Abstracts
for the Seminar

Fall 2016

**Speaker: **Hailun Zheng, University of Washington

**Title: **A characterization of simplicial manifolds with $g_2 \leq 2$

**Time:** 2:30 PM, Monday, September 26, 2016

**Place:** Malott 206

**Abstract:** The celebrated lower bound theorem states that any simplicial manifold of dimension $\geq 3$ satisfies $g_2 \geq 0$, and equality holds if and only if it is a stacked sphere. Furthermore, more recently, the class of all simplicial spheres with $g_2=1$ was characterized by Nevo and Novinsky, by an argument based on rigidity theory for graphs. In this talk, I will first define three different retriangulations of simplicial complexes that preserve the homeomorphism type. Then I will show that all simplicial manifolds with $g_2 \leq 2$ can be obtained by retriangulating a polytopal sphere with a smaller $g_2$. This implies Nevo and Novinsky's result for simplicial spheres of dimension $\geq 4$. More surprisingly, it also implies that all simplicial manifolds with $g_2=2$ are polytopal spheres.

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