Artin Salvetti Complex

The Artin Salvetti complex of a Coxeter matrix on a finite set is the standard CW-complex for its Artin group, with one cell of dimension for each subset whose generated Coxeter subgroup is finite. The empty subset contributes one 0-cell, singletons contribute generating loops, and two-element subsets with finite Coxeter label contribute the Artin relation cells. The higher cells retain the corresponding finite Coxeter subgroup data. Its fundamental group is the Artin group.

For two commuting generators, the complex is a torus with one 0-cell, two 1-cells, and one two-dimensional cell. The Artin K(pi,1) conjecture asserts that this full complex is an aspherical space, so that it is a classifying space for the Artin group.