Non-existence of tight neighborly manifolds with ${\beta }_1=2$

Abstract

For $d\ge 2$, Walkup's class $\Kd$ consists of the $d$-dimensional simplicial complexes whose vertex-links are stacked $(d-1)$-spheres. Recently Lutz, Sulanke and Swartz have shown that all $\mathbb{F}$-orientable triangulated $d$-manifolds satisfy the inequality $\left(\genfrac{}{}{0ex}{}{f_0-d-1}{2}\right)\ge \left(\genfrac{}{}{0ex}{}{d+2}{2}\right){\beta }_1$ for $d\ge 3$. They call a $d$-manifold \emph{tight neighborly} if it attains the equality in the bound. For $d\ge 4$, tight neighborly $d$-manifolds are precisely the 2-neighborly members of $\Kd$. In this paper we show that there does not exist any tight neighborly $d$-manifold with ${\beta }_1=2$.

arXiv comment8 pages. arXiv admin note: text overlap with arXiv:1102.0856, arXiv:1207.5599 by other authors