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Tight triangulations of some 4-manifolds

Walkup's class ${\cal K}(d)$ 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 $χ$ satisfies $f_1 \geq 5f_0 - 15/2 χ$, with equality only for $X \in {\cal K}(4)$. Kühnel observed that this implies $f_0(f_0 - 11) \geq -15χ$, with equality only for 2-neighborly members of ${\cal K}(4)$. For $n = 6, 11$ and 15, there are triangulated 4-manifolds with $f_0=n$ and $f_0(f_0 - 11) = -15χ$. In this article, we present triangulated 4-manifolds with $f_0 = 21, 26$ and 41 which satisfy $f_0(f_0 - 11) = -15χ$. All these triangulated manifolds are tight and strongly minimal.