Operads and knot spaces. (English) Zbl 1112.57004
Let \(\mathit{E}_m\) denote the space of embeddings of the unit interval into the \(m\)-dimensional unit cube with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, and equipped with a homotopy through immersions to the unknot. For an operad with multiplication \(\mathcal{O}\), J. McClure and J. Smith [Recent progress in homotopy theory. Proceedings of a conference, Baltimore, MD, USA, March 17-27, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. AMS 293, 153–193 (2002; Zbl 1009.18009)] have defined a cosimplicial object \(\mathcal{O}^{\bullet}\). The (homotopy invariant) totalization of \(\mathcal{O}^{\bullet}\) is denoted by \(\widetilde{\mathrm{Tot}}(\mathcal{O}^{\bullet})\). Furthermore, let \(\mathcal{K}_m\) denote the \(m\)th Kontsevich operad introduced by M. Kontsevich [Lett. Math. Phys. 48, 35–72 (1999; Zbl 0945.18008)].
Main Theorem: The totalization of the Kontsevich operad \(\widetilde{\mathrm{Tot}}(\mathcal{K}^{\bullet})\) is homotopy equivalent to the inverse limit of the Taylor tower approximations for \(\mathit{E}_m\) in the calculus of embeddings. Moreover, \(\widetilde{\mathrm{Tot}}^n(\mathcal{K}^{\bullet})\) is the \(n\)th degree approximation. As corollaries, the author derives that for \(m>3\), \(\mathit{E}_m\) is weakly equivalent to \(\widetilde{\mathrm{Tot}}(\mathcal{K}^{\bullet})\). For \(m>3\), he also finds a spectral sequence with \(\mathit{E}^2\)-term given by the Hochschild cohomology of the degree \(m-1\) Poisson operad which converges to the homology of \(\mathit{E}_m\). In another noteworthy theorem, the author uses a result of McClure and Smith to establish that for any \(m\), there is a little two-cubes action on \(\widetilde{\mathrm{Tot}}(\mathcal{K}^{\bullet})\), and that for \(m>3\), \(\mathit{E}_m\) is a two-fold loop space.
Main Theorem: The totalization of the Kontsevich operad \(\widetilde{\mathrm{Tot}}(\mathcal{K}^{\bullet})\) is homotopy equivalent to the inverse limit of the Taylor tower approximations for \(\mathit{E}_m\) in the calculus of embeddings. Moreover, \(\widetilde{\mathrm{Tot}}^n(\mathcal{K}^{\bullet})\) is the \(n\)th degree approximation. As corollaries, the author derives that for \(m>3\), \(\mathit{E}_m\) is weakly equivalent to \(\widetilde{\mathrm{Tot}}(\mathcal{K}^{\bullet})\). For \(m>3\), he also finds a spectral sequence with \(\mathit{E}^2\)-term given by the Hochschild cohomology of the degree \(m-1\) Poisson operad which converges to the homology of \(\mathit{E}_m\). In another noteworthy theorem, the author uses a result of McClure and Smith to establish that for any \(m\), there is a little two-cubes action on \(\widetilde{\mathrm{Tot}}(\mathcal{K}^{\bullet})\), and that for \(m>3\), \(\mathit{E}_m\) is a two-fold loop space.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)
MSC:
| 57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |
| 18D50 | Operads (MSC2010) |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
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