In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces are all non-free and noncompact, but still built via simple convex polytopes. We obtain an explicit formula of Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the $h$-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of Euler characteristic for the classical configuration space, which generalizes the Félix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retract of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which turns out that we are able to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.
Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.
We describe the $K$-ring of a quasi-toric manifold in terms of generators and relations. We apply our results to describe the $K$-ring of Bott-Samelson varieties.
For a simplicial complex $K$ with $m$ vertices, there is a canonical $\mathbb Z_2^m$-space known as a real moment angle complex $\mathbb R \mathcal Z_K$. In this paper, we consider the quotient spaces $Y=\mathbb R \mathcal Z_K / \mathbb Z_2^{k}$, where $K$ is a pure shellable complex and $\mathbb Z_2^k \subset \mathbb Z_2^m$ is a maximal free action on $\mathbb R \mathcal Z_K$. A typical example of such spaces is a small cover, where a small cover is known as a topological analog of a real toric manifold. We compute the integral cohomology group of $Y$ by using the PL cell decomposition obtain…
In this paper, we do the two things. 1. We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for other coefficient $\mathbb{Z}_q = \mathbb{Z}/q\mathbb{Z}$, where $q$ is a positive odd integer. 2. We construct infinitely many real toric manifolds and small covers whose integral cohomology have a $q$-torsion for any positive odd integer $q$.
We construct toric manifolds of complex dimension $\geq 4$, whose orbit spaces by the action of the compact torus are not homeomorphic to simple polytopes (as manifolds with corners). These provide the first known examples of toric manifolds which are not quasitoric manifolds.
We present two examples in toric geometry concerning the relationship between toric and quasitoric manifolds, and provide the sufficient conditions on the base polytope and characteristic map so that the resulting quasitoric manifold is almost complex.
A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or, simply, a real toric manifold), that is, a real locus of projective toric manifold. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method finding toric spaces called puzzles.
We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties.
In this paper we consider the real topological K-groups of a toric manifold M, which turns out to be closely related to the topology of the small cover MR, the fixed points under the canonical conjugation on M . Following the work of Bahri and Bendersky [2], we give an explicit formula for the KO-groups of toric manifolds, and then we characterize the two extreme classes of toric manifolds according to the two A(1) modules shown in [2].
We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The problem of cohomological rigidity for small covers is also studied and we have found a whole new series of weakly cohomologically rigid simple polytopes. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have …
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.
The K-rings of non-singular complex pro jective varieties as well as quasi- toric manifolds were described in terms of generators and relations in an earlier work of the author with V. Uma. In this paper we obtain a similar description for complete non-singular toric varieties. Indeed, our approach enables us to obtain such a description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope.
The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group $\mathbb{Z}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…
We show that any topological toric manifold can be covered by finitely many open charts so that all the transition functions between these charts are Laurent monomials of $z_j$'s and $\bar{z}_j$'s. In addition, we will describe toric manifolds and some special class of topological toric manifolds in terms of transition functions of charts up to (weakly) equivariant diffeomorphism.
A quasitoric manifold is a smooth 2n-manifold M^{2n} with an action of the compact torus T^n such that the action is locally isomorphic to the standard action of T^n on C^n and the orbit space is diffeomorphic, as manifold with corners, to a simple polytope P^n. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to that 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 co…
We introduce the notion of a topological toric manifold and a topological fan and show that there is a bijection between omnioriented topological toric manifolds and complete non-singular topological fans. A topological toric manifold is a topological analogue of a toric manifold and the family of topological toric manifolds is much larger than that of toric manifolds. A topological fan is a combinatorial object generalizing the notion of a simplicial fan in toric geometry. Prior to this paper, two topological analogues of a toric manifold have been introduced. One is a quasitoric manifold and…
By analogy with algebraic geometry, we define a category of non-linear sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety o…
A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. 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 fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and 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 …
We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions >8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.
We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as …
Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting "twisted" diagram of modules satisfies a certain gluing condition, stating that the data is compatible with restriction to smaller open sets. In case X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category D(X) of quasi-coherent sheaves on X can be obtained from a category of twisted diagrams which do not n…
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 exp…
In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equiva…
