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Models for knot spaces and Atiyah duality

Syunji Moriya

Algebraic & Geometric Topology 24 (2024) 183–250

Let Emb (S1,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 (S1,M)) for a simply connected closed manifold M of dimension 4 or more, which has an explicit E1–page and a computable E2–page. As applications, we compute some part of the cohomology for M = Sk × Sl with some conditions on the dimensions k and l, and prove that the inclusion Emb (S1,M) Imm (S1,M) to the immersions induces an isomorphism on π1 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.

embedding calculus, operad, knot space
Mathematical Subject Classification
Primary: 18M75, 55P43, 55T99, 57R40
Secondary: 18N40
Received: 7 April 2021
Revised: 26 August 2022
Accepted: 18 October 2022
Published: 18 March 2024
Syunji Moriya
Department of Mathematics and Information Sciences
Osaka Metropolitan University

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