Subword Complexes and Kalai's Conjecture on Reconstruction of Spheres

TL;DR

This paper proves Kalai's conjecture for a specific class of simplicial spheres called subword complexes, which are related to Coxeter groups, and demonstrates that not all manifolds are reconstructible from their facet-ridge graphs.

Contribution

It confirms Kalai's conjecture for subword complexes and provides explicit examples of non-reconstructible manifolds.

Findings

Kalai's conjecture holds for subword complexes.

Not all manifolds are reconstructible from facet-ridge graphs.

Subword complexes are related to Coxeter groups and are conjectured to be polytopal.

Abstract

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani in 1987, via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai in 2009. In this paper, we show that Kalai's conjecture holds in the particular case of Knutson and Miller's spherical subword complexes. This family of simplicial spheres arises in the context of Coxeter groups, and is conjectured to be polytopal. In contrast, not all manifolds are reconstructible. We show two explicit examples, namely the torus and the projective plane.