On Bredon Cohomology with Mackey Functor in Morava $K$-Theory
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Keywords:
Morava K-theory, equivariant CW complex, Bredon HomologyAbstract
Utilizing the algebraic framework of Mackey functors, we analyze the cochain complex and explicitly compute the cellular differentials governed by restrictions and transfer maps within Morava $K$-theory $K(n)^*$. We focus on the geometric construction of the octahedral surface 2-complex, isolating purely equivariant phenomena driven by stabilizer jumps. By analyzing the complex directly over the Morava ring $K(n)_*[x,y]$, we determine the zero-th and first codifferentials. We show that while the underlying space is geometrically trivial in degree one, the equivariant chromatic classes persist due to the cluster's sector geometry. Finally, we provide a complete description of the first Bredon cohomology group $H_{Q_8}^1(X; \underline{K(n)}^*)$ expressed as a quotient of the coefficient ring. These results contribute to the structural classification of equivariant cohomology rings for finite $p$-groups with concentrated even degrees.
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