Equivariant Topology and Derived Algebra

This volume contains eight research papers inspired by the 2019 'Equivariant Topology and Derived Algebra' conference, held at the Norwegian University of Science and Technology, Trondheim in honour of Professor J. P. C. Greenlees' 60th birthday. These papers, written by experts in the field, are intended to introduce complex topics from equivariant topology and derived algebra while also presenting novel research. As such this book is suitable for new researchers in the area and provides an excellent reference for established researchers. The inter-connected topics of the volume include: algebraic models for rational equivariant spectra; dualities and fracture theorems in chromatic homotopy theory; duality and stratification in tensor triangulated geometry; Mackey functors, Tambara functors and connections to axiomatic representation theory; homotopy limits and monoidal Bousfield localization of model categories.

The dominant dimension of cohomological Mackey functors

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category [Formula: see text] of cohomological Mackey functors of a [Formula: see text]-block [Formula: see text] of a finite group with a nontrivial defect group is [Formula: see text].

The slice spectral sequence for the C4$C_{4}$ analog of real K-theory

We describe the slice spectral sequence of a 32-periodic $C_{4}$-spectrum $K_{[2]}$ related to the $C_{4}$ norm ${\mathrm{MU}^{((C_{4}))}=N_{C_{2}}^{C_{4}}\mathrm{MU}_{\mathbb{R}}}$ of the real cobordism spectrum $\mathrm{MU}_{\mathbb{R}}$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor $\underline{\pi}_{*}K_{[2]}$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K-theory spectrum $K_{\mathbb{R}}$ was first analyzed by Dugger. The $C_{8}$ analog of $K_{[2]}$ is 256-periodic and detects the Kervaire invariant classes $\theta_{j}$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that $\theta_{j}$ does not exist for ${j\geq 7}$.