29 research outputs found
Calculus of functors, operad formality, and rational homology of embedding spaces
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.Comment: 35 pages. An erroneous definition in the last section was corrected,
as well as several misprints. The introduction was somewhat reworked. The
paper was accepted for publication in Acta Mathematic
Embedding obstructions in from the Goodwillie-Weiss calculus and Whitney disks
Given an -dimensional CW complex , we use a version of the
Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into a
Euclidean space . For -complexes in a
geometric analogue is also introduced, based on intersections of Whitney disks
and more generally on the intersection theory of Whitney towers developed by
Schneiderman and Teichner. The focus in this paper is on the first obstruction
beyond the classical embedding obstruction of van Kampen. In this case we show
the two approaches give the same result, and also relate it to the Arnold class
in the cohomology of configuration spaces. The obstructions are shown to be
realized in a family of examples. Conjectures are formulated, relating higher
versions of these homotopy-theoretic, geometric and cohomological theories.Comment: 44 page
Bredon Homology of Partition Complexes
We prove that the Bredon homology or cohomology of the partition complex with
fairly general coefficients is either trivial or computable in terms of
constructions with the Steinberg module. The argument involves developing a
theory of Bredon homology and cohomology approximation.Comment: 48 pages. Minor revisions. A typo in the statement of Corollary 1.2
was corrected, along with other typos. Some references have been adde