Equivariant triangulations of tori of compact Lie groups and hyperbolic extension to non-crystallographic Coxeter groups (Q6051032)

The paper is part of a program to construct \(W\)-equivariant cellular structures in Lie theory, and more precisely for the flag manifold \(K/T\) and the classifying space for \(T\) where \(K\) denotes a simple compact Lie group, \(T < K\) a maximal torus and \(W = N_K(T)/T\) the Weyl group acting on \(T\). ``Our first aim is to provide an explicit \(W\)-equivariant triangulation of \(T\) and to describe the associated cellular homology cochain complex, as a \(W\)-dg-ring.'' ``For a non-crystallographic Coxeter group \(W\), using compact hyperbolic extensions rather than affine ones, we construct a compact \(W\)-manifold \(T(W)\), which is an analogue of a torus for \(W\). We exhibit a \(W\)-equivariant triangulation of \(T(W)\) and compute the associated \(W\)-dg-ring. Also, we derive its homology representation.'' There arise some interesting hyperbolic manifolds from the construction; e.g., the compact hyperbolic 4-manifold \(T(H_4)\) is the hyperbolic 4-manifold constructed by \textit{M. W. Davis} by identifying faces of the 4-dimensional regular hyperbolic 120-cell [Proc. Am. Math. Soc. 93, 325--328 (1985; Zbl 0533.51015)], the hyperbolic 3-manifold \(T(H_3)\) is constructed by the reviewer from a regular hyperbolic dodecahedron [Math. Proc. Camb. Philos. Soc. 113, No. 1, 87--90 (1993; Zbl 0802.57004)], the case of the flag manifold of \(SL_3(\mathbb R)\) is considered in a paper by \textit{R. Chirivì} et al. [Expo. Math. 40, No. 3, 572--604 (2022; Zbl 1502.57016)]. So the present paper offers a systematic conceptual approach which applies to any finite Coxeter group.