BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Malkhaz Bakuradze (Iv. Javakhishvili Tbilisi State University)
DTSTART:20260513T113000Z
DTEND:20260513T123000Z
DTSTAMP:20260526T034527Z
UID:PHK-cohomology-seminar/160
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PHK-cohomolo
 gy-seminar/160/">On Equivariant Bredon Cohomologies with Mackey Functor in
  Morava K-theory</a>\nby Malkhaz Bakuradze (Iv. Javakhishvili Tbilisi Stat
 e University) as part of Prague-Hradec Kralove seminar Cohomology in algeb
 ra\, geometry\, physics and statistics\n\nLecture held in blue lecture roo
 m + ZOOM meeting.\n\nAbstract\nOur consideration of the Mackey functor $\\
 underline{M}$ for Bredon cohomology relies on the explicit Morava K-theory
  ring calculations for ’good’ finite p-groups provided in [Bakuradze\,
  M. Morava K-theory rings for finite groups. J. Homotopy Relat. Struct. 20
 \, 567–630 (2025)]\n\nLet $G$ be a finite group and $X$ a finite $G$-CW 
 complex. We define the contravariant Bredon $\\underline{K(n)}^*$ module o
 n the orbit category $\\mathcal{O}_G$ by setting $M(G/H) = K(n)^*(BH)$ for
  each subgroup $H \\subseteq G$. The $n$-th Bredon cochain group is define
 d as the direct sum over the $n$-dimensional orbit representatives:\n\n  \
 \[ C_G^n(X\; \\underline{K(n)}^*) = \\bigoplus_{\\sigma \\in \\text{Orbits
 }_n(X)} K(n)^*(BG_\\sigma) \\]\n\n  The coboundary operator $\\delta^n: C_
 G^n \\to C_G^{n+1}$ is given by:\n\n  \\[ (\\delta^n \\alpha)(\\psi) = \\s
 um_{\\sigma \\in \\text{Orbits}_n} \\sum_{g \\in G} [\\psi : g\\sigma] \\c
 dot \\text{Res}_{G_\\psi}^{gG_\\sigma g^{-1}}(\\alpha_\\sigma) \\]\n\n  wh
 ere $[\\psi : g\\sigma]$ are the degrees of the corresponding attaching ma
 ps and $\\text{Res}$ is the restriction map in Morava $K$-theory. The $n$-
 th Bredon cohomology group of $X$ with coefficients in $\\underline{K(n)}^
 *$ is:\n\n  \\[ H_G^n(X\; \\underline{K(n)}^*) = \\frac{\\ker(\\delta^n)}{
 \\text{im}(\\delta^{n-1})} \\]\n\n  While the evenness of the Bredon cohom
 ology is a sufficient rather than a necessary criterion for the goodness o
 f $X$\, it remains the most computationally viable path. If this $E_2$-con
 dition is met\, the equivariant Atiyah-Hirzebruch spectral sequence collap
 ses immediately.\n
LOCATION:https://researchseminars.org/talk/PHK-cohomology-seminar/160/
END:VEVENT
END:VCALENDAR