Structure and Cohomology in Fusion Systems

University Of Louisiana At Lafayette, Lafayette LA

Investigators

Abstract

The investigator will study problems at the interface between finite group theory and topology. Finite group theory is the study of the symmetry of finite objects by algebraic means. While the project involves fundamental research, group theory is applicable in many of the natural sciences where highly symmetric objects are found, including in biology, chemistry, and physics, and in the study of communication networks and cryptographic schemes. A finite simple group is akin to an atom in that it cannot be broken down into smaller groups. Each finite group of symmetries is made up of simple groups stacked on top one another analogously to the way molecules are built out of atoms. One of the premier mathematical achievements of the 20th century was the description and classification of all the finite simple groups (CFSG). Its proof is however long and difficult, currently spanning around 10,000-15,000 pages, and it is desirable to have significantly simpler proofs. The PI will investigate this problem by using strategies that come form topology, a branch of mathematics that studies properties of objects that are invariant under continuous transformations. One bridge between group theory and topology is given by the classifying space of a group. When the group is finite, there is an associated set of prime numbers, and both the group and its classifying space can be studied "one prime at a time". This strategy has been abstracted into the notion of a p-fusion system, the basic object of study. The project arises out of two recent major developments in fusion systems: a program for the classification of simple 2-fusion systems of component type (CSFS) whose ultimate aim is to give a substantially simpler proof of the CFSG, and the recent solution of the existence and uniqueness of centric linking systems, which provide the bridge from fusion systems to topology. Both developments are brought together by their connections with the cohomology of various functors defined on the orbit category and other related categories. The investigator will: (1) investigate fusion systems at the prime 2 whose linking systems support a noninner rigid automorphism by using methods from the CSFS, (2) work to find necessary and sufficient conditions for the existence and uniqueness of centralizers of fusion subsystems through the definition and computation of the cohomology of functors which obstruct internal rigid actions on centric linking systems, and (3) work directly within the nearly-completed CSFS to solve certain outstanding problems, and to generalize others for use in related problems such as in (3). The three interrelated projects aim to gain a better understanding of some aspect of the p-local structure of finite groups and/or the homotopy theory of p-completed classifying spaces by exploiting their interplay. Project (1) uses finite group theoretic methods to provide a better understanding of the group of self-homotopy equivalences of the p-completed classifying space of a finite group, while project (2) provides an application of functor cohomology to the open problem of constructing centralizers. The construction of centralizers is a fundamental problem with applications within the CSFS, e.g. in the defining and understanding of standard subsystems of fusion systems, and it also has potential applications to the open problem of describing maps between different p-completed classifying spaces. Project (3) sits within the CSFS proper, simultaneously supporting the goals in (1) and the completion of the classification program. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.