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Register a new userStrong cohomological rigidity of quasitoric manifolds over the 3-cube
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Sho Hasui
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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 $I^3$.
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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.
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.
In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds $X$ and $X$, the following conditions are equivalent: (1) the upper triangular matrix associated to $X$ can be transformed into that of $X$ by those three operations; (2) $X$ and $X$ are diffeomorphic; (3) the integral cohomology rings of $X$ and $X$ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.
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