Quasitoric Manifolds Research Articles

On quasitoric orbifolds

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen-Ruan cohomology groups of an almost complex quasitoric orbifold.

Toric cohomological rigidity of simple convex polytopes

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.

Quasitoric manifolds over a product of simplices

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.

Characterization of homogeneous torus manifolds

This is the first of a series of papers which will be devoted to the study of the extended $G$-actions on torus manifolds $(M^{2n}, T^{n})$, where $G$ is a compact, connected Lie group whose maximal torus is $T^{n}$. The goal of this paper is to characterize codimension $0$ extended $G$-actions up to essential isomorphism. For technical reasons, we do not assume that torus manifolds are omnioriented. The main result of this paper is as follows: a homogeneous torus manifold $M^{2n}$ is (weak equivariantly) diffeomorphic to a product of complex projective spaces $\prod\mathbb{C}P(l)$ and quotient spaces of a product of spheres $\bigl(\prod S^{2m}\bigr)/\mathcal{A}$ with standard torus actions, where $\mathcal{A}$ is a subgroup of $\prod \mathbb{Z}_{2}$ generated by the antipodal involutions on $S^{2m}$. In particular, if the homogeneous torus manifold $M^{2n}$ is a compact (non-singular) toric variety or a quasitoric manifold, then $M^{2n}$ is just a product of complex projective spaces $\prod \mathbb{C}P(l)$.

Toric Genera

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T^k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus \Phi, which was introduced independently by Krichever and Loeffler in 1974, albeit from radically different viewpoints. In fact \Phi is a version of tom Dieck's bundling transformation of 1970, defined on T^k-equivariant complex cobordism classes and taking values in the complex cobordism algebra of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera, and refer to the index theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework, and to introduce parametrised versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever's generalised elliptic genus as universal amongst genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.

Equivariant almost complex structures on quasitoric manifolds

It is proved that there exists an equivariant almost complex structure on any quasitoric manifold that admits a positive omniorientation. This gives an answer to the question raised by M. Davis and T. Januszkiewicz: Find a criterion for the existence of an equivariant almost complex structure on a quasitoric manifold in terms of its characteristic function.

Semifree circle actions, Bott towers and quasitoric manifolds

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.

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.

$K$-rings of smooth complete toric varieties and related spaces

The $K$-rings of non-singular complex projective varieties as well as quasi-toric manifolds were described in terms of generators and relations in earlier work of the author with V. Uma. In this paper we obtain a similar description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope, which includes the class of all smooth complete complex toric varieties.

K -THEORY OF QUASI-TORIC MANIFOLDS

In this note we shall give a description of the K -ring of a quasi-toric manifolds in terms of generators and relations. We apply our results to describe the K -ring of Bott-Samelson varieties.

Spaces of Polytopes and Cobordism of Quasitoric Manifolds

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details.

Hirzebruch genera of manifolds with torus action

A quasitoric manifold is a smooth -manifold with an action of the compact torus such that the action is locally isomorphic to the standard action of on and the orbit space is diffeomorphic, as a manifold with corners, to a simple polytope . The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to those of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex. However, they always admit a stably (or weakly almost) complex structure, and their cobordism classes generate the complex cobordism ring. Buchstaber and Ray have recently shown that the stably complex structure on a quasitoric manifold is determined in purely combinatorial terms, namely, by an orientation of the polytope and a function from the set of codimension-one faces of the polytope to primitive vectors of the integer lattice. We calculate the -genus of a quasitoric manifold with a fixed stably complex structure in terms of the corresponding combinatorial data. In particular, this gives explicit formulae for the classical Todd genus and the signature. We also compare our results with well-known facts in the theory of toric varieties.

Three-Dimensional Manifolds Defined by Coloring a Simple Polytope

In the present paper we introduce and study a class of three-dimensional manifolds endowed with the action of the group ℤ23 whose orbit space is a simple convex polytope. These manifolds originate from three-dimensional polytopes whose faces allow a coloring into three colors with the help of the construction used for studying quasitoric manifolds. For such manifolds we prove the existence of an equivariant embedding into Euclidean space ℝ4. We also describe the action on the set of operations of the equivariant connected sum and the equivariant Dehn surgery. We prove that any such manifold can be obtained from a finitely many three-dimensional tori with the canonical action of the group ℤ23 by using these operations.

Classification Problem for Quasitoric Manifolds over a Given Simple Polytope

The classification problem for quasitoric manifolds over a given polytope Pn is considered. The known construction of weights, which was used in the study of the similar problem for toric manifolds, is modified. The construction thus obtained is applied to the solution of the above problem in small dimensions. Classification results are also obtained for polytopes that are products of finitely many simplices of arbitrary dimensions.

Local face rings and diffeomorphisms of quasitoric manifolds

Local face rings and diffeomorphisms of quasitoric manifolds

On the homotopy groups of toric spaces

Given a certain class of simple polyhedral complexes P and the associated Borel space BTP we compute the E2-term of the Unstable Adams Novikov Spectral Sequence for BTP through a range. As a result, through a range, the higher homotopy groups of BTP are isomorphic to the homotopy groups of a wedge of spheres whose dimensions depend on the combinatorics of P. This paper provides a unified approach to attacking the problem of computing the higher homotopy groups of complements of arbitrary complex coordinate subspace arrangements. We extend all higher homotopy group computations in the cases where the homotopy type of a complement of a complex coordinate subspace arrangement is unknown. If K is a simplicial complex that defines a triangulation of a sphere that is dual to a simple convex polytope P, then, in many cases, the homotopy groups of the quasi-toric manifold M 2n (‚) can be computed through a range that was previously unknown. As an application, the homotopy type of a family of moment angle complexes ZK will be determined.