Non-existence of tight neighborly manifolds with $\beta_1=2$

TL;DR

This paper proves that there are no tight neighborly manifolds with first Betti number equal to 2 for dimensions greater than or equal to 4, clarifying the limitations of such geometric structures.

Contribution

It establishes the non-existence of tight neighborly manifolds with 1=2 in dimensions , advancing understanding of their topological and combinatorial properties.

Findings

No tight neighborly 1=2 manifolds exist for d.

Results restrict possible configurations of neighborly manifolds.

Clarifies the structure of 1=2 manifolds in higher dimensions.

Abstract

For $d\geq 2$, 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 $\binom{f_0-d-1}{2} \geq \binom{d+2}{2}\beta_1$ for $d\geq 3$. They call a $d$-manifold \emph{tight neighborly} if it attains the equality in the bound. For $d\geq 4$, 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 $\beta_1=2$.