Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area

Toric Topology

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.

Toric topology

If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P(E) of E is again a toric manifold. Starting with B as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, "Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.

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Topological classification of generalized Bott towers

A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^{1})^{n}$ (resp. $(\mathbb{Z}_{2})^{n}$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\mathbb{F}$, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, is a sequence of projective bundles of the Whitney sum of $\mathbb{F}$-line bundles starting with a point. Each stage of the tower over $\mathbb{F}$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\mathbb{F}=\mathbb{C}$) and small covers (when $\mathbb{F}=\mathbb{R}$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\mathbb{Q}$ coefficients.

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Quasitoric manifolds over a product of simplices

A Bott tower is the total space of a tower of fibre bundles with base and fibres . 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 the fixed points. We show that a quasitoric manifold over a combinatorial n-cube admitting a semifree action of a 1-dimensional subtorus with 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 Il'inskiĭ, who showed that a smooth compact toric variety admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of Hattori's problem of semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to this product. In the case of Bott towers the homeomorphism is actually a diffeomorphism.Bibliography: 18 titles.

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Semifree circle actions, Bott towers and quasitoric manifolds

Several rigidity problems in toric topology are addressed in the survey paper by the second and third authors, “Classification Problems of Toric Manifolds via Topology” (in Toric Topology, Am. Math. Soc., Providence, RI, 2008, pp. 273–286). In the present paper, we survey the results on those problems including recent developments.

Rigidity problems in toric topology: A survey

A simple convex polytope P is cohomologically rigid if its combinatorial structure is determined
by the cohomology ring of a quasitoric manifold over P. Not every P has this property, but
some important polytopes such as simplices or cubes are known to be cohomologically rigid. In
this paper we investigate the cohomological rigidity of polytopes and establish it for several new
classes of polytopes, including products of simplices. The cohomological rigidity of P is related
to the bigraded Betti numbers of its Stanley�Reisner ring, another important invariant coming
from combinatorial commutative algebra

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Toric cohomological rigidity of simple convex polytopes

We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.

Dong Youp Suh

Papers

Topological classification of generalized Bott towers

Quasitoric manifolds over a product of simplices

Rigidity problems in toric topology: A survey

Toric cohomological rigidity of simple convex polytopes

Classification problems of toric manifolds via topology