Cohomological rigidity of manifolds associated to ideal right-angled hyperbolic 3-polytopes

Nikolai Erokhovets, Moscow State University

Online Talk 

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Passcode: 114700

Toric topology assigns to each n-dimensional combinatorial simple convex polytope P with m facets an (m+n)-dimensional moment-angle manifold Z_P with an action of a compact torus T^m such that Z_P/T^m is a convex polytope of combinatorial type P. A simple  n-polytope is called B-rigid, if any isomorphism of graded rings H*(Z_P,Z)= H*(Z_Q,Z) for a simple n-polytope Q implies that P and Q are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial 3-polytope obtained by cutting off all the ideal vertices of an ideal right-angled 3-polytope in the Lobachevsky (hyperbolic) space L^3. These polytopes are exactly the polytopes obtained from any, not necessarily simple, convex 3-polytopes by cutting off all the vertices followed by cutting off all the "old" edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We will discuss the following Theorem.  Any ideal almost Pogorelov polytope is B-rigid. A family of manifolds is called cohomologically rigid over the ring R, if for any two manifolds M and N  from the family any isomorphism of graded rings H*(M,R)= H*(N,R) implies that M and N are diffeomorphic. Any ideal almost Pogorelov polytope P has a canonical colouring of facets in 3 colours  corresponding to vertices, edges and facets of the polytope that gives P via cutting off vertices and "old" edges. This colouring produces the 6-dimensional quasitoric manifold M(P) and the 3-dimensional small cover N(P), which are known as "pullbacks from the linear model". Corollary. The families {Z_P}, {M(P)}, and {N(P)} indexed by ideal right-angled hyperpolic 3-polytopes are cohomologically rigid over  Z,  Z and Z_2 respectively.

We will also find the Thurston's geometric decomposition of the 3-manifold N(P): each quadrangle arising from a vertex of the ideal polytope gives an incompressible torus in N(P), and N(P) is divided by these tori into two equal parts homeomorphic to a hyperbolic manifold of finite volume glued of 4 copies of the ideal polytope.