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(2003 / vol.6 / no.1)
Two multiplications in infinite braid-permutation group
Chan-Seok Jeong, Yongjin Song
Pages. 79-86     



The braid-permutation group $BP_n$ of rank $n$, which is a special
kind of Artin group, is a group of welded braids. It is also
described as a subgroup of the automorphism group of a free group.
The disjoint union of $BP_n$'s forms a symmetric monoidal category
$mathcal{BP}$ so that the group completion of the classifying
space of $mathcal{BP}$ is an infinite loop space. Since the
abelianization of $BP_n$ equals $mathbb{Z} oplus mathbb{Z}/2$
we can conjecture that
$$BBP^+ simeq S^1 imes BSigma_{infty}^+ imes Y,$$ for some
infinite loop space $Y$. V. Vershinin claimed that he constructed
of a splitting map, though a significant gap has been found. In
order to fill the gap we should verify a certain compatibility of
two distinct multiplications on $BBP^+$.



1. Introduction
2. The braid-permutation group
3. Infinite loop space and a splitting conjecture



braid-permutation group, classifying space, group completion, infinite loop space.



20F36, 55R35