Abstract:
A combinatorial 3-dimensional polytope $P$ can be realized in
Lobachevsky 3-space with right dihedral angles if and only if it is
simple, flag and does not have 4-belts of facets. This criterion was
proved in the works of A. Pogorelov and E. Andreev of the 1960s. We
refer to combinatorial 3 polytopes admitting a right-angled realisation
in Lobachevsky 3-space as Pogorelov polytopes. The Pogorelov class
contains all fullerenes, i.e. simple 3-polytopes with only 5-gonal and
6-gonal facets. There are two families of smooth manifolds associated
with Pogorelov polytopes. The first family consists of 3-dimensional
small covers (in the sense of M. Davis and T. Januszkiewicz) of
Pogorelov polytopes $P$, also known as hyperbolic 3-manifolds of Loebell
type. These are aspherical 3-manifolds whose fundamental groups are
certain extensions of abelian 2-groups by hyperbolic right-angled
reflection groups in the facets of $P$. The second family consists of
6-dimensional quasi toric manifolds over Pogorelov polytopes. These are
simply connected 6-manifolds with a 3-dimensional torus action and orbit
space $P$. Our main result is that both families are cohomologically
rigid, i.e. two manifolds $M$ and $M'$ from either family are
diffeomorphic if and only if their cohomology rings are isomorphic. We
also prove that a cohomology ring isomorphism implies an equivalence of
characteristic pairs; in particular, the corresponding polytopes $P$ and
$P'$ are combinatorially equivalent. This leads to a positive solution
of a problem of A. Vesnin (1991) on hyperbolic Loebell manifolds, and
implies their full classification. Our results are intertwined with
classical subjects of geometry and topology such as combinatorics of
3-polytopes, the Four Colour Theorem, aspherical manifolds, a
diffeomorphism classification of 6-manifolds and invariance of
Pontryagin classes. The proofs use techniques of toric topology. This is
a joint work with V. Buchstaber, N. Erokhovets, M. Masuda and S. Park.
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