Formality of Sinha’s cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Tsopméné, Songhafouo; Arnaud, Paul

doi:10.2140/agt.2013.13.2193

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…Few years later, the author [14] and Moriya [9] prove independently that the collapsing result still holds for d ≥ 3, and simplify the proof of the main result of [6]. Notice that, in this paper, we produce (using a completely different approach than that of [6,9,14]) another proof of the collapsing result.…”

Section: Introductionmentioning

confidence: 92%

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The rational homology of spaces of long links

Tsopméné

Arnaud²

2016

Algebr. Geom. Topol.

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We provide a complete understanding of the rational homology of the space of long links of m strands in R d , when d ≥ 4. First, we construct explicitly a cosimplicial chain complex, L• * , whose totalization is quasi-isomorphic to the singular chain complex of the space of long links. Next we show, using the fact that the Bousfield-Kan spectral sequence associated to L• * collapses at the E 2 page, that the homology Bousfield-Kan spectral sequence associated to the Munson-Volić cosimplicial model for the space of long links collapses at the E 2 page rationally, solving a conjecture of Munson-Volić. Our method enables us also to determine the rational homology of high dimensional analogues of spaces of long links. Our last result states that the radius of convergence of the Poincaré series for the space of long links (modulo immersions) tends to zero as m goes to the infinity.

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Section: Introductionmentioning

confidence: 92%

“…The author [14] and Moriya [9] independently discover the multiplicative formality (for d ≥ 3) of the Kontsevich operad K d (•). This result has two strong consequences: the first one is immediate and says that Sinha's cosimplicial space K • d is formal, when d ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

The rational homology of spaces of long links

Tsopméné

Arnaud²

2016

Algebr. Geom. Topol.

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“…We use the case of n = 1 of this theorem to prove Theorem 1.0.1 (2). As the both E n -actions have many applications [14,20,17,39,37], this theorem will be useful in other context. Theorem 1.0.2 is not so trivial as it looks since the E n -operad actions on Tot and Tot are realized by different operads and there is no obvious morphism between them.…”

Section: Introductionmentioning

confidence: 98%

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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“…is essentially a simple observation for the number of generators and degrees. Note that formality as a multiplicative operad implies degeneracy of the spectral sequence at E 2 -page as in [15,27,22], so in turn, non-degeneracy of the spectral sequence implies non-formality as a multiplicative operad. But it is weaker than non-formality as a non-symmetric operad.…”

Section: Introductionmentioning

confidence: 99%