On homological properties of singular braids

Vershinin, Vladimir

doi:10.1090/S0002-9947-98-02048-0

On homological properties of singular braids

Author: Vladimir V. Vershinin
Journal: Trans. Amer. Math. Soc. 350 (1998), 2431-2455
MSC (1991): Primary 20J05, 20F36, 20F38, 18D10, 55P35
DOI: https://doi.org/10.1090/S0002-9947-98-02048-0
MathSciNet review: 1443895
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Abstract: Homology of objects which can be considered as singular braids, or braids with crossings, is studied. Such braids were introduced in connection with Vassiliev's theory of invariants of knots and links. The corresponding algebraic objects are the braid-permutation group $BP_{n}$ of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid $SB_{n}$ which embeds into the singular braid group $SG_{n}$ . The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group

$\begin{equation*}\mathbb{Z}\times BBP_{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }\times S^{1} \times Y, \end{equation*}$

$\begin{equation*}\mathbb{Z}\times BSG_{\infty }^{+}\simeq S^{1}\times \Omega ^{2} S^{2}\times W, \end{equation*}$

where $Y$ is an infinite loop space and $W$ is a double loop space.

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