160 research outputs found
h-vectors of Gorenstein* simplicial posets
As is well known, h-vectors of simple (or simplicial) convex polytopes are
characterized. In fact, those h-vectors must satisfy Dehn-Sommerville equations
and some other inequalities. Simple convex polytopes determine Gorenstein*
simplicial posets and h-vectors are defined for simplicial posets. It is known
that h-vectors of Gorenstein* simplicial posets must satisfy Dehn-Sommerville
equations and that every component in the h-vectors must be non-negative. In
this paper we will show that h-vectors of Gorenstein* simplicial posets must
satisfy one more subtle condition conjectured by R. Stanley and complete
characterization of those h-vectors. Our proof is purely algebraic but the idea
of the proof stems from topology.Comment: 12 page
Toric topology
We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in
"Sugaku" vol. 62 (2010), 386-41
Equivariant cohomology distinguishes toric manifolds
The equivariant cohomology of a space with a group action is not only a ring
but also an algebra over the cohomology ring of the classifying space of the
acting group. We prove that toric manifolds (i.e. compact smooth toric
varieties) are isomorphic as varieties if and only if their equivariant
cohomology algebras are weakly isomorphic. We also prove that quasitoric
manifolds, which can be thought of as a topological counterpart to toric
manifolds, are equivariantly homeomorphic if and only if their equivariant
cohomology algebras are isomorphic
Semifree circle actions, Bott towers, and quasitoric manifolds
A Bott tower is the total space of a tower of fibre bundles with base CP^1
and fibres CP^1. Every Bott tower of height n is a smooth projective toric
variety whose moment polytope is combinatorially equivalent to an n-cube. A
circle action is semifree if it is free on the complement to fixed points. We
show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an
n-cube with a semifree circle action and isolated fixed points is a Bott tower.
Then we show that every Bott tower obtained in this way is topologically
trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent
result of Ilinskii, who showed that a smooth compact toric variety with a
semifree circle action and isolated fixed points is homeomorphic to a product
of 2-spheres, and makes a further step towards our understanding of a problem
motivated by Hattori's work on semifree circle actions. Finally, we show that
if the cohomology ring of a quasitoric manifold (or Bott tower) is isomorphic
to that of a product of 2-spheres, then the manifold is homeomorphic to the
product.Comment: 22 pages, LaTEX; substantially revise
Lattice multi-polygons
We discuss generalizations of some results on lattice polygons to certain
piecewise linear loops which may have a self-intersection but have vertices in
the lattice . We first prove a formula on the rotation number of
a unimodular sequence in . This formula implies the generalized
twelve-point theorem in [12]. We then introduce the notion of lattice
multi-polygons which is a generalization of lattice polygons, state the
generalized Pick's formula and discuss the classification of Ehrhart
polynomials of lattice multi-polygons and also of several natural subfamilies
of lattice multi-polygons.Comment: 21 pages, 7 figures, Kyoto J. Math. to appea
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