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A partial order on the symmetric group and new \(K(\pi,1)\)’s for the braid groups. (English) Zbl 1011.20040
The author defines a certain partial order in the symmetric group \(S_n\) using the Cayley graph structure with respect to the set of all transpositions. Then a certain group \(\Gamma_n\) is introduced, which turns out to be isomorphic to the Artin braid group with \(n\) strands, so as to be generated by the elements geodesically governed by the \(n\)-cycle \((1,2,\dots,n)\in S_n\) with explicit defining relations. A certain simplicial complex induced on the vertex set \(\Gamma_n\) is shown to be contractible.

20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20B30 Symmetric groups
57M07 Topological methods in group theory
Full Text: DOI
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