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arXiv · 2003.03815

Models for knot spaces and Atiyah duality

Syunji Moriya
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Abstract

Let $\mathrm{Emb}(S^1,M)$ be the space of smooth embeddings from the circle to a closed manifold $M$ of dimension $\geq 4$. 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 $H^*(\mathrm{Emb}(S^1,M))$ for simply connected $M$ and for major coefficient rings. Using this, we compute $H^*(\mathrm{Emb}(S^1, S^k\times S^l))$ 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 $π_1$ 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.

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BibTeXRIS

Syunji Moriya. 2021-11-02. Models for knot spaces and Atiyah duality. https://doi.org/10.2140/agt.2024.24.183

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