mackey functor. Time permitting, we will further discuss how to use this knowledge to derive a multiplicative structure for the coefficients for any ring Mackey functor.