Theory of multi-fans
By A. Hattori, M. Masuda. 2001
148 citations recorded by the source corpus.
We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define ``topological invariants'' of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/050399ad3a340cd21548e7b4e07adce006040ce6
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Citations in the corpus
- Classification problems of toric manifolds via topology (2007, background): “A torus manifold M introduced in [18] is a closed smooth orientable manifold of dimension 2n with a smooth effective action of (S(1)) having a fixed point.”
- Surveys in Contemporary Mathematics: Cohomology of face rings, and torus actions (2005, background): “Finally, we mention that if K is an arbitrary (not necessarily polytopal) triangulation of sphere, then the manifold M2nK (Λ) is a torus manifold in the sense of Hattori–Masuda [19].”
- On Quasitoric Orbifolds (2008, unknown): “Later, in Section 6, it will be evident that our definitions of a quasitoric orbifold are more general than the original defin-ition in [8], as the quotient Z ( P ) = T K of a higher dimensional manifold by a smooth torus action.”
- Toric vector bundles, branched covers of fans, and the resolution property (2006, background): “degrees d and d′, respectively, then let w · w′ be the weight function on ∆×Σ ∆′ given by w · w′((x,y)) = w(x) · w′(y). Then (∆ ×Σ ∆′,w · w′) is a branched cover of Σ of degree d· d′. Remark 3.20. In [HM03], a weighted multifan in NR is called complete if the projection to NR satisfies the trace cond”
- Topological toric manifolds (2010, background): “¶ Obviously a In [11] an omniorientation is incorporated in the definition of a torus manifold.”
- Torus Actions and their Applications in Topology and Combinatorics (2002, unknown): “For more information about multi-fans see [72].”
- Toric topology (2012, background): “We call such a manifold M a torus manifold ([35])(1).”
- Rigidity problems in toric topology: A survey (2011, background): “Hattori and Masuda [24] introduced a torus manifold (or unitary toric manifold in an earlier terminology [32]), which is a closed, smooth manifold of dimension 2n with an effective smooth action of (S1)n having a fixed point, and developed an analogous theory to toric geometry for torus manifolds to”
- On the cohomology of torus manifolds (2003, unknown): “In Section 2 we establish the notation concerning torus actions on manifolds and prove three pivotal statements (Lemmas 2.1–2.3) describing different properties of fixed point sets.”
- Torus graphs and simplicial posets (2005, unknown): “A T ] 1 2 O ct 2 00 6”