Equivariant triangulations of tori of compact Lie groups and hyperbolic extension to non-crystallographic Coxeter groups

Abstract

Given a simple connected compact Lie group K and a maximal torus T of K, the Weyl group W = NK (T )/T naturally acts on T .First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit W -equivariant triangulation of T . We describe the associated cellular homology chain complex and give a formula for the cup product on its dual cochain complex, making it a Z[W ]-dg-algebra.Next, remarking that the combinatorics of this dg-algebra is still valid for Coxeter groups, we associate a closed compact manifold T(W ) to any finite irreducible Coxeter group W , which coincides with a torus if W is a Weyl group and is hyperbolic in other cases. Of course, we focus our study on non-crystallographic groups, which are I2(m) with m = 5 or m ≥ 7, H3 and H4.The manifold T(W ) comes with a W -action and an equivariant triangulation, whose related Z[W ]-dg-algebra is the one mentioned above. We finish by computing the homology of T(W ), as a representation of W .