Models for knot spaces and Atiyah duality

Moriya, Syunji

doi:10.2140/agt.2024.24.183

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Models for knot spaces and Atiyah duality

Syunji Moriya

Algebraic & Geometric Topology 24 (2024) 183–250

DOI: 10.2140/agt.2024.24.183

Abstract

Let $Emb (S^{1}, M)$ be the space of smooth embeddings from the circle to a closed manifold $M$ . We introduce a new spectral sequence converging to $H^{*} (Emb (S^{1}, M))$ for a simply connected closed manifold $M$ of dimension $4$ or more, which has an explicit $E_{1}$ –page and a computable $E_{2}$ –page. As applications, we compute some part of the cohomology for $M = S^{k} \times S^{l}$ with some conditions on the dimensions $k$ and $l$ , and prove that the inclusion $Emb (S^{1}, M) \to Imm (S^{1}, M)$ to the immersions induces an isomorphism on $π_{1}$ for some simply connected $4$ –manifolds. This gives a restriction on a question posed by Arone and Szymik. The idea to construct the spectral sequence is to combine a version of Sinha’s cosimplicial model for the knot space and a spectral sequence for a configuration space by Bendersky and Gitler. The cosimplicial model consists of configuration spaces of points (with a tangent vector) in $M$ . We use Atiyah duality to transfer the structure maps on the configuration spaces to maps on Thom spectra of the quotient of a direct product of $M$ by the fat diagonal. This transferred structure is the key to defining our spectral sequence, and is also used to show that Sinha’s model can be resolved into simpler pieces in a stable category.

Keywords

embedding calculus, operad, knot space

Mathematical Subject Classification

Primary: 18M75, 55P43, 55T99, 57R40

Secondary: 18N40

References

Publication

Received: 7 April 2021

Revised: 26 August 2022

Accepted: 18 October 2022

Published: 18 March 2024

Authors

Syunji Moriya
Department of Mathematics and Information Sciences Osaka Metropolitan University Sakai Japan

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Volume 24, issue 1 (2024)