2007/06/08 by Hirotaka Tamanoi, Tamanoi, Hirotaka
Mathematics · Medicine · #55P35 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #math.AT #msc:55P35
paper · pdf · doi:10.48550/arxiv.0706.1277
13 pages
arxiv created 2007/06/08 · openalex publication_date 2007/06/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological proof of the BV identity connecting the loop product, the loop bracket, and the BV operator. The proof is based on an observation that the loop bracket and the BV derivation are given by the same cycle in the free loop space, except that they differ by parametrization of loops.