On the g2-number of various classes of spheres and manifolds

Abstract

For a $(d-1)$-dimensional simplicial complex $\Delta$, we let $f_i=f_i(\Delta)$ be the number of $i$-dimensional faces of $\Delta$ for $-1\leq i\leq d-1$. One classic problem in geometric combinatorics is the following: for a given class of simplicial complexes, find tight upper and lower bounds on the face numbers and characterize the complexes that attain these bounds. This dissertation studies these questions in various classes of simplicial complexes including balanced manifolds, flag manifolds and simplicial spheres. A $(d-1)$-dimensional simplicial complex is called balanced if its graph is $d$-colorable. In Chapter 2, we determine the minimum number of vertices needed to provide balanced triangulations of $\Sp^{d-2}$-bundles over $\Sp^1$. Similar results apply to all balanced triangulated manifolds with $\beta_1\neq 0$ and $\beta_2=0$. In Chapter 3, we turn to the Upper Bound Conjecture for balanced simplicial spheres. We find the first two examples of non-octahedral balanced 2-neighborly spheres. Each construction is of dimension 3 and with 16 vertices. Along the way, we show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition. A simplicial complex is flag if it is the clique complex of its graph. In Chapter 4, we settle the Upper Bound Conjecture for flag 3-manifolds, establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. In Chapter 5, we characterize homology manifolds with $g_2\leq 2$. We prove that every prime homology manifold with $g_2=2$ is obtained by centrally retriangulating a polytopal sphere with $g_2\leq 1$ along a certain subcomplex. This implies that all homology spheres with $g_2=2$ are polytopal spheres. In Chapter 6, we prove that for any prime homology $(d-1)$-sphere $\Delta$ ($d\geq 4$) with $g_2(\Delta)\geq 1$ and any edge $e\in \Delta$, the graph $G(\Delta)-e$ is generically $d$-rigid. This confirms a conjecture of Nevo and Novinsky.