A spectral sequence for cohomology of knot spaces

Abstract: This talk is based on the preprint arXiv:2003.03815. Let $Emb(S^1,M)$ be the space of embeddings from $S^1$ to a closed manifold $M$ (space of knots in $M$). Recently, this space is studied by Arone-Szymik, Budney-Gabai, and Kupers, using Goodwillie-Weiss embedding calculus. In this talk, we introduce a spectral sequence for cohomology of $Emb(S^1,M)$ whose $E_2$-term has an algebraic presentation, using Sinha's cosimplicial model which is derived from the calculus. This converges to the correct target if $M$ is simply connected and of dimension $\geq 4$ for general coefficient ring. Using this, we see a computation of $H^*(Emb(S^1,S^k\times S^l))$ in low degrees under some assumption on $k,l$ and an isomorphism $\pi_1(Emb(S^1,M))\cong H_2(M,\mathbb{Z})$ for some simply connected $4$-dimensional $M$.

Our main idea of the construction is to replace configuration spaces in the cosimplicial model with fat diagonals via Poincaré Lefschetz duality. To do this, we use a notion of a (co)module over an operad. A somewhat curious point is that we need spectra (in stable homotopy) even though our concern is singular cohomology.

algebraic topologycategory theoryquantum algebra

Audience: learners

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