On Equivariant Bredon Cohomologies with Mackey Functor in Morava K-theory

Abstract: 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.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( slides | video )

---

Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.

---