

Geometric Topology Seminar
October 19, 2017 14:00–16:00, Moscow, Math Department of the Higher School of Economics (Usachyova, 6), Room 209






Manifolds defined by rightangled 3dimensional polytopes
T. E. Panov^{} 
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Abstract:
A combinatorial 3dimensional polytope P can be realised in Lobachevsky (hyperbolic) 3space with right dihedral angles if and only if it is simple, flag and does not have 4belts of facets. This criterion was proved in the works of Pogorelov and Andreev of the 1960s. We refer to combinatorial 3polytopes admitting a rightangled realisation in Lobachevsky 3space as Pogorelov polytopes. The Pogorelov class contains all fullerenes, i.e. simple 3polytopes with only 5gonal and 6gonal facets.
There are two families of smooth manifolds associated with Pogorelov polytopes. The first family consists of 3dimensional small covers of Pogorelov polytopes P, also known as hyperbolic 3manifolds of Loebell type. These are aspherical 3manifolds whose fundamental groups are certain finite abelian extensions of hyperbolic rightangled reflection groups in the facets of P. The second family consists of 6dimensional quasitoric manifolds over Pogorelov polytopes. These are simply connected 6manifolds with a 3dimensional 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 of the families 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. These results are intertwined with the classical subjects of geometry and topology, such as combinatorics of 3polytopes, the Four Colour Theorem, aspherical manifolds, diffeomorphism classification of 6manifolds 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.

