Models for knot spaces and Atiyah duality

Abstract

Let $\mathrm{Emb}(S^1,M)$ be the space of smooth embeddings from the circle to a closed manifold $M$ of dimension $\ge 4$. We study a cosimplicial model of $\mathrm{Emb}(S^1,M)$ in stable categories, using a spectral version of Poincaré-Lefschetz duality called Atiyah duality. We actually deal with a notion of a comodule instead of the cosimplicial model, and prove a comodule version of the duality. As an application, we introduce a new spectral sequence converging to $H^{\ast }(\mathrm{Emb}(S^1,M))$ for simply connected $M$ and for major coefficient rings. Using this, we compute $H^{\ast }(\mathrm{Emb}(S^1,S^k\times S^l))$ in low degrees with some conditions on $k$, $l$. We also prove the inclusion $\mathrm{Emb}(S^1,M)\to \mathrm{Imm}(S^1,M)$ to the immersions induces an isomorphism on ${\pi }_1$ for some simply connected $4$-manifolds, related to a question posed by Arone and Szymik. We also prove an equivalence of singular cochain complex of $\mathrm{Emb}(S^1,M)$ and a homotopy colimit of chain complexes of a Thom spectrum of a bundle over a comprehensible space. Our key ingredient is a structured version of the duality due to R. Cohen.

arXiv comment56 pages, errors corrected

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Cited by in corpus (2)

• On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold
• Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad