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Non-existence of tight neighborly manifolds with $β_1=2$

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 $\binom{f_0-d-1}{2} \geq \binom{d+2}{2}β_1$ 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 $β_1=2$.