[PHK_seminar] [Cohomology]Wednesday 13 May, Malkhaz Bakuradze

[Petr Somberg]

 This is just a gentle reminder of our seminar for today!

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> Cohomology in algebra, geometry, physics and statistics
> Wednesday, 13 May 2026 - 13:30 to 14:30
> Place: Institute of Mathematics of ASCR, Žitná 25, Praha 1, the  blue
> lecture  room + ZOOM meeting
>
> Speaker: Malkhaz Bakuradze, Iv. Javakhishvili Tbilisi State University
> Title: On Equivariant Bredon Cohomologies with Maskey Functor in Morava
> K-theory
> Abstract
>
>
> Abstract in the attached  pdf file.  Here  is a latex transcription.
>
>
>
>
>
> Our 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)]
>
>
>
>
> Let $G$ be a finite group and $X$ a finite $G$-CW complex. We define the
> contravariant Bredon $\underline{K(n)}^*$ module on 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 defined as the direct
> sum over the $n$-dimensional orbit representatives:
>
> \[ C_G^n(X; \underline{K(n)}^*) = \bigoplus_{\sigma \in
> \text{Orbits}_n(X)} K(n)^*(BG_\sigma) \]
>
>
> The coboundary operator $\delta^n: C_G^n \to C_G^{n+1}$ is given by:
>
>
> \[ (\delta^n \alpha)(\psi) = \sum_{\sigma \in \text{Orbits}_n} \sum_{g
> \in G} [\psi : g\sigma] \cdot \text{Res}_{G_\psi}^{gG_\sigma
> g^{-1}}(\alpha_\sigma) \]
>
> where $[\psi : g\sigma]$ are the degrees of the corresponding attaching
> maps 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:
>
>
> \[ H_G^n(X; \underline{K(n)}^*) =
> \frac{\ker(\delta^n)}{\text{im}(\delta^{n-1})} \]
>
>
> While the evenness of the Bredon cohomology is a sufficient rather than a
> necessary criterion for the goodness of $X$, it remains the most
> computationally viable path. If this $E_2$-condition is met, the
> equivariant Atiyah-Hirzebruch spectral sequence collapses immediately.
>
> -------------------------------------------------------------------------
> ----------------------
> We shall open the seminar room and ZOOM meeting at 13.15 for coffee and
> close ZOOM at 15.30
>
>
> Join Zoom Meeting
>
>
>
> https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
>
>
>
> Meeting ID:99598413922
> Passcode:Galois
> -----------------------------------------------------
> On May 20  Igor Khavkine shall give a talk  Compatibility complex for the
> conformal-to-Einstein operator
>
>
>
>
>
>
>
> For more information see the seminar web page at
> https://www.math.cas.cz/index.php/events/seminar/16
>
>
> Cohomology in algebra, geometry, physics and statistics mailing list
> phk_seminar at math.cas.cz
> https://list.math.cas.cz/listinfo/phk_seminar@math.cas.cz
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