Right-angled polytopes, hyperbolic manifolds and torus actions

• 主讲人：Taras Panov (Lomonosov Moscow State University)
• 举办方： Beijing-Saint Petersburg Mathematics Colloquium
• 时间： 2023-06-08 21:00 - 2023-06-08 22:00
• 地点： online

Recording: https://disk.pku.edu.cn:443/link/AA3FBCE528DCEDB51ACB6A1C97BCB41A
Valid Until: 2027-07-31 23:59

 

Abstract: A combinatorial 3-dimensional polytope P can be realised in Lobachevsky 3-space with right dihedral angles if and only if it is simple, flag and does not have 4-belts (4-prismatic circuits 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 quasitoric 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.

 

Bio: Taras Panov is a Professor at Higher School of Economics in Moscow and Moscow State University (Faculty of Mechanics and Mathematics). He is a renowned expert in algebraic geometry, algebraic topology, and combinatorial geometry author of three monographs and a big number of papers in these areas.

Prof. Panov graduated from Moscow State University in 1996, defended his PhD thesis in 1999 (supervisor: Prof. Buchstaber), and doctoral thesis in 2009. Before getting his Professor position at MSU, Taras Panov has got postdoc positions at Osaka City University and University of Manchester.

Besides, Prof. Panov is laureat of various prestigeous prizes and awards on national and international levels.