2017/11/09 by Dylan Wilson, Wilson, Dylan
Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1711.03472
openalex publication_date 2017/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we give an algebraic description of the category of n-slices for an arbitrary group G, in the sense of Hill-Hopkins-Ravenel. Specifically, given a finite group G and an integer n, we construct an explicit G-spectrum W (called an isotropic slice n-sphere) with the following properties: (i) the n-slice of a G-spectrum X is equivalent to the data of a certain quotient of the Mackey functor \underline[W,X] as a module over the endomorphism Green functor \underline[W,W]; (ii) the category of n-slices is equivalent to the full subcategory of right modules over \underline[W,W] for which certain restriction maps are injective. We use this theorem to recover the known results on categories of slices to date, and exhibit the utility of our description in several new examples. We go further and show that the Green functors \underline[W,W] for certain slice n-spheres have a special property (they are "geometrically split") which reduces the amount of data necessary to specify a \underline[W,W]-module. This step is purely algebraic and may be of independent interest.