Calculus of functors, operad formality, and rational homology of embedding spaces

Abstract

Let M be a smooth manifold and V a Euclidean space. Let \( \overline{{{\text{Emb}}}} \)(M,V) be the homotopy fiber of the map Emb(M,V) → Imm(M,V). This paper is about the rational homology of \( \overline{{{\text{Emb}}}} \)(M,V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M,V)↦ HQ ∧ \( \overline{{{\text{Emb}}}} \)(M,V)+. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$ (where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$ The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of \( \overline{{{\text{Emb}}}} \)(–,V).