We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of R m into R n . We view the space of embeddings as the value of a certain functor at R m , and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when 2m + 1 < n, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.is a weak homotopy equivalence if m + 3 ≤ n. Combining this with Theorem 0.1, we conclude that the following natural map is an equivalence when m + 3 ≤ nThe case m = 1 of Theorem 0.1 is related to Sinha's cosimplicial model for the space of long knots [23]. Indeed, the homotopy totalization of Sinha's cosimplicial space can be

We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on different ways how the calculations can be done. In particular we describe three different graph-complexes computing these rational homotopy groups. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

We construct a homological approximation to the partition complex, and identify it as the Tits building. This gives a homological relationship between the symmetric group and the affine group, leads to a geometric tie between symmetric powers of spheres and the Steinberg idempotent, and allows us to use the self‐duality of the Steinberg module to study layers in the Goodwillie tower of the identity functor. 2000 Mathematics Subject Classification: 55N25, 55S15, 20B30, 55P25.

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 ).

Abstract. Given a special Γ-category C satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to C and the Eilenberg-Mac Lane spectrum H Z. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of H Z by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, {Am}, that interpolate between bu and H Z. We think of Am as a "bu-analogue" of Sp m (S) and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the mth subquotient is contractible unless m is a power of a prime, and in v k -periodic homotopy the filtration has only k + 2 nontrivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weiss's orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {Am}, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of bu. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture.