Quasitoric manifolds over a product of simplices
By Suyoung Choi, M. Masuda, D. Suh. 2008
91 citations recorded by the source corpus.
A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^1)^n$ (resp. $(\mathbb Z_2)^n$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\F$, where $\F=\C$ or $\R$, is a sequence of projective bundles of the Whitney sum of $\F$-line bundles starting with a point. Each stage of the tower over $\F$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\F=\C$) and small covers (when $\F=\R$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz \cite{DJ}) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\Q$ coefficients.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/68574cd778cad0285c99ca6faf815e5f604f076a
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
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Citations in the corpus
- Cohomological rigidity of real Bott manifolds (2008, background): “it space is an n-cube as observed in Section 8; so a real Bott manifold of dimension nbecomes a small cover over an n-cube and the converse is known to be true up to homeomorphism. Theorem 9.1 ([13], [3]). A small cover over an n-cube is homeomorphic to a real Bott manifold of dimension n. The numbe”