Signalizer functors, existence, and applications to the fundamental group

TL;DR

This paper constructs explicit signalizer functors to solve longstanding problems in group theory, establishing the existence of centric linking systems and exploring applications to fundamental groups with numerous examples.

Contribution

It provides an explicit construction of signalizer functors for various group models, solving key problems in fusion systems and linking systems.

Findings

Solved Oliver's seventh problem using explicit signalizer functors.

Proved existence of centric linking systems in certain group models.

Applied results to fundamental groups with illustrative examples.

Abstract

We solve the seventh problem of Oliver's list [M.\ Aschbacher, R.\ Kessar, B.\ Oliver, \textit{Fusion systems in algebra and topology}, LMS Lecture Note Series: 31, Cambridge University Press, 2011] via an explicit signalizer functor construction in the sense of Aschbacher-Chermak for various group models. Moreover we prove the existence of centric linking systems via group models in certain cases which is the first problem and give applications to the fundamental group which is the eighth problem of the list respectively. We illustrate with many examples.