Models for knot spaces and Atiyah duality

Abstract

Let Emb(S 1 , M ) be the space of smooth embeddings from the circle to a closed manifold M of dimension ≥ 4. We study a cosimplicial model of 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 in Theorem 1.1. As an application, we introduce a new spectral sequence converging to H * (Emb(S 1 , M )) for simply connected M and for major coefficient rings as in Theorem 1.2. Using this, we compute H * (Emb(S 1 , S k × S l )) in low degrees with some conditions on k, l. We also prove the inclusion Emb(S 1 , M ) → Imm(S 1 , M ) to the immersions induces an isomorphism on π1 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 Emb(S 1 , M ) and a homotopy colimit of chain complexes of a Thom spectrum of a bundle over a comprehensible space as in Theorem 1.4. Our key ingredient is a structured version of the duality due to R.

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

“…This answers in negative a prediction of Arone and Szymik [AS20] that for a simply connected 4-manifold the inclusion of embedded circles into immersed ones can fail to be injective on π 1 . Injectivity was shown for a certain class of 4-manifolds by [Mor20].…”