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Models for knot spaces and Atiyah duality

Syunji Moriya

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.

Models for knot spaces and Atiyah duality

Abstract

Let be the space of smooth embeddings from the circle to a closed manifold of dimension . We study a cosimplicial model of 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 for simply connected and for major coefficient rings. Using this, we compute in low degrees with some conditions on , . We also prove the inclusion to the immersions induces an isomorphism on for some simply connected -manifolds, related to a question posed by Arone and Szymik. We also prove an equivalence of singular cochain complex of 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.

Paper Structure

This paper contains 23 sections, 47 theorems, 126 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and terminologies
  4. $\check{\text{C}}$ech complex and homotopy colimit
  5. Goodwillie-Weiss embedding calculus and Sinha's cosimplicial model
  6. Operads, comodules, and Hochschild complex
  7. Comodule $\mathcal{TH}_M$
  8. A model of a Thom spectrum
  9. Construction of a comodule $\widetilde{\mathcal{TH}}_M$
  10. Construction of the comodule $\mathcal{TH}_M$
  11. Atiyah duality for comodules
  12. Spectral sequences
  13. A chain functor
  14. Construction of $\check{\text{C}}$ech spectral sequence
  15. Convergence
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exists a left $\mathcal{D}_1$-comodule $\mathcal{TH}_M$ of non-unital commutative symmetric ring spectra as follows.

Figures (3)

  • Figure 1: intuition of the coface map $d^i$ : $y_i$ is the vector at $x_i$
  • Figure 2: the map $\Delta'$ : the geodesic segment is devided into the pieces of rate of length $|c_1|:|c_2|:|c_3|$
  • Figure 3: the first inclusion of Lemma \ref{['Ldiagonalinclusion']} (1) with $n=2$ : The bold line is a part of the geodesic segment used to define $\Delta'$. $(x',y')$ is the $i$-th component of $\pi_{e'}(u)\in \widehat{M}^{\times n}$. $x_i$ and $x_{i+1}$ exist in the interior of the disks at $x_i'$ and $x_{i+1}'$ if $(\mathfrak{c} \circ_i\phi)(\mathfrak{d})\not=*$.

Theorems & Definitions (110)

  • Theorem 1.1: Theorem \ref{['TAtiyahdual']}
  • Theorem 1.2: Theorems \ref{['Tam']}, \ref{['Tconvergence']} and \ref{['Talgebraicss']}
  • Corollary 1.3
  • Theorem 1.4: Theorem \ref{['Tthomcolimit']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 100 more