The homotopy invariance of the string topology loop product and string bracket

Cohen, Ralph L.; Klein, John R.; Sullivan, Dennis

doi:10.1112/jtopol/jtn001

Cited by 18 publications

(33 citation statements)

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“…The definition we use here is due to Cohen and Jones [15]. See also Chas and Sullivan [10], Sullivan [49], Baas, Cohen and Ramírez [6], Cohen [13], Cohen, Hess and Voronov [14], Ramírez [41], Sullivan [50], Cohen, Klein and Sullivan [17] and Cohen and Schwarz [18] for more information and for other interpretations of this product.…”

Section: Remark 11mentioning

confidence: 99%

“…be a smooth partition of unity of C satisfying (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20). Then the sequences .…”

Section: The Fredholm Propertymentioning

confidence: 99%

“…By (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and (4-21), u 2 is constant, so by (4-21) and (4-22) u 1 is a periodic solution of (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). Since˛is positive, u 1 is zero and by (4-21) so is u 2 .…”

Section: The Left-hand Square Is Homotopy Commutativementioning

confidence: 99%

“…Let H 2 C 1 .R=2Z T M / be defined by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). Given and the asymptotic conditions…”

Section: Factorization Of the Pair-of-pants Productmentioning

confidence: 99%

“…This object is a Riemann surface with boundary, with the holomorphic coordinates (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) at .0; 1/ .0; 0 / and at .0; 0 C / .0; 1/, and with the holomorphic coordinates (3-13) and (3-21) (translated by˛) at .˛; 0/ and at .˛; 1/ .˛; 1/. The resulting object is a Riemann surface with three cylindrical ends and one boundary circle.…”

Section: Factorization Of the Pair-of-pants Productmentioning

confidence: 99%

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Floer homology of cotangent bundles and the loop product

2010

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We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M . We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique. 53D40, 57R58; 55N45 IntroductionLet M be a closed manifold, and let H be a time-dependent smooth Hamiltonian on T M , the cotangent bundle of M . We assume that H is 1-periodic in time and grows asymptotically quadratically on each fiber. Generically, the corresponding Hamiltonian system (0-1) x 0 .t/ D X H .t; x.t// has a discrete set ᏼ.H / of 1-periodic orbits. The free abelian group F .H / generated by the elements in ᏼ.H /, graded by their Conley-Zehnder index, supports a chain complex, the Floer complex .F .H /; @/. The boundary operator @ is defined by an algebraic count of the maps u from the cylinder R T to T M , solving the Cauchy-Riemann type equation (0-2) @ s u.s; t/ C J.u.s; t// @ t u.s; t/ X H .t; u.s; t// D 0; 8.s; t/ 2 R T ;and converging to two 1-periodic orbits of (0-1) for s ! 1 and s ! C1. Here J is the almost-complex structure on T M induced by a Riemannian metric on M , and (0-2) can be seen as the negative L 2 -gradient equation for the Hamiltonian action functional.This construction is due to A Floer (eg [21;22;23;24]) in the case of a closed symplectic manifold P , in order to prove a conjecture of Arnold on the number of Since the space W 1;2 .T ; M / is homotopy equivalent to ƒ.M /, we get the required isomorphismGeometry & Topology, Volume 14 (2010) Floer homology of cotangent bundles and the loop product 1571 from the singular homology of the free loop space of M to the Floer homology of T M .Additional interesting algebraic structures on the Floer homology of a symplectic manifold are obtained by considering other Riemann surfaces than the cylinder as domain for the Cauchy-Riemann type equation (0-2). By considering the pair-of-pants surface, a noncompact Riemann surface with three cylindrical ends, one obtains the pairof-pants product in Floer homology (see the second author's thesis [46] and McDuff and Salamon [38]). When the symplectic manifold P is closed and symplectically aspherical, this product corresponds to the standard cup product from topology, after identifying the Floer homology of P with its singular cohomology by Poincaré duality, while when the manifold P can carry J -holomorphic spheres, the pair-of-pants product corresponds to the quantum cup product of P (see Piunikhin, Salamon and Schwarz [39] and Liu and Tian [37]).The main result of this paper is that in the case of cotangent bundles, the pair-of-pants product is also equivalent to a product on H .ƒ.M // coming from topology, but a more interesting one than the simple cup produ...

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Section: Remark 11mentioning

confidence: 99%

“…be a smooth partition of unity of C satisfying (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20). Then the sequences .…”

Section: The Fredholm Propertymentioning

confidence: 99%

Section: The Left-hand Square Is Homotopy Commutativementioning

confidence: 99%

“…Let H 2 C 1 .R=2Z T M / be defined by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). Given and the asymptotic conditions…”

Section: Factorization Of the Pair-of-pants Productmentioning

confidence: 99%

Section: Factorization Of the Pair-of-pants Productmentioning

confidence: 99%

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Floer homology of cotangent bundles and the loop product

2010

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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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