Models for knot spaces and Atiyah duality

Abstract

M / be the space of smooth embeddings from the circle to a closed manifold M . We introduce a new spectral sequence converging to H .Emb.S 1 ; M // for a simply connected closed manifold M of dimension 4 or more, which has an explicit E 1 -page and a computable E 2 -page. As applications, we compute some part of the cohomology for M D S k S l with some conditions on the dimensions k and l, and prove that the inclusion Emb.S 1 ; M / ! Imm.S 1 ; M / to the immersions induces an isomorphism on 1 for some simply connected 4-manifolds. This gives a restriction on a question posed by Arone and Szymik. The idea to construct the spectral sequence is to combine a version of Sinha's cosimplicial model for the knot space and a spectral sequence for a configuration space by Bendersky and Gitler. The cosimplicial model consists of configuration spaces of points (with a tangent vector) in M . We use Atiyah duality to transfer the structure maps on the configuration spaces to maps on Thom spectra of the quotient of a direct product of M by the fat diagonal. This transferred structure is the key to defining our spectral sequence, and is also used to show that Sinha's model can be resolved into simpler pieces in a stable category.

On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

“…For an oriented compact smooth d-manifold X with d ≥ 4 we have studied in [KT21;Kos24c] the first interesting homotopy group π d−3 Emb(S 1 , X) of the space of smooth embeddings of the circle into X, following the seminal work of Dax [Dax72] in more general settings. The groups π 1 (Emb(S 1 , X); c) for various 4-manifolds X and basepoints c : S 1 → X have been studied by Arone and Szymik [AS20], Moriya [Mor24], by Gabai in [Gab21] and together with Budney in [BG19]. In this paper we extend the results from [Kos24c] for the case d = 4 in two directions.…”