Orbifold homeomorphism finiteness based on geometric constraints

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman’s stability theorem must preserve orbifold structure.     

Double Ramification Cycles with Orbifold Targets

Acta Mathematica Sinica, English Series - In this paper, we consider double ramification cycles with orbifold targets. An explicit formula for double ramification cycles with orbifold targets,...     

Orbifold fibrations of Eschenburg spaces

Most of the few known examples of compact Riemannian manifolds with positive sectional curvature are the total space of a Riemannian submersion. In this article we show that this is true for all known examples, if we enlarge the category to orbifold fibrations. For this purpose we study all almost free isometric circle actions on positively curved Eschenburg spaces, which give rise to principle orbifold bundle structures, and we examine in detail their geometric properties. In particular, we obtain a new family of 6-dimensional orbifolds with positive sectional curvature whose singular locus consists of just two points.      

辛群胚的弱Morita等价与orbifold基本群

本文研究辛orbifold群胚的弱Morita等价,证明了两个辛orbifold群胚弱Morita等价当且仅当其orbifold基本群同构.     

Orbifold Gromov-Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid (X, ω). Let X_a be the weight-a blowup of X along S, and D_a = PN_a be the exceptional divisor, where N is the normal bundle of S in X. In this paper we show that the absolute orbifold Gromov-Witten theory of X_α can be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X, S and D_α, the natural restriction homomorphism H*_CR(X) → H*_CR(S) and the first Chern class of the tautological line bundle over D_α. To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of (X_α | D_α) and (N_α | D_α). As applications of these results, we prove an orbifold version of a conjecture of Maulik and Pandharipande (Topology, 2006) on the Gromov-Witten theory of blowups along complete intersections, a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You (J Pure Appl Algebra, 2016).

     

Orbifold Triangulations and Crystallographic Groups

Given a triangulation of a 3-dimensional euclidean orbifold, e.g. in terms of the Delaney symbol of a periodic tiling, a method is discussed for identifying the isomorphism type of the corresponding space group. Of the 219 types of groups, 175 can be recognized solely by considering the orbifold graph associated with the given triangulation. Simple algebraic invariants distinguish between the remaining 44 cases. The graphs and invariants are listed.     

辛轨形群胚的辛邻域定理

本文给出一种几何的子轨形群胚的定义,还给出了判定子轨形群胚的依据,并证明了紧子轨形群胚的轨形管状邻域、紧辛子轨形群胚的辛邻域和紧Lagrangian子轨形群胚的Lagrangian邻域的存在性.     

Orbifold genera, product formulas and power operations

We generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H_∞-map into the Morava-Lubin-Tate theory E_h, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus [R. Dijkgraaf et al., Elliptic genera of symmetric products and second quantized strings Comm. Math. Phys. 185(1) (1997) 197-209]. It depends only on h and not on the genus.     

Orbifold Gromov–Witten Invariants of Weighted Blow-up at Smooth Points

In this paper, one considers the change of orbifold Gromov–Witten invariants under weighted blow-up at smooth points. Some blow-up formula for Gromov–Witten invariants of symplectic orbifolds is proved. These results extend the results of manifolds case to orbifold case.     

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant stable almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds, the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold. The authors show that for a finite group G and a G-equivariant stable almost complex manifold X, the G-invariant part of the stringy cohomology ring of (X, G) is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold [X/G]. Similar result holds when G is a torus and the action is locally free. Moreover, for a compact presentable stable almost complex orbifold, they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.     

Instanton Counting and Wall-Crossing for Orbifold Quivers

Noncommutative Donaldson–Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative ${{\mathcal N}=2}$ gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated with the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson–Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.     

Local formulae for Stiefel-Whitney classes

We construct an explicit Čech cocycle representing the k-th Stiefel-Whitney class of a vector bundle. This construction involves only the transition functions of the bundle. We also give local formulae for the secondary Stiefel-Whitney classes. These may be useful in determining whether the Stiefel-Whitney numbers of a flat bundle are zero.     

Spectral Bounds on Orbifold Isotropy

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Petersen, as well as comparison geometry techniques developed by Grove and Petersen.This research was partially supported by NSF grant DMS 0072534.     

A Model for the Orbifold Chow Ring of Weighted Projective Spaces

We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.     

Orbifold Euler characteristics and the number of commutingm-tuples in the symmetric groups

Generating functions for the number of commutingm-tuples in the symmetric groups are obtained. We define a natural sequence of orbifold Euler characteristics for a finite groupG acting on a manifoldX. Our definition generalizes the ordinary Euler characteristic ofX/G and the string-theoretic orbifold Euler characteristic. Our formulae for commutingm-tuples underlie formulae that generalize the results of Macdonald and Hirzebruch-Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products.Supported partly by a grant from the Ford Foundation.     

Orbifold construction for topological field theories

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G?H$ of finite groups assigns in a functorial way to a G-equivariant topological field theory an H-equivariant topological field theory, the pushforward theory. When H is the trivial group, this yields an orbifold construction for G-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.     

$CP(n)\times CP(m)$上向量丛的全Stiefel-Whitney类

The possible form of the total Stiefel-Whitney classes of vector bundles on CP（n） × CP（m） is determined in this paper.     

RP(j)×CP(k)上向量丛的全Stiefel-Whitney类

本文利用Steenrod上同调运算及吴公式决定了RP(j)×CP(k)上的向量丛的全Stiefel-Whitney类的可能的形状．     

Construction de classes de Chern éguivariantes pour un fibré vectoriel Réel

On définit des classes de Chern pour un fibré vectoriel “réel” au sens d'Atiyah [1], à valeurs dans la cohomologie équivariante paire de la base à coefficients entiers tordus. Dans le cas d'un fibré réel au sens usuel, ces classes redonnent les classes de Pontrjagin et les classes de Stiefel-Whitney.     

Stiefel-Whitney classes and the conormal cycle of a singular variety

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety is given by means of the conormal cycle of an embedding of in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.