# On Walkup's class ${cal K}d$ and a minimal triangulation of $S^3 imes otatebox{90}{ltimes} S^1^{#3}$ - Mathematics > Geometric Topology

On Walkup's class ${cal K}d$ and a minimal triangulation of $S^3 imes otatebox{90}{ltimes} S^1^{#3}$ - Mathematics > Geometric Topology - Download this document for free, or read online. Document in PDF available to download.

Abstract: For $d \geq 2$, Walkup-s class ${\cal K}d$ consists of the $d$-dimensionalsimplicial complexes all whose vertex-links are stacked $d-1$-spheres. Kalaishowed that for $d \geq 4$, all connected members of ${\cal K}d$ are obtainedfrom stacked $d$-spheres by finitely many elementary handle additions.According to a result of Walkup, the face vector of any triangulated 4-manifold$X$ with Euler characteristic $\chi$ satisfies $f 1 \geq 5f 0 - {15-2} \chi$,with equality only for $X \in {\cal K}4$. K\-{u}hnel observed that thisimplies $f 0f 0 - 11 \geq -15\chi$, with equality only for 2-neighborlymembers of ${\cal K}4$. K\-{u}hnel also asked if there is a triangulated4-manifold with $f 0 = 15$, $\chi = -4$ attaining equality in his lowerbound. In this paper, guided by Kalai-s theorem, we show that indeed there issuch a triangulation. It triangulates the connected sum of three copies of thetwisted sphere product $S^3 \times {-2.8mm} {-} S^1$. Because of K\-{u}hnel-sinequality, the given triangulation of this manifold is a vertex-minimaltriangulation. By a recent result of Effenberger, the triangulation constructedhere is tight. Apart from the neighborly 2-manifolds and the infinite family of$2d+ 3$-vertex sphere products $S^{d-1} \times S^1$ twisted for $d$ odd,only fourteen tight triangulated manifolds were known so far. The presentconstruction yields a new member of this sporadic family. We also present aself-contained proof of Kalai-s result.

Author: ** Bhaskar Bagchi, Basudeb Datta**

Source: https://arxiv.org/