Orbifold string topology

Lupercio, Ernesto; Uribe, Bernardo; Xicoténcatl, Miguel A.

doi:10.2140/gt.2008.12.2203

Cited by 31 publications

(61 citation statements)

References 19 publications

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“…More recently the theory was successfully extended to topological orbifolds [45] and topological stacks [8]. K. Gruher and P. Salvatore also studied a Pro-spectrum version of string topology for the classifying space of a compact Lie group [33].…”

Section: Introductionmentioning

confidence: 99%

String topology of classifying spaces

Chataur

Menichi

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG := map(S 1 , BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H * (LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology H * (LBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH

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Section: Introductionmentioning

confidence: 99%

String topology of classifying spaces

Chataur

Menichi

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The construction in [28] furthermore commutes with the functor B from groupoids to spaces defined in the first section, as shown in [31], in the sense that there is an homotopy equivalence BLX ≃ LBX.…”

Section: Loop Orbifoldsmentioning

confidence: 95%

A localization principle for orbifold theories

Uribe¹,

Fernex²,

Nevins³

et al. 2006

Recent Developments in Algebraic Topology

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Abstract. In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as K-theory, index theory, motivic integration and elliptic genera. OrbifoldsIn this paper we will attempt to explain a general localization principle that appears frequently under several guises in the study of orbifolds. We will begin by reminding the reader what we mean by an orbifold.The most familiar situation in physics is that of an orbifold of the type X = [M/G], where M is a smooth manifold and G is a finite group acting 1 smoothly on M ; namely, we give ourselves a homomorphism G → Diff(M ). We make a point of distinguishing the orbifold X = [M/G] from its quotient space (also called orbit space) X = M/G. As a set, as we know, a point in X is an orbit of the action: that is, a typical element of M/G is Orb(x) = {xg | g ∈ G}.For us an orbifold X = [M/G] is a smooth category 2 (actually a topological groupoid) whose objects are the points of M , X 0 = Obj(X) = M , and we insist on remembering that X 0 = Obj(X) is a smooth manifold. The arrows of this category are X 1 = Mor(X) = M × G again thinking of it as a smooth manifold. A typical arrow in this category is

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“…Others have considered String Topology operations for orbifolds and manifold stacks in a more abstract setting [1], [3], [18]. It would be interesting to relate those results to the concrete results here.…”

Section: Introductionmentioning

confidence: 92%

The extended Goldman bracket determines intersection numbers for surfaces and orbifolds

Chas

Gadgil

2016

Algebr. Geom. Topol.

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Abstract. In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically, in terms of the same Lie structure. We show how the Goldman bracket answers these questions for all finite type surfaces. In fact we count self-intersection numbers and mutual intersection numbers for all finite type orientable orbifolds in terms of a new Lie bracket operation, extending Goldman's. The arguments are purely topological, or based on elementary ideas from hyperbolic geometry.These results are intended to be used to recognize hyperbolic and Seifert vertices and the gluing graph in the geometrization of three manifolds. The recognition is based on the structure of the String Topology bracket of three manifolds.This work is dedicated with deep and grateful admiration to Bill Thurston (1946Thurston ( -2012.

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Orbifold string topology

Abstract: In this paper we study the string topology (à la Chas-Sullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties and do some explicit calculations. 55P35; 18D50, 55R35

Cited by 31 publications

References 19 publications

String topology of classifying spaces

String topology of classifying spaces

A localization principle for orbifold theories

The extended Goldman bracket determines intersection numbers for surfaces and orbifolds

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