Title: Calculus of functors, operad formality, and rational homology of
embedding spaces
Authors:
Gregory Arone,
Department of Mathematics, University of Virginia, Charlottesville, VA, USA.
zga2m@virginia.edu
Pascal Lambrechts
Institut Math\'{e}matique, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
lambrechts@math.ucl.ac.be
Ismar Voli\'c
Department of Mathematics, University of Virginia, Charlottesville, VA, USA
ismar@virginia.edu
Abstract:
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) homotopy 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 homotopy type of M.