Cai, Mingzhong; Chen, Xin; Lü, Zhi Small covers over prisms. (English) Zbl 1125.52013 Topology Appl. 154, No. 11, 2228-2234 (2007). M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417–451 (1991; Zbl 0733.52006)] introduced small covers as smooth \(n\)-dimensional manifolds \(M(P)\) with a locally standard \((\mathbb Z_2)^n\) action such that the quotient is a simple convex polytope \(P\). An \(n\)-dimensional polytope is simple if each vertex is contained in exactly \(n\) facets, that is, faces of codimension \(1\). The manifold \(M(P)\) is determined by \(P\) and additional combinatorial data (a \(({\mathbb Z}_2)^n\)-coloring of the facets). More precisely, the equivariant diffeomorphism classes \(\mathbb D(P)\) of small covers over \(P\) correspond to the orbits of the induced action of the combinatorial automorphism group Aut\((P)\) on the set of \(({\mathbb Z}_2)^n\)-colorings. The purpose of the paper under review is to determine the cardinality of \(\mathbb D(P_m)\) where \(P_m\) is a prism over an \(m\)-gon. In particular, we have \(\dim M(P)=\dim P=3\). The proofs are direct computations based on Burnside’s Lemma on the number of orbits of a finite group action. The authors give an explicit formula which, however, is too long to repeat here. For the cube \(P_4\) (which is the most symmetric case), for instance, they obtain \(\#\mathbb D(P_4)=259\). Reviewer: Michael Joswig (Darmstadt) Cited in 11 Documents MSC: 52B70 Polyhedral manifolds 57S10 Compact groups of homeomorphisms 57M60 Group actions on manifolds and cell complexes in low dimensions 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 05C10 Planar graphs; geometric and topological aspects of graph theory 52B11 \(n\)-dimensional polytopes 57Q91 Equivariant PL-topology Keywords:small cover; equivariant diffeomorphism; polytope; coloring Citations:Zbl 0733.52006 PDFBibTeX XMLCite \textit{M. Cai} et al., Topology Appl. 154, No. 11, 2228--2234 (2007; Zbl 1125.52013) Full Text: DOI arXiv References: [1] Alperin, J. L.; Bell, R. B., Groups and Representations, Graduate Texts in Mathematics, vol. 162 (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0839.20001 [2] Buchstaber, V. M.; Panov, T. E., Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series, vol. 24 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0996.52012 [3] Davis, M., Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. Math., 117, 293-324 (1983) · Zbl 0531.57041 [4] Davis, M.; Januszkiewicz, T., Convex polytope, Coxeter orbifolds and torus actions, Duke Math. J., 62, 417-451 (1991) · Zbl 0733.52006 [5] Garrison, A.; Scott, R., Small covers of the dodecahedron and the 120-cell, Proc. Amer. Math. Soc., 131, 963-971 (2002) · Zbl 1009.57019 [6] Z. Lü, M. Masuda, Equivariant classification of 2-torus manifolds, preprint; Z. Lü, M. Masuda, Equivariant classification of 2-torus manifolds, preprint · Zbl 1165.57023 [7] Ziegler, G. M., Lectures on Polytopes, Graduate Texts in Math. (1994), Springer-Verlag: Springer-Verlag Berlin This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.