Shellable Simplicial Spheres: Enumeration and Reconstruction

Abstract

This thesis studies shellable simplicial spheres, which are triangulations of topological spheres with a special combinatorial decomposition property. In Chapter~3 we present enumeration results for shellable simplicial spheres. Nevo, Santos, and Wilson constructed $2^{\Omega(n^k)}$ combinatorially distinct simplicial $(2k-1)$-spheres with $n$ vertices. We prove that all spheres produced by one of their constructions are shellable. Combining this with earlier results of Kalai, Lee, and Benedetti--Ziegler, we conclude that for all $d \ge 3$ there exist $2^{\Theta(n^{\lceil d/2 \rceil})}$ shellable simplicial $d$-spheres with $n$ vertices. In Chapter~4 we show that the facet-ridge graph of a shellable simplicial sphere uniquely determines its entire combinatorial structure. This generalizes the celebrated results of Blind--Mani and Kalai on reconstructing simple polytopes from their graphs. Our proof uses the notion of good acyclic orientations from Kalai's proof together with $k$-systems introduced by Joswig, Kaibel, and K\"orner. In Chapter~5 we study flag shellable simplicial spheres arising as independence complexes of Gorenstein planar graphs with no induced cycles of length divisible by three. We analyze transformations among these flag spheres using edge subdivisions and contractions and prove that their $h$-polynomials are real-rooted.