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

Abstract

Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V ) be the homotopy fiber of the map Emb(M, V ) −→ Imm(M, V ). This paper is about the rational homology of Emb(M, V ). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M, V ) → HQ ∧ Emb(M, V )+. Our main theorem states that if dim V ≥ 2 ED(M ) + 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 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 functorThe 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 Emb(−, V ).

Hodge-type decomposition in the homology of long knots

“…One should mention that besides the long knots (Theorem 2.1) this rational homology collapse result holds also for the spaces of embeddings modulo immersions Emb(M, R d ) of any compact manifold into an affine space of a sufficiently high dimension d [2].…”

“…Proposition 9.2. The assignment (2) are the product and the bracket of the operad A d of (d − 1)-Poisson algebras, defines an inclusion of operads…”

Multivariable manifold calculus of functors

“…Although statements and proofs here are usually combinatorially more complex, many definitions and techniques used in [26] carry over nicely to our multivariable setting. Manifold calculus has had many applications in the past decade [1,14,17,16,22,21,24]. With an eye toward extending some of them, we wish to generalize this theory to the setting where M breaks up as a disjoint union of manifolds, say M D`m i D1 P i .…”

Coformality and rational homotopy groups of spaces of long knots

“…• are also well understood [18,20] so it is not hard to write explicit formulas for the differential d 1 . Some low degree calculations in the E 2 page were carried out in [18].…”