BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Igor Sikora (University of Warwick) DTSTART:20211004T103000Z DTEND:20211004T113000Z DTSTAMP:20260422T140001Z UID:BilTop/25 DESCRIPTION:Title: $RO(C_2)$-graded coefficients of $C_2$-Eilenberg-MacLane spectra\nby I gor Sikora (University of Warwick) as part of Bilkent Topology Seminar\n\n Lecture held in SB-Z11.\n\nAbstract\nIn non-equivariant topology the ordin ary homology of a point is described by the dimension axiom and is quite s imple - namely\, it is concentrated in degree zero. The situation in $G$-e quivariant topology is different. This is due to the fact that Bredon homo logy - the equivariant counterpart of the ordinary homology - is naturally graded over $RO(G)$\, the ring of $G$-representations. Whereas the equiva riant dimension axiom describes the part of the Bredon homology of a point which is graded over trivial representations\, it does not put any requir ements on the rest of the grading - in which the homology may be quite com plicated.\n\nThe $RO(G)$-graded Bredon homology theories are represented b y $G$-Eilenberg-MacLane spectra\, and thus the Bredon homology of a point is the same thing as coefficients of these spectra. During the talk I will present the method of computing the $RO(C_2)$-graded coefficients of $C_2 $-Eilenberg-MacLane spectra based on the Tate square. As demonstrated by G reenlees\, the Tate square gives an algorithmic approach to computing the coefficients of equivariant spectra. In the talk we will discuss how to us e this method to obtain the $RO(C_2)$-graded coefficients of a $C_2$-Eilen berg-MacLane spectrum as a $RO(C_2)$-graded abelian group. We will also pr esent the multiplicative structure of the $C_2$-Eilenberg-MacLane spectrum associated to the Burnside Mackey functor. This allows us to further desc ribe the $RO(C_2)$-graded coefficients of any $C_2$-Eilenberg-MacLane spec trum as a module over the coefficients of the $C_2$-Eilenberg-MacLane spec trum of the Burnside Mackey functor. Finally\, we will discuss the $RO(C_2 )$-graded ring structure of coefficients of spectra associated to ring Mac key functors.\n LOCATION:https://researchseminars.org/talk/BilTop/25/ END:VEVENT END:VCALENDAR