The purpose of this paper is to represent the cohomology ring of a moment-angle manifold over an m-gon explicitly in terms of the quotient of an exterior algebra,and to count the Betti numbers of the cohomology groups...The purpose of this paper is to represent the cohomology ring of a moment-angle manifold over an m-gon explicitly in terms of the quotient of an exterior algebra,and to count the Betti numbers of the cohomology groups of a special class of quotients of moment-angle manifolds.展开更多
In this paper,we study the topology of moment-angle manifolds and prove a conjecture of S.Gitler and S.Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery’cutting off a vertex...In this paper,we study the topology of moment-angle manifolds and prove a conjecture of S.Gitler and S.Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery’cutting off a vertex’on a simple polytope.Let P be a simple polytope of dimension n with m facets and Pv be a poly tope obtained from P by cutting off one vertex v.Let Z=Z(P)and Zv=Z(Pv)be the corresponding moment-angle manifolds.S.Gitler and S.Lopez de Medrano conjectured that:Zv is diffeomorphic to δ[(Z(-D^n+m)×D^2]##^m-n/j=1(m-n/j)(S^j+2×S^m+n-j-1),and they proved the conjecture in the case m<3 n.In this paper we prove the conjecture in the general case.展开更多
In this paper, the authors consider the problem of which(generalized) momentangle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polyto...In this paper, the authors consider the problem of which(generalized) momentangle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope P_v, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to P_v also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.展开更多
The author constructs a family of manifolds, one for each n ≥ 2, having a nontrivial Massey n-product in their cohomology for any given n. These manifolds turn out to be smooth closed 2-connected manifolds with a com...The author constructs a family of manifolds, one for each n ≥ 2, having a nontrivial Massey n-product in their cohomology for any given n. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus Tm-action called momentangle manifolds ZP, whose orbit spaces are simple n-dimensional polytopes P obtained from an n-cube by a sequence of truncations of faces of codimension 2 only(2-truncated cubes). Moreover, the polytopes P are flag nestohedra but not graph-associahedra. The author also describes the numbers β^(-i,2(i+1))(Q) for an associahedron Q in terms of its graph structure and relates it to the structure of the loop homology(Pontryagin algebra)H_*(?ZQ), and then studies higher Massey products in H*(ZQ) for a graph-associahedron Q.展开更多
The connected sum is a fundamental operation in geometric topology and combinatorics.In this paper,we study the connection between connected sums of simplicial spheres and the algebraic topology of their corresponding...The connected sum is a fundamental operation in geometric topology and combinatorics.In this paper,we study the connection between connected sums of simplicial spheres and the algebraic topology of their corresponding moment-angle manifolds.The cohomology rings of moment-angle manifolds corresponding to connected sums of simplicial spheres are computed,which leads to a conjecture on the topology of such moment-angle manifolds.展开更多
In this paper, we generalize the conception of characteristic function in toric topology and construct many new smooth manifolds by using it. As an application, we classify the Moment-Angle manifolds and the partial q...In this paper, we generalize the conception of characteristic function in toric topology and construct many new smooth manifolds by using it. As an application, we classify the Moment-Angle manifolds and the partial quotients manifolds of them over a polygon. In the appendix we give a simple new proof for Orlik-Raymond＇s theorem in terms of characteristic function which gives the classification for quasitoric manifolds of dimension 4.展开更多
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 manifo...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 have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the 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 the 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 retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.展开更多
The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical a...The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology. In addition, they give a shorter proof of a well known criterion on this subject.展开更多
基金Partially supported by the NSFC(11971112)China ScholarshipCouncil(202106100095).
文摘The purpose of this paper is to represent the cohomology ring of a moment-angle manifold over an m-gon explicitly in terms of the quotient of an exterior algebra,and to count the Betti numbers of the cohomology groups of a special class of quotients of moment-angle manifolds.
基金supported by National Natural Science Foundation of China(Grant Nos.11571186,11701411,11801580 and 11871284)。
文摘In this paper,we study the topology of moment-angle manifolds and prove a conjecture of S.Gitler and S.Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery’cutting off a vertex’on a simple polytope.Let P be a simple polytope of dimension n with m facets and Pv be a poly tope obtained from P by cutting off one vertex v.Let Z=Z(P)and Zv=Z(Pv)be the corresponding moment-angle manifolds.S.Gitler and S.Lopez de Medrano conjectured that:Zv is diffeomorphic to δ[(Z(-D^n+m)×D^2]##^m-n/j=1(m-n/j)(S^j+2×S^m+n-j-1),and they proved the conjecture in the case m<3 n.In this paper we prove the conjecture in the general case.
基金supported by the National Natural Science Foundation of China(Nos.11471167,11571186,11701411,11801580)
文摘In this paper, the authors consider the problem of which(generalized) momentangle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope P_v, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to P_v also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.
基金supported by the General Financial Grant from the China Postdoctoral Science Foundation(No.2016M601486)
文摘The author constructs a family of manifolds, one for each n ≥ 2, having a nontrivial Massey n-product in their cohomology for any given n. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus Tm-action called momentangle manifolds ZP, whose orbit spaces are simple n-dimensional polytopes P obtained from an n-cube by a sequence of truncations of faces of codimension 2 only(2-truncated cubes). Moreover, the polytopes P are flag nestohedra but not graph-associahedra. The author also describes the numbers β^(-i,2(i+1))(Q) for an associahedron Q in terms of its graph structure and relates it to the structure of the loop homology(Pontryagin algebra)H_*(?ZQ), and then studies higher Massey products in H*(ZQ) for a graph-associahedron Q.
基金supported by National Natural Science Foundation of China(Grant Nos.11801580 and 11871284)supported by National Natural Science Foundation of China(Grant Nos.11871284 and 11761072).
文摘The connected sum is a fundamental operation in geometric topology and combinatorics.In this paper,we study the connection between connected sums of simplicial spheres and the algebraic topology of their corresponding moment-angle manifolds.The cohomology rings of moment-angle manifolds corresponding to connected sums of simplicial spheres are computed,which leads to a conjecture on the topology of such moment-angle manifolds.
文摘In this paper, we generalize the conception of characteristic function in toric topology and construct many new smooth manifolds by using it. As an application, we classify the Moment-Angle manifolds and the partial quotients manifolds of them over a polygon. In the appendix we give a simple new proof for Orlik-Raymond＇s theorem in terms of characteristic function which gives the classification for quasitoric manifolds of dimension 4.
基金supported by National Natural Science Foundation of China(Grant Nos.11371093,11431009 and 11661131004)supported by National Natural Science Foundation of China(Grant No.11028104)。
文摘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 have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the 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 the 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 retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.
基金supported by the National Natural Science Foundation of China(No.11871266)the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology. In addition, they give a shorter proof of a well known criterion on this subject.
