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 $\F$, where $\F=\C$ or $\R$, is a sequence of projective bundles of the Whitney sum of $\F$-line bundles starting with a point. Each stage of the tower over $\F$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\F=\C$) and small covers (when $\F=\R$) 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 \cite{DJ}) 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 $\Q$ coefficients.
In this paper, we classify the characteristic matrices associated to quasitoric manifolds over a vertex cut of a finite product of simplices. We discuss the integral cohomology rings of these quasitoric manifolds with possibly minimal generators and show several relations among the product of these generators. We classify integral cohomology rings (up to isomorphism as graded rings) of the quasitoric manifolds over the vertex cut of a finite product of simplices.
Homotopy rigidity for quasitoric manifolds over a product of simplices
Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Institute of Mathematical Science, Pusan National University（釜山国立大学数学科学研究所）；Department of Mathematics Education, Pusan National University（釜山国立大学数学教育系）；Mathematical Sciences, University of Southampton（南安普顿大学数学科学系）
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.
The aim of this article is to establish the notion of bundle-type quasitoric manifolds and provide two classification results on them: (1) ($\mathbb{C}P^2\sharp\mathbb{C}P^2$)-bundle type quasitoric manifolds are weakly equivariantly homeomorphic if their cohomology rings are isomorphic, and (2) quasitoric manifolds over $I^3$ are homeomorphic if their cohomology rings are isomorphic. In the latter case, there are only four quasitoric manifolds up to weakly equivariant homeomorphism which are not bundle-type.
Quasitoric manifolds are manifolds that admit an action of the torus that is locally as the standard action of T^n on C^n. It is known that the quotients of such actions are nice manifolds with corners. We prove that such manifolds are equivariantly rigid i.e., that any other manifold that is T^n-homotopy equivalent to a quasitoric manifold, is T^n-homeomorphic to it.
We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.
It is proved that if two quasitoric manifolds of dimension $\le 2p^2-4$ for a prime $p$ have isomorphic cohomology rings, then they have the same $p$-local stable homotopy type.
For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a $2n$-dimensional smooth manifold with a locally standard action of the $n$-dimensional torus for which the orbit space is identified with $P$. This paper shows the topological classification of quasitoric manifolds over the dual cyclic polytope $C^n(m)^*$, when $n>3$ or $m-n=3$. Besides, we classify small covers, the "real version" of quasitoric manifolds, over all dual cyclic polytopes.
This note shows that for each $n\geq 5$ with only $n\not= 6$, there exists a $2n$-dimensional specially omnioriented quasitoric manifold $M^{2n}$ which represents a nonzero element in $Ω_*^U$. This provides the counterexamples of Buchstaber--Panov--Ray conjecture.
The study of embeddings of smooth manifolds into Euclidean and projective spaces has been for a long time an important area in topology. In this paper we obtain improvements of classical results on embeddings of smooth manifolds, focusing on the case of quasitoric manifolds. We give explicit constructions of equivariant embeddings of a quasitoric manifold described by combinatorial data $(P,Λ)$ into Euclidean and complex projective space. This construction provides effective bounds on the dimension of the equivariant embedding.
A quasitoric manifold is a $2n$-dimensional compact smooth manifold with a locally standard action of an $n$-dimensional torus whose orbit space is a simple polytope. In this article, we classify quasitoric manifolds with the second Betti number $β_2=2$ topologically. Interestingly, they are distinguished by their cohomology rings up to homeomorphism.
We give three sufficient criteria for two quasitoric manifolds (M,M') to be (weakly) equivariantly homeomorphic. We apply these criteria to count the weakly equivariant homeomorphism types of quasitoric manifolds with a given cohomology ring.
A quasitoric manifold is a smooth manifold with a locally standard torus action for which the orbit space is identified with a simple polytope. For a class of topological spaces, the class is called strongly cohomologically rigid if any isomorphism of cohomology rings can be realized as a homeomorphism. This paper shows the strong cohomological rigidity of the class of quasitoric manifolds over $I^3$.
We show a homotopy decomposition of $p$-localized suspension $ΣM_{(p)}$ of a quasitoric manifold $M$ by constructing power maps. As an application we investigate the $p$-localized suspension of the projection $π$ from the moment-angle complex onto $M$, from which we deduce its triviality for $p>\dim M/2$. We also discuss non-triviality of $π_{(p)}$ and $Σ^\inftyπ$.
We show that all generalized (real) Bott manifolds which are (small covers) quasitoric manifolds over a product of simplices $Δ^{n_1}\times\cdots\timesΔ^{n_r}\timesΔ^{1}$ are always boundaries of some manifolds. But these manifolds with the natural $(\mathbb{Z}_2)^n$ action do not necessarily bound equvariantly. In addition, we can construct some examples of null-cobordant but not orientedly null-cobordant manifolds among quasitoric manifolds.
This article mainly aims to give combinatorial characterizations and topological descriptions of quasitoric manifolds with string property. We provide a necessary and sufficient condition for a simple polytope in dimension 2 and 3 to be realizable as the orbit polytope of a string quasitoric manifold. In particular, a complete description of string quasitoric manifolds over prisms is obtained. On the other hand, we characterize string quasitoric manifolds over $n$-dimensional simple polytopes with no more than $2n+2$ facets. Further results are available when the orbit polytope is the connecte…
Let $X(Q,Λ)$ be a quasitoric manifold associated to a simple convex polytope $Q$ and characteristic function $Λ$. Let $T\cong (\mathbb{S}^1)^n$ denote the compact $n$-torus acting on $X=X(Q,Λ)$. The main aim of this article is to give a presentation of the $T$-equivariant $K$-ring of $X$, as a Stanley-Reisner ring over $K^*(pt)$. We also derive the presentation for the ordinary $K$-ring of $X$.
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 prove that any quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if $M$ admits a positive omniorientation. In particular, we show that all obstructions to existence of $T^n$-invariant almost complex structure on $M^{2n}$ arise from cohomology of underlying polytope - and hence are trivial.
We construct small covers and quasitoric manifolds over $n$-dimensional simple polytopes which allow proper colorings of facets with $n$ colors. We calculate Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case $n$ is a power of two.
We establish a vanishing result for indices of certain twisted Dirac operators on $\text{Spin}^c$-manifolds with non-abelian Lie-group actions. We apply this result to study non-abelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.
In this paper we develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the BKK Theorem, the Stanley-Reisner and Pukhlikov-Khovanskii type descriptions for cohomology rings of generalized quasitoric manifolds.
We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The obtained sets of integers are interesting, e. g. those representable as the sum of two squares D (C P^2#C P^2, C P^2) or the sum of three squares D (C P^2 # C P^2 # C P^2, C P^2). Beside the general results about the map degrees between quasitoric 4-manifolds, the connections among Duan-Wang's approach, the quadratic forms, the number theory and the lattices is es…
Baker and Richter construct a remarkable $A_\infty$ ring-spectrum $MΞ$ whose elements possess characteristic numbers associated to quasisymmetric functions; its relations, on one hand to the theory of noncommutative formal groups, and on the other to the theory of omnioriented (quasi)toric manifolds [in the sense of Buchstaber, Panov, and Ray], seem worth investigating.
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 …
A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS-manifold, for short, resp.) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex linear bundles, resp. In this paper we construct manifolds $M$ s.t. any complex vector bundle over $M$ is stably equivalent to a Whitney sum of complex linear bundles. A quasitoric manifold shares this property iff it is a TNS-manifold. We establish a new criterion of the TNS-property for a quasitoric manifold $M$ via non-semidefiniteness of certain higher-degree …
We present some classification results for quasitoric manifolds (M) with (p_1(M)=-\sum a_i^2) for some (a_i\in H^2(M)) which admit an action of a compact connected Lie-group (G) such that (\dim M/G \leq 1). In contrast to Kuroki's work we do not require that the action of (G) extends the torus action on (M).
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…
Totally skew embeddings are introduced by Ghomi and Tabachnikov. They are naturally related to classical problems in topology, such as the generalized vector field problem and the immersion problem for real projective spaces. In recent paper Topological obstructions to totally skew embeddings, {totaly skew} embeddings are studied by using the Stiefel-Whitney classes In the same paper it is conjectured that for every $n$-dimensional, compact smooth manifold $M^n$ $(n>1)$, $$N(M^n)\leq 4n-2α(n)+1,$$ where $N(M^n)$ is defined as the smallest dimension $N$ such that there exists a {\em totally ske…
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…
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 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.
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 …
We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(\mathbb{R}^n)$. In particular, we show that the number of top-dimensional simplices grows exponentially with $n$. More precise estimates are given for $k=2,3,4$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.
In this note, we show that the homotopy type of a complex manifold X satisfying the Oka property is captured by holomorphic maps from the affine spaces C^n, n\geq 0, into X. Among such X are all complex Lie groups and their homogeneous spaces. We present generalisations of this result, one of which states that the homotopy type of the space of continuous maps from any smooth manifold to X is given by a simplicial set whose simplices are holomorphic maps into X.
We propose a modification of the three-manifold invariant based on the use of Euclidean metric values ascribed to the elements of manifold triangulation. We thus obtain a nontrivial invariant that can, in particular, distinguish non-homeomorphic lens spaces.
The cohomological rigidity problem for toric manifolds asks whether the cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these resul…
We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably c…
We consider simple polytopes $P=vc^{k}(Δ^{n_{1}}\times\ldots\timesΔ^{n_{r}})$, for $n_1\ge\ldots\ge n_r\ge 1,r\ge 1,k\ge 0$, that is, $k$-vertex cuts of a product of simplices, and call them {\emph{generalized truncation polytopes}}. For these polytopes we describe the cohomology ring of the corresponding moment-angle manifold $\mathcal Z_P$ and explore some topological consequences of this calculation. We also examine minimal non-Golodness for their Stanley--Reisner rings and relate it to the property of $\mathcal Z_P$ being a connected sum of sphere products.
For a small cover Q^n and any principal (Z_2)^m-bundle M^n over Q^n, it was shown in a previous work of the author that the total sum of Z_2-Betti numbers of M^n is at least 2^m. In this paper, we prove that when M^n is connected, the total sum of Z_2-Betti numbers of such an M^n exactly equals 2^m if and only if M^n is homeomorphic to a product of spheres, and Q^n in this case must be a generalized real Bott manifold (or equivalent, Q^n is a small cover over a product of simplices).
The Geometry of Cochains on Sampled Vietoris-Rips Complexes
University of Edinburgh（爱丁堡大学）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology（系统生物学中心）；Technische Universitat Dresden（德累斯顿工业大学）
