Models for knot spaces and Atiyah duality

Let Emb.S 1 ; 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.18M75, 55P43, 55T99, 57R40; 18N40 1. Introduction 183 2. Preliminaries 189 3. The comodule T M 201 4. Atiyah duality for comodules 209 5. Spectral sequences 214 6. Algebraic presentations of the E 2 -page of the Čech spectral sequence 225 7. Examples 236 8. Precise statement and proof of Theorem 1.5