AbstractsMathematics

Gerstenhaber algebras and the homology of spaces of long knots and long links

by Paul Arnaud Songhafouo Tsopméné




Institution: Université Catholique de Louvain
Department: Institut de recherche en mathématique et physique
Year: 2014
Keywords: Operads; Long knots and links; Gerstenhaber algebras
Record ID: 1077165
Full text PDF: http://hdl.handle.net/2078.1/146519


Abstract

The goal of this thesis is to better understand the homology of spaces of long knots and long links. Our approach is based on the one hand on the Goodwillie and Weiss calculus of functors, and on the other hand on Kontsevich formality theorems. In the study of the space of long knots, we determine the natural algebraic structure of its real homology. More precisely we prove that this homology is isomorphic as a Gerstenhaber algebra to the E2 page of the homology Bousfield-Kan spectral sequence associated to Sinha’s cosimplicial model for the space of long knots. To obtain that result, we show that the Kontsevich operad is formal over reals as a multiplicative operad using arguments of model categories for nonsymmetric operads. In the study of the space of long links, we prove that the spectral sequence computing its rational homology collapses at the E2 page, thus solving a conjecture of Munson-Volic. Our method enables us also to determine the rational homology of high dimensional analogues of the space of long links. (SC - Sciences)  – UCL, 2014