Equivariant Stable Stems

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

Spheres are simple yet important objects of study in topology. One of the central questions of algebraic topology is the classification of all possible mappings of a high-dimensional sphere onto a sphere of lower dimension. It turns out that this classification of mappings of spheres has wide-ranging repercussions in geometry and in physics. Recently, this question has received attention in other contexts: when the spheres are considered in the realm of algebraic geometry, or when the spheres have specified symmetries which must be preserved by the mappings in question. More recently, greater understanding of how these various contexts impact each other has emerged. The research supported by this award will employ these newfound connections to expand the range in which these questions are understood, especially in the setting of spheres with a twofold symmetry. This project provides and funds research training for graduate students. The principal investigator will continue joint work with Dan Isaksen on computations of the motivic and C2-equivariant stable homotopy groups of spheres. The R-motivic computations are more approachable, and these determine a portion of the C2-equivariant stable homotopy groups. The main tools will be the rho-Bockstein spectral sequence and the Adams spectral sequence. Various techniques will be employed to run these spectral sequences, including the use of Massey products. The PI and collaborators will also investigate v1-periodicity in the R-motivic and C2-equivariant settings, producing finite complexes that support periodicity operators. This will lead to periodic families of elements in the stable homotopy groups of spheres in these contexts. In another direction, another collaboration will analyze additive power operations for equivariant cohomology theories. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.