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<title>Cohomology in algebra, geometry, physics and statistics</title>
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Wednesday, 13 May 2026 - 13:30 to 14:30 <br />
Place: Institute of Mathematics of ASCR, Žitná 25, Praha 1, the blue lecture room + ZOOM meeting
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Speaker: Malkhaz Bakuradze, Iv. Javakhishvili Tbilisi State University<br />
Title: On Equivariant Bredon Cohomologies with Maskey Functor in Morava K-theory
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Abstract <br />
<p>Abstract in the attached pdf file. Here is a latex transcription.</p><p><br></p><p>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)]</p><p><br></p><p>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:</p><p> \[ C_G^n(X; \underline{K(n)}^*) = \bigoplus_{\sigma \in \text{Orbits}_n(X)} K(n)^*(BG_\sigma) \]</p><p> The coboundary operator $\delta^n: C_G^n \to C_G^{n+1}$ is given by:</p><p> \[ (\delta^n \alpha)(\psi) = \sum_{\sigma \in \text{Orbits}_n} \sum_{g \in G} [\psi : g\sigma] \cdot \text{Res}_{G_\psi}^{gG_\sigma g^{-1}}(\alpha_\sigma) \]</p><p> 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:</p><p> \[ H_G^n(X; \underline{K(n)}^*) = \frac{\ker(\delta^n)}{\text{im}(\delta^{n-1})} \]</p><p> 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. </p><p>-----------------------------------------------------------------------------------------------</p><p class="ql-align-justify"><span style="background-color: rgb(255, 255, 255); color: rgb(34, 34, 34);">We shall open the seminar room and ZOOM meeting at 13.15 for coffee and close ZOOM at 15.30</span></p><p class="ql-align-justify"><br></p><p class="ql-align-justify"><span style="background-color: rgb(255, 255, 255); color: rgb(34, 34, 34);">Join Zoom Meeting</span></p><p class="ql-align-justify"><br></p><p class="ql-align-justify"><a href="https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09" rel="noopener noreferrer" target="_blank" style="background-color: rgb(255, 255, 255); color: rgb(17, 85, 204);">https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09</a></p><p class="ql-align-justify"><br></p><p class="ql-align-justify"><span style="background-color: rgb(255, 255, 255); color: rgb(34, 34, 34);">Meeting ID:99598413922</span></p><p class="ql-align-justify"><span style="background-color: rgb(255, 255, 255); color: rgb(34, 34, 34);">Passcode:Galois</span></p><p class="ql-align-justify"><span style="background-color: rgb(255, 255, 255); color: rgb(34, 34, 34);">-----------------------------------------------------</span></p><p class="ql-align-justify">On May 20 Igor Khavkine shall give a talk <a href="https://www.math.cas.cz/index.php/events/event/4137" rel="noopener noreferrer" target="_blank">Compatibility complex for the conformal-to-Einstein operator </a></p><p class="ql-align-justify"><br></p><p><br></p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/16
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Cohomology in algebra, geometry, physics and statistics mailing list <br />
phk_seminar@math.cas.cz <br />
https://list.math.cas.cz/listinfo/phk_seminar@math.cas.cz
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