Computational Motivic and Equivariant Homotopy Theory

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 to classify all of the possible mappings of a sphere of a given dimension onto a sphere of different dimension. More 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. Even more recently, greater understanding of how these various contexts impact each other has emerged. The PI will endeavor to expand the range in which these questions are understood, especially in the setting of spheres with a twofold symmetry. Another component of the project focuses on an invariant known as K-theory, which lies at the interface of topology and algebra. The PI will develop a version of K-theory which is simultaneously equipped with symmetries and multiplications. The PI will continue his joint work with Dan Isaksen on computations in the motivic and C2-equivariant stable homotopy groups of spheres. The computation of motivic stable homotopy groups over C is fairly similar to that of classical stable homotopy groups. From there, the Bockstein spectral sequence recovers the motivic stable homotopy groups over R. Finally, accounting for the "negative cone" in the C2-equivariant homology of a point leads to the equivariant stable homotopy groups. In a different direction the PI, together with May, Merling, and Osorno, will continue to investigate the presentation of G-spectra as spectral Mackey functors. A centerpiece of this theory is an equivariant infinite loop space machine with good multiplicative properties. A number of tools that figure centrally in applications, such as equivariantly commutative multiplications, norm maps, and geometric fixed points, are not well understood in this model of G-spectra.