Finite \(K(\pi, 1)\)s for Artin groups. (English) Zbl 0930.55006

Charney, Ruth; Davis, Michael W.

Quinn, Frank (ed.), Prospects in topology. Proceedings of a conference in honor of William Browder, Princeton, NJ, USA, March 1994. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 138, 110-124 (1995).

It is not so widely known that there is a beautiful and simple description of a certain finite CW-complex \(Z\) which is a \(K(\pi,1)\) for the braid group on \(n+1\) strands. The complex \(Z\) is obtained by identifying certain faces on an \(n\)-dimensional convex polytope called a “permutohedron”. The resulting CW complex has exactly one \(k\)-cell for each \(k\)-element subset of \(\{1,\dots,n\}\). We are not sure who first discovered this complex, but we first heard it described by C. Squirer in the mid 1980’s and later by K. Tatsuoka. It has also been known to J. Milgram for some time. The purpose of this paper is to give the details of the construction of this complex and its generalizations to other Artin groups.
For the entire collection see [Zbl 0833.00037].

MSC:

55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology
57M25	Knots and links in the \(3\)-sphere (MSC2010)
57M07	Topological methods in group theory
20F36	Braid groups; Artin groups
55P20	Eilenberg-Mac Lane spaces

Keywords:

permutohedron; \(K(\pi,1)\) for the braid group on \(n+1\) strands