2015/03/15 by Niket Gowravaram, Gowravaram, Niket, Uma Roy +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1503.04372
openalex publication_date 2015/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate a novel diagrammatic approach to examining strict actions of a Coxeter group or a braid group on a category. This diagrammatic language, which was developed in a series of papers by Elias, Khovanov and Williamson, provides new tools and methods to attack many problems of current interest in representation theory. In our research we considered a particular problem which arises in this context. To a Coxeter group W one can associate a real hyperplane arrangement, and can consider the complement of these hyperplanes in the complexification YW. The celebrated K(π,1) conjecture states that YW should be a classifying space for the pure braid group, and thus a natural quotient YW/W should be a classifying space for the braid group. Salvetti provided a cell complex realization of the quotient, which we refer to as the Salvetti complex. In this paper we investigate a part of the K(π,1) conjecture, which we call the K(π,1) conjecturette, that states that the second homotopy group of the Salvetti complex is trivial. In this paper we present a diagrammatic proof of the K(π,1) conjecturette for a family of braid groups as well as an analogous result for several families of Coxeter groups.