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Beijing–Moscow Mathematics Colloquium
20 ноября 2020 г. 11:00–12:00, г. Москва, online
 


Right-angled polytopes, hyperbolic manifolds and torus actions

T. E. Panov

Lomonosov Moscow State University
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Аннотация: 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.

Материалы: 2020bmmc_panov_handout.pdf (547.7 Kb)

Язык доклада: английский

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