2012/09/21 by John Guaschi, Guaschi, John, Daniel Juan-Pineda +3
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.1209.4791
openalex publication_date 2012/09/21 · openalex created_date 2018/07/10 · openalex updated_date 2026/07/28
We study K-theoretical aspects of the braid groups B_n(\mathbbS2) on n strings of the 2-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture~\citeJM. In light of this, in order to determine the algebraic K-theory of the group ring ℤ[B_n(\mathbbS2)], one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gon\c calves and the first author. We calculate the Whitehead and K_-1-groups of the group rings of the finite subgroups (dicyclic and binary polyhedral) of B_n(\mathbbS2) for all 4≤ n≤ 11. Some new phenomena occur, such as the appearance of torsion for the K_-1-groups. We then go on to study the case n=4 in detail, which is the smallest value of n for which B_n(\mathbbS2) is infinite. We show that B_n(\mathbbS2) is an amalgamated product of two finite groups, from which we are able to determine a universal space for proper actions of the group B_n(\mathbbS2). We also calculate the algebraic K-theory of the infinite virtually cyclic subgroups of B_n(\mathbbS2), including the Nil groups of the quaternion group of order 8. This enables us to determine the lower algebraic K-theory of ℤ[B_n(\mathbbS2)].