Behavior of the Eilenberg–Moore spectral sequence in derived string topology

Kuribayashi, Katsuhiko; Menichi, Luc; Naito, Takahito

doi:10.1016/j.topol.2013.12.003

Cited by 5 publications

(5 citation statements)

References 21 publications

(35 reference statements)

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“…In particular, we consider spaces with exterior cohomology. The (co)homology of free loop spaces of simply connected spaces with mod p exterior cohomology has been considered, for instance, in [24], [22], [32], and [23]. Our approach yields new results, as consequences of the following general collapse result for the coBökstedt spectral sequence, which we prove later in this section.…”

Section: Proposition 71 ([3]mentioning

confidence: 83%

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Topological coHochschild Homology and the Homology of Free Loop Spaces

Bohmann¹,

Gerhardt²,

Shipley³

2021

Preprint

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We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.

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Section: Proposition 71 ([3]mentioning

confidence: 83%

“…We consider in detail the homology of LX when X is a simply connected space with exterior cohomology. The (co)homology of LX in such cases has been considered, for instance, in [24], [22], and [23]. Our approach yields new results.…”

Section: Introductionmentioning

confidence: 95%

Topological coHochschild Homology and the Homology of Free Loop Spaces

Bohmann¹,

Gerhardt²,

Shipley³

2021

Preprint

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“…Let N be a simply-connected space whose cohomology is of finite dimension and is generated by a single element. Then explicit calculations of the dual EMSS made in the sequel [20] to this paper yield that the loop homology of N is isomorphic to the Hochschild cohomology of H * (N ; K) as an algebra. This illustrates computability of our spectral sequence in Theorem 2.11.…”

Section: Introductionmentioning

confidence: 95%

Derived string topology and the Eilenberg-Moore spectral sequence

Kuribayashi

Menichi²,

Naito

2015

Isr. J. Math.

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Let M be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space H * (LM ). We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively.We also define a generalized cup product on the Hochschild cohomology HH * (A, A ∨ ) of a commutative Gorenstein algebra A and show that over Q, HH * (A P L (M ), A P L (M ) ∨ ) is isomorphic as algebras to H * (LM ). Thus, when M is a Poincaré duality space, we recover the isomorphism of algebras H * (LM ; Q) ∼ = HH * (A P L (M ), A P L (M )) of Félix and Thomas.

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“…It is a study of certain algebraic structures on the homology of LM , which can be seen as a generalization of Goldman's Lie algebra for a Riemann surface. These structures are studied in many ways e.g., relation to counting problem of closed geodesics (see Goresky-Hingston [27]), generalization to Gorenstein spaces (see Félix-Thomas [25], Kuribayashi-Menichi-Naito [35,38], and Naito [36]). Another interesting subject is the relationship between string topology operations and intrinsic operations on Hochschild cohomology of the cochain due to Gerstenhaber [2] and Jones [6].…”

Section: Introductionmentioning

confidence: 99%

On Cohen-Jones isomorphism in string topology

Moriya¹

2020

Preprint

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The loop product is an operation in string topology. Cohen and Jones [15,16] gave a homotopy theoretic realization of the loop product as a classical ring spectrum LM −T M for a manifold M . Using this, they presented a proof of the statement that the loop product is isomorphic to the Gerstenhaber cup product on the Hochschild cohomology HH * (C * (M ) ; C * (M )) for simply connected M . However, some parts of their proof is technically difficult to justify. The main aim of the present paper is to give detailed modification to a geometric part of their proof. To do so, we set up an "up to higher homotopy" version of McClure-Smith's cosimplicial product. We prove a structured version of Cohen-Jones isomorphism in the category of symmetric spectra.

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Behavior of the Eilenberg–Moore spectral sequence in derived string topology

Cited by 5 publications

References 21 publications

Topological coHochschild Homology and the Homology of Free Loop Spaces

Topological coHochschild Homology and the Homology of Free Loop Spaces

Derived string topology and the Eilenberg-Moore spectral sequence

On Cohen-Jones isomorphism in string topology

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