Abstract. If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P (E) of E is again a toric manifold. Starting with B as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, "Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.
A simple convex polytope P is cohomologically rigid if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over P . Not every P has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this article we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes including products of simplices. Cohomological rigidity of P is related to the bigraded Betti numbers of its Stanley-Reisner ring, another important invariant coming from combinatorial commutative algebra.
In this paper, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the i-th (rational) Betti number and Euler characteristic of the real toric variety associated to a graph associahedron P B(G) . They can be calculated by a purely combinatorial method (in terms of graphs) and are named ai(G) and b(G), respectively. To our surprise, for specific families of the graph G, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.
Abstract. In this paper, we do the two things.(1) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for other coefficient Zq = Z/qZ, where q is a positive odd integer. (2) We construct infinitely many real toric manifolds and small covers whose integral cohomology have a q-torsion for any positive odd integer q.
In the present paper we find a bijection between the set of small covers over
an $n$-cube and the set of acyclic digraphs with $n$ labeled nodes. Using this,
we give a formula of the number of small covers over an $n$-cube (generally, a
product of simplices) up to Davis-Januszkiewicz equivalence classes and
$\mathbf{Z}^n$-equivariant diffeomorphism classes. Moreover we prove that the
number of acyclic digraphs with $n$ unlabeled nodes is an upper bound of the
number of small covers over an $n$-cube up to diffeomorphism.Comment: 8 page
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