On the invariance of the string topology coproduct

Hingston, Nancy; Wahl, Nathalie

doi:10.48550/arxiv.1908.03857

Cited by 5 publications

(8 citation statements)

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“…This homology coproduct and cohomology product have further been studied by Hingston and Wahl in [HW17]. In [HW19] the same authors show that the homology coproduct is a homotopy invariant just like the Chas-Sullivan product.…”

Section: Introductionmentioning

confidence: 88%

On the String Topology Coproduct for Lie Groups

Stegemeyer¹

2021

Preprint

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The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a so-called based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups.

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Section: Introductionmentioning

confidence: 88%

On the String Topology Coproduct for Lie Groups

Stegemeyer¹

2021

Preprint

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“…Here Theorem 2.26 perfectly fulfils its role as a bridge between the loop product and the cohomology product, since it allows us to prove in one sweep the homotopy invariance of both. Let us mention that homotopy invariance of the cohomology product was also proved using different methods by Hingston and Wahl in [47].…”

Section: Theorem 226 ([19]mentioning

confidence: 98%

Poincaré duality for loop spaces

Cieliebak¹,

Hingston²,

Oancea³

2020

Preprint

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We prove a Poincaré duality theorem with products between Rabinowitz Floer homology and cohomology, for both closed and open strings. This lifts to a duality theorem between open-closed TQFTs. Specializing to the case of cotangent bundles, we define extended loop homology and cohomology and explain from a unified perspective pairs of dual results which have been observed over the years in the context of the search for closed geodesics. These concern critical levels, relations to the based loop space, manifolds all of whose geodesics are closed, Bott index iteration, level-potency, and homotopy invariance. We extend the loop cohomology product to include constant loops. We prove a relation conjectured by Sullivan between the loop product and the extended loop homology coproduct as a consequence of associativity for the product on extended loop homology. ContentsKAI CIELIEBAK, NANCY HINGSTON, AND ALEXANDRU OANCEA 3.1. Rabinowitz Floer homology 3.2. Poincaré duality 4. Poincaré duality in symplectic homology 4.1. Recollections on Poincaré duality and exact sequences 4.2. TQFT operations on Floer homology 4.3. Products and the mapping cone 4.4. Poincaré duality with products 4.5. Proof of Theorems 1.1, 1.11, and 1.13 4.6. A Morse theoretic description of the Gysin sequence 4.7. Canonical splitting of the duality sequence and proof of Theorem 1.4 5. Open-closed TQFT structures 5.1. Primary open-closed TQFT structure in homology 5.2. Canonical operations 5.3. Duality of open-closed TQFTs 5.4. Topological interpretation of the open-closed and closed-open maps 6. BV structures and Poincaré duality 6.1. Twisted BV-structures 6.2. BV structure on loop homology 6.3. BV structure on extended loop homology Appendix A. Poincaré duality product on SH ă0 ˚pV, BV q via homotopies Appendix B. Grading conventions References

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“…) is induced by the homotopy retract (10). To calculate H i (D * (A, A)) in terms of Hochschild homology and cohomology we analyze the double complex D * , * (A, A) in the case A 0 ∼ = K and A 1 = 0: .…”

Section: 2mentioning

confidence: 99%

“…An interesting question is to give a geometric or topological explanation of the extension of the operation on relative homology H * (LM, M ) to reduced homology H * (LM ) so that it coincides with the algebraic extension. A geometric proof of the homotopy invariance of the Goresky-Hingston operation at the level of H * (LM, M ) (with any coefficients) has been announced in [10]. They also prove that such operation is natural in the sense that if ϕ : M → M ′ is a homotopy equivalence between oriented closed manifolds of the same dimension then H * (ϕ) :…”

Section: Introductionmentioning

confidence: 96%

Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

Rivera¹,

Wang²

2019

Preprint

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We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky-Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky-Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold M of dimension k, the Goresky-Hingston algebra on reduced Hochschild homology induces an algebra structure of degree k−1 on H * (LM ; Q), the reduced rational cohomology of the free loop space of M . As a consequence of our algebraic result, we deduce that the isomorphism class of the induced algebra structure on H * (LM ; Q) is an invariant of the homotopy type of M .

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On the invariance of the string topology coproduct

Abstract: We show that the Goresky-Hingston coproduct in string topology, just like the Chas-Sullivan product, is homotopy invariant.

Cited by 5 publications

References 11 publications

On the String Topology Coproduct for Lie Groups

On the String Topology Coproduct for Lie Groups

Poincaré duality for loop spaces

Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

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