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Register a new userOn The Topological Period-Index Problem over 8-manifolds
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We establish upper bounds of the indices of topological Brauer classes over a closed orientable 8-manifolds. In particular, we verify the Topological Period-Index Conjecture (TPIC) for topological Brauer classes over closed orientable 8-manifolds of order not congruent to 2 mod 4. In addition, we provide a counter-example which shows that the TPIC fails in general for closed orientable 8-manifolds.
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We prove the topological analogue of the period-index conjecture in each dimension away from a small set of primes.
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.
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 study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application, we show that the u-invariant of the function field of a real algebraic surface is equal to 4, answering questions of Lang and Pfister. Our strategy relies on a new Hodge-theoretic approach to de Jongs period-index theorem on complex surfaces.
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