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

For $d\ge 2$, Walkup’s class $\text{Kd}$ 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$d\geq 3$\mathrm{.}Theycalla$d$-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 $\text{Kd}$. In this paper we show that there does not exist any tight neighborly $d$-manifold with ${\beta }_1=2$.