Toric topology

By M. Masuda. 2012

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Toric geometry is a bridge connecting algebraic geometry and combinatorics. Through this bridge, some problems and results in combinatorics can be interpreted in terms of algebraic geometry and vice versa. For example, counting lattice points in a lattice convex polytope and KouchnirenkoBernstein’s theorem describing the relation between the number of solutions of a system of Laurent polynomials and Minkowski’s mixed volume of the Newton polytopes associated with the equations are such a problem or a result (see [22], [72], [46]). Among them, the solution of McMullen’s conjecture ([65]) by Stanley [81] must have been striking. McMullen’s conjecture says that the face numbers of simplicial polytopes would be characterized by three conditions. Stanley proved the necessity of these three conditions by applying the Poincaré duality theorem (topology), the hard Lefschetz theorem (algebraic geometry) and Macaulay’s theorem (commutative algebra) to the projective toric orbifolds associated with the simplicial polytopes. On the other hand, Billera-Lee [4] constructed simplicial polytopes satisfying those three conditions almost at the same time, so that McMullen’s conjecture was affirmatively solved dramatically. It was around 1980. This result is now called the g-theorem (see [22], [37]). The theory of toric geometry can be developed using topology instead of algebraic geometry to some extent ([17], [9], [35], [55]). Namely, one can construct a bridge between topology and combinatorics. Since the method is topological, the geometrical object treated there need not be algebraic varieties. Moreover, the combinatorial object which appears is a wider class than that in toric geometry and one can see a natural generalization of results on convex polytopes in combinatorics to the wider class. One can also see the difference between algebraic geometry and topology through the generalization. In this article, the author will explain toric topology from this point of view. Nigel Ray is probably the first person who initiated the word toric topology but he uses this word in a wider meaning than used here. We restrict our concern to smooth manifolds or orbifolds in this article but Ray uses the word toric topology for the study of not only manifolds or orbifolds but also topological spaces with symmetry (especially torus actions) which fit well to combinatorics.

Subfields: Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/d1bfeece6c2fe4af37d25bc9d87eae4869c8f017

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In Mechanical Engineering

  1. L. Cai

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