Three-dimensional normal pseudomanifolds with relatively few edges

Abstract: From the Lower Bound Theorem, we know that if $\Delta$ is a $d$-dimensional normal pseudomanifold then $g_2(\Delta):= f_1(\Delta)-(d+1)f_0(\Delta) + \binom{d+2}{2}\geq 0$ and equality holds if and only if $\Delta$ is a stacked sphere for $d\geq 3$. Thus, Lower Bound Theorem classifies normal pseudomanifolds of dimension $d\geq 3$ with $g_2=0$. Later, Nevo and Novinsky have classified homology $d$-spheres with $g_2=1$ for $d\geq 3$. Zheng has shown that homology manifolds of dimension $d\geq 3$ with $g_2=2$ are polytopal spheres. From the works of Kalai and Fogelsanger it follows that $g_2(\Delta) \geq g_2({\rm lk}(v, \Delta))$ for any vertex $v$ of $\Delta$.

In this talk, I shall show that the topological and combinatorial classification of normal $3$-pseudomanifolds $\Delta$ when $\Delta$ has at most two singularity and $g_2(\Delta) = g_2({\rm lk}(v, \Delta))$ for some vertex $v$ of $\Delta$. In particular, I shall show that normal $3$-pseudomanifolds with $g_2=3$ are either sphere or suspension of $\mathbb{RP}^2$.

commutative algebraalgebraic topologycombinatorics

Audience: researchers in the topic

Applications of Combinatorics in Algebra, Topology and Graph Theory

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