Homotopy and geometric perspectives on string topology

Let M be a manifold and V a Euclidean space. We use calculus of functors to study the rational homology and homotopy of the space Emb(M, V ), which is defined to be the homotopy fiber of the map Emb(M, V ) → Imm(M, V ). If M is one-dimensional, we get, essentially, the space of knots in R. One general point that we want to make is that quite a bit of what had been done for knots can be done for general embedding spaces. Technically, our main theme is the interplay between embedding calculus, orthogonal calculus, and the rational formality of the little balls operad. The formality theorem implies that the spectral sequence for H∗(Emb(M, V );Q) that arises from embedding calculus collapses at E2. We get an even better statement when we translate this into the framework of orthogonal calculus. Namely, the orthogonal calculus Taylor tower of HQ ∧ Emb(M, V ) splits as the product of its layers. We write explicit formulas for the layers in the orthogonal towers of the functors HQ ∧ Emb(M, V ) and Emb(M, V ) as a twisted cohomology of certain spaces of trees and graphs “grafted” on to M . The formulas show, in particular, that the homotopy groups of the layers depend only on the homology of M . Combining this with the splitting of the orthogonal tower, we conclude that as long as 2 dim(M) + 1 < dim(V ), the rational homology groups of Emb(M, V ) depend only on the rational homology of M . Finally, we show that there also are collapsing results for spectral sequences computing the rational homotopy of Emb(M, V ). For example, the rational homotopy spectral sequence arising from orthogonal calculus collapses at E1. It follows that if dim(M)+2 < dim(V ) then the rational homotopy groups of Emb(M, V ) (in positive dimensions) depend only on the rational homology of M . The main ingredient of the proof here is the coformality of the little balls operad. This is a report on joint work with Pascal Lambrechts and Ismar Volić. LetM be a smooth manifold (possibly with a boundary) of dimensionm and let V denote a Euclidean space, so V ∼= RN for some N . We are interested in the space of embeddings Emb(M,V ) or rather the space Emb(M,V ) := hofiber(Emb(M,V ) → Imm(M,V )), where Imm(M,V ) denotes the space of immersions of M into V . Most of the time, we will work with the suspension spectrum Σ∞Emb(M,V ), and our results are actually about the rationalization of this spectrum. In other words, our results are about the rational homology (and sometimes rational homotopy) of Emb(M,V ). Our main tools are calculus of functors and the formality of the little balls operad. There are two versions of calculus of functors (both developed by M. Weiss) that are relevant to us: embedding calculus and orthogonal calculus. Embedding calculus [4, 1] is the more elementary of the two, as it relies on familiar constructions from category theory. Embedding calculus is designed for studying contravariant functors on manifolds, such as F (M) = Emb(M,V ). Let O∞(M) be the category of open subsets of M that are homeomorphic to a finite union of open balls. Morphisms are inclusions, and F is assumed to be a contravariant functor from O∞(M) to spaces. The category O∞(M) is filtered by an increasing sequence of full subcategories Ok(M),