String topology for loop stacks

Behrend, Kai; Ginot, Grégory; Noohi, Behrang; Xu, Ping

doi:10.1016/j.crma.2006.10.006

Cited by 28 publications

(67 citation statements)

References 40 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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“…As an application, if G acts on a Poincaré duality space M then the homotopy quotient EG × G M is a Gorenstein space since it is the total space of the Borel fibration M → EG × G M → BG. String operations of some of this quotients have been studied in [24, §7] and using stacks and bivariant homology theory in [2]. For an oriented Poincaré duality space M of dimension m, the construction of the cohomological (g, p + q)-string operations on L M were depending only on the existence and the uniqueness (up to homotopy), in the derived category of C * (M t )-modules, of a map of degree m(t − r ), !…”

Section: Introductionmentioning

confidence: 99%

String topology on Gorenstein spaces

Félix

Thomas

2009

Math. Ann.

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The purpose of this paper is to describe a general and simple setting for defining (g, p + q)-string operations on a Poincaré duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincaré duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each (g, p + q)-string operation as well as it leads to explicit computations. IntroductionShriek maps play a central role in string topology and its generalizations. Following Dold, various presentations have been given [1,3,5,10,30]. Usually shriek maps are defined in (co)homology for maps f : N → M from a closed oriented n-manifold to a closed oriented m-manifold. Here we will consider shriek maps at the cochain level and in a more general setting.More precisely we work in the category of (left or right) differential graded modules over a differential graded lk-algebra (R, d) (lk is a fixed field), that we call for sake of simplicity the category of (R, d)-modules. Its associated derived category [20] is obtained by formally inverting quasi-isomorphisms, i.e. the homomorphisms of (R, d)-modules that induce isomorphisms in homology and hereafter denoted by in the diagrams. In the derived category the vector space of Y. Félix

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“…The definition and properties of LX can be found in eg [9]. Loop rotation yields an S 1 -action on LX .…”

Section: Loop Space Interpretationmentioning

confidence: 99%

Orbifold quantum Riemann–Roch, Lefschetz and Serre

2010

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Given a vector bundle F on a smooth Deligne-Mumford stack X and an invertible multiplicative characteristic class c, we define orbifold Gromov-Witten invariants of X twisted by F and c. We prove a "quantum Riemann-Roch theorem" (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.

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String topology for loop stacks

Cited by 28 publications

References 40 publications

String topology of classifying spaces

String topology of classifying spaces

String topology on Gorenstein spaces

Orbifold quantum Riemann–Roch, Lefschetz and Serre

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