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Small covers over prisms. (English) Zbl 1125.52013

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 \(\#\mathbb D(P_4)=259\).

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

Citations:

Zbl 0733.52006
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References:

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