Toric Topology.

Intro -- Contents -- Preface -- List of Participants -- An invitation to toric topology: Vertex four of a remarkable tetrahedron -- Cohomological aspects of torus actions -- A counterexample to a conjecture of Bosio and Meersseman -- Symplectic quasi-states and semi-simplicity of quantum homology -- Miraculous cancellation and Pick's theorem -- Freeness of equivariant cohomology and mutants of compactified representations -- Weighted hyperprojective spaces and homotopy invariance in orbifold cohomology -- Homotopy theory and the complement of a coordinate subspace arrangement -- 1. Introduction -- 2. Homotopy theory -- 3. Homotopy decompositions -- 4. Toric Topology - main definitions and constructions -- 5. The homotopy type of the complement of an arrangement -- 6. Examples -- 7. Topological extensions -- 8. Applications -- References -- The quantization of a toric manifold is given by the integer lattice points in the moment polytope -- Invariance property of orbifold elliptic genus for multi-fans -- Act globally, compute locally: group actions, fixed points, and localization -- Introduction -- 1. A brief review of the symplectic category -- 2. Equivariant cohomology and localization theorems -- 3. Using localization to compute equivariant cohomology -- 4. Combinatorial localization and polytope decompositions -- References -- Tropical toric geometry -- The symplectic volume and intersection pairings of the moduli spaces of spatial polygons -- Logarithmic functional and reciprocity laws -- Orbifold cohomology reloaded -- The geometry of toric hyperkähler varieties -- Graphs of 2-torus actions -- Classification problems of toric manifolds via topology -- The quasi KO-types of certain toric manifolds -- Categorical aspects of toric topology -- A survey of hypertoric geometry and topology -- On asymptotic partition functions for root systems.

Torus actions of complexity one -- Permutation actions on equivariant cohomology of flag varieties -- K-theory of torus manifolds -- On liftings of local torus actions to fiber bundles.

Toric topology is the study of algebraic, differential, symplectic-geometric, combinatorial, and homotopy-theoretic aspects of a particular class of torus actions whose quotients are highly structured. The combinatorial properties of this quotient and the equivariant topology of the original manifold interact in a rich variety of ways, thus illuminating subtle aspects of both the combinatorics and the equivariant topology. Many of the motivations and guiding principles of the field are provided by (though not limited to) the theory of toric varieties in algebraic geometry as well as that of symplectic toric manifolds in symplectic geometry. This volume is the proceedings of the International Conference on Toric Topology held in Osaka in May-June 2006. It contains about 25 research and survey articles written by conference speakers, covering many different aspects of, and approaches to, torus actions, such as those mentioned above. Some of the manuscripts are survey articles, intended to give a broad overview of an aspect of the subject; all manuscripts consciously aim to be accessible to a broad reading audience of students and researchers interested in the interaction of the subjects involved. We hope that this volume serves as an enticing invitation to this emerging field.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.