On reduced complexity of closed piecewise linear 5-manifolds (Q2052780)

Crystallization theory is a representation method for closed PL \(n\)-manifolds \(M\) by means of a particular class of edge-colored graphs called crystallizations. \textit{J. Bracho} and \textit{L. Montejano} [Geom. Dedicata 22, 303--328 (1987; Zbl 0631.57017)] defined the \textit{reduced complexity} \(\tilde c (M) = c(M) - 2\) of \(M\) within crystallization theory (here \(c(M) \ge 2\) denotes the \textit{complexity} of \(M\), the minimum number of \(n\)-simplices of a contracted triangulation of \(M\)) (for example, for a closed connected surface \(M\), \(\tilde c (M) = 4 - 2\chi(M)\) where \(\chi(M)\) denotes the Euler characteristic). Concentrating on dimension 5 in the present paper, the authors classify the closed connected PL (or smooth) 5-manifolds \(M\) up to reduced complexity 20: the only \(M\) with \(\tilde c (M) = 0\) is the 5-sphere \(\mathbb{S}^5\); there are no \(M\) with \(0 < \tilde c (M) < 10\) and exactly two manifolds \(\mathbb{S}^1 \times \mathbb{S}^4\) and the twisted version \(\mathbb{S}^1 \tilde \times \mathbb{S}^4\) with \(\tilde c (M) = 10\), no \(M\) with \(10 < \tilde c (M) < 20\), and finally a few explicitly described manifolds with \(\tilde c (M) = 20\) (among them \(\mathbb{S}^2 \times \mathbb{S}^3\)). ``In particular, this gives a combinatorial characterization of \(\mathbb{S}^2 \times \mathbb{S}^3\) among closed connected spin PL 5-manifolds.'' The paper starts with a short survey on complexities in general, and in particular in dimensions 3 and 4.