Motivic and Equivariant Stable Homotopy Groups

Wayne State University, Detroit MI

Investigators

Abstract

Spheres are the basic building blocks of all geometric objects. More complicated geometric objects can be constructed by fitting these spheres together, but spheres of different dimensions can fit together in only certain combinations. Enumerating these combinations of spheres is one of the fundamental questions of stable homotopy theory. This problem is known as the computation of homotopy groups of spheres. The project uses spectral sequences to carry out these computations. They are delicate, intricate, subtle, and complicated, but they are also understandable with enough insight and patience. Each time a new part of the machinery is understood, another layer of complexities becomes accessible for further study. Computer calculations play a large supporting role. The project promotes the use of videoconferencing to collaborate with peers, to advise graduate students, and to host online seminars. These efforts build towards a self-sustaining virtual mathematical research community. One major benefit of these new modes of interaction is that they erode traditional barriers to entry. This is especially beneficial for people in remote geographical locations and for those with non-traditional or non-prestigious backgrounds who are not typically afforded access to traditional departments of mathematics. The project will compute classical, C-motivic, R-motivic, and C2-equivariant stable homotopy groups. The key tools are the Adams spectral sequence, the Adams-Novikov spectral sequence, and the effective slice spectral sequence. The project consists of a series of interlocking problems, both algebraic and homotopical. Many of the problems suggest specific methods for obtaining calculational data about stable homotopy groups. Other problems address related structural issues, such as exotic periodicity and Mahowald invariants. A key idea is to use C-motivic calculations to see deeper into classical structure. Preliminary results suggest that this is a surprisingly powerful technique that allows for the computation of new stable homotopy groups in a range. The project also includes a series of techniques for computing R-motivic and C2-equivariant stable homotopy groups. This represents the first serious effort to grapple with equivariant versions of tools like the Adams spectral sequence at a computational level. The key point is to build up to the C2-equivariant computations gradually, through C-motivic and R-motivic intermediate steps. One example of the kind of payoff for this work are computations of new values of notoriously difficult Mahowald invariants. Finally, the project will study the effective slice spectral sequence, especially in R-motivic homotopy theory but also over arbitrary fields. This spectral sequence is a motivic replacement for the Adams-Novikov spectral sequence. The R-motivic calculation is more accessible, while the arbitrary fields involve interesting arithmetic. Inspired by the motivic effective slice filtration, the project will also attempt to study a new C2-equivariant filtration that ought to be useful for studying C2-equivariant stable homotopy groups. 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.