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

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

2006-07-20

arxiv.org/abs/math/0607486v2

Mathematics

Algebraic Topology

35 pages. An erroneous definition in the last section was corrected, as well as several misprints. The introduction was somewh

Scientific paper

Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice 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 E^1. 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 /\Ebar(M,V)_+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homology type of M. This, together with our rational splitting theorem, implies that under the above assumption on codimension, the rational homology groups of Ebar(M,V) are determined by the rational homology type of M.

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