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Register a new userOn the cohomology equivalences between bundle-type quasitoric manifolds over a cube
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Sho Hasui
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The aim of this article is to establish the notion of bundle-type quasitoric manifolds and provide two classification results on them: (1) ($mathbb{C}P^2sharpmathbb{C}P^2$)-bundle type quasitoric manifolds are weakly equivariantly homeomorphic if their cohomology rings are isomorphic, and (2) quasitoric manifolds over $I^3$ are homeomorphic if their cohomology rings are isomorphic. In the latter case, there are only four quasitoric manifolds up to weakly equivariant homeomorphism which are not bundle-type.
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A quasitoric manifold is a smooth manifold with a locally standard torus action for which the orbit space is identified with a simple polytope. For a class of topological spaces, the class is called strongly cohomologically rigid if any isomorphism of cohomology rings can be realized as a homeomorphism. This paper shows the strong cohomological rigidity of the class of quasitoric manifolds over $I^3$.
For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a $2n$-dimensional smooth manifold with a locally standard action of the $n$-dimensional torus for which the orbit space is identified with $P$. This paper shows the topological classification of quasitoric manifolds over the dual cyclic polytope $C^n(m)^*$, when $n>3$ or $m-n=3$. Besides, we classify small covers, the real version of quasitoric manifolds, over all dual cyclic polytopes.
We show a homotopy decomposition of $p$-localized suspension $Sigma M_{(p)}$ of a quasitoric manifold $M$ by constructing power maps. As an application we investigate the $p$-localized suspension of the projection $pi$ from the moment-angle complex onto $M$, from which we deduce its triviality for $p>dim M/2$. We also discuss non-triviality of $pi_{(p)}$ and $Sigma^inftypi$.
It is proved that if two quasitoric manifolds of dimension $le 2p^2-4$ for a prime $p$ have isomorphic cohomology rings, then they have the same $p$-local stable homotopy type.
We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $mathrm{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Odas $3$-fold, the simplest non-projective toric manifold, is over $mathrm{vc}(I^n)$. In this paper, we classify toric manifolds over $mathrm{vc}(I^n)$ $(nge 3)$ as varieties and also as smooth manifolds. As a consequence, it turns out that (1) there are many non-projective toric manifolds over $mathrm{vc}(I^n)$ but they are all diffeomorphic, and (2) toric manifolds over $mathrm{vc}(I^n)$ in some class are determined by their cohomology rings as varieties among toric manifolds.
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