Tight triangulations of some 4-manifolds

Walkup’s class $\mathcal{K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex links are stacked $(d-1)$-spheres. According to a result of Walkup, the face vector of any triangulated 4-manifold $X$ with Euler characteristic $\chi$ satisfies $f_1\ge 5f_0-15\mathrm{/}2\chi$, with equality only for $X\in \mathcal{K}(4)$. K"{u}hnel observed that this implies $f_0(f_0 - 11) \geq -15\chi$,withequalityonlyfor2-neighborlymembersof${\cal K}(4)$. For $n=6,11$ and 15, there are triangulated 4-manifolds with $f_0=n$ and $f_0(f_0-11)=-15\chi$. In this article, we present triangulated 4-manifolds with $f_0=21,26$ and 41 which satisfy $f_0(f_0-11)=-15\chi$. All these triangulated manifolds are tight and strongly minimal.