Stable Homotopy Groups: Theory and Computation

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Spheres are the basic building blocks of topology. More complicated topological objects can be constructed by fitting these spheres together. Spheres of different dimensions can fit together in only a few different ways. Enumerating these combinations of spheres is one of the fundamental questions of a branch of mathematics called homotopy theory. This project focuses on the computation of the stable homotopy groups of spheres, using three general categories of techniques: theory, manual computation, and machine computation. These categories are mutually reinforcing. Theory establishes new techniques of computation; manual computation reveals new structure that points toward theoretical advances; machine computation provides independent verification of manual computation and carries it even further. Broader impacts of the project include development and growth of the electronic Computational Homotopy Theory (eCHT) virtual research community. With a long-term goal of self-sustainability, the eCHT community will experiment with different types of online engagement, such as foundational courses for early-career graduate students, seminar series on focused topics, career development events, online conferences, and building connections to online networks in neighboring subjects. These new modes of interaction aim to overcome traditional barriers to entry, especially for people in remote geographical locations and for those with non-traditional backgrounds. The project's main research goal is to compute stable homotopy groups of spheres in the classical, C-motivic, R-motivic, and C2-equivariant contexts. It breaks into three main topics: the Adams spectral sequence, the effective slice spectral sequence, and deformations of stable homotopy theory. The effective spectral sequence is a replacement for the Adams-Novikov spectral sequence in the R-motivic and C2-equivariant situations. The first two topics are essentially computational, although they point towards a number of more theoretical questions. Two of the many possible outcomes of these computations are: new information about Mahowald (root) invariants; and settling the last unknown case of the Kervaire invariant one problem. The third topic is a theoretical perspective that was recently uncovered by computational exploration. Deformations will play a growing role in computational stable homotopy theory.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.