Visibility complexes and the Baues problem for triangulations in the plane (Q1269448)

Let \(S\) be finite set of points in \(d\)-dimensional space. The polytopal subdivisions of \(S\) form a partial order whose maximal chains yield a simplicial complex. The generalized Baues conjecture by \textit{L. J. Billera, M. M. Kapranov} and \textit{B. Sturmfels} [Proc. Am. Math. Soc. 122, No.~2, 549-555 (1994; Zbl 0812.52007)] states that this simplicial complex has the homotopy type of a sphere. The conjecture was proven for \(d=1\), but disproven in general by \textit{J. Rambau} and \textit{G. M. Ziegler} [Discrete Comput. Geom. 16, No.~3, 215-237 (1996; Zbl 0865.52008)], who gave counterexamples for \(d=5\). In this paper, the authors consider the most natural and important case of polytopal subdivisions, i.e., triangulations of a set of points in general position in \(d=2\). They prove that for this case, the generalized Baues conjecture does hold. The proof is quite intricate and uses deletion-contraction techniques, as well as a newly-defined visibility complex for a simplicial complex.