
TL;DR
This paper introduces a new family of groups called vector braids, generalizing classical braid groups to points in complex spaces and projective spaces, and explores their algebraic and homological properties.
Contribution
It defines the vector braid groups, provides presentations, and investigates their homology, extending classical braid theory to higher-dimensional and projective settings.
Findings
Presented a group $PL_n$ surjecting onto $P_n^2$
Showed $PL_n$ induces isomorphism on first and second homology with $P_n^2$
Derived an infinitesimal presentation of $P_n^2$
Abstract
In this paper we define a new family of groups which generalize the {\it classical braid groups on} . We denote this family by where . The family is the set of classical braid groups on strings. The group is the set of motions of unordered points in , so that at any time during the motion, each of the points span the whole of as an affine space. There is a map from to the symmetric group on letters. We let denote the kernel of this map. In this paper we are mainly interested in understanding . We give a presentation of a group which maps surjectively onto . We also show the surjection induces an isomorphism on first and second integral homology and conjecture that it is an isomorphism. We then find an infinitesimal presentation of…
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics
