Theorem 1.4. For any m, there is a little two-cubes action on Tot(K • m ). For m > 3, E m is a two-fold loop space.We conjecture that this two-cubes action on Tot(K • m ) is compatible with a two-cubes action on the space of framed knots which has been recently defined by Budney [7], who goes on two show that long knots in dimension three are free over the two-cubes action. In future work we plan to investigate analogues of this freeness result in higher dimensions. A first step will be to construct operations compatible with this the two-cubes structure in the homology spectral sequence for an operad with multiplication, as McClure and Smith currently plan to do. On the E 1 -term such operations will presumably coincide with Tourtchine's bracket, defined combinatorially in [40], but through their space-level construction would also be compatible with differentials and extend to further terms. Some of the technical results developed in this paper may be of independent interest. We fully develop the operad structure, with multiplication, on the simplicial compactification of configurations in Euclidean spaces. An operad structure on the canonical (Axelrod-Singer) compactification is known [14,25] but does not yield an operad with multiplication. Instead there is a map from Stasheff's A ∞ operad. Our approach to the operad structure on the simplicial variant blends geometry and combinatorics, revealing an operad structure on the standard simplicial model for the two-sphere.1.1. Acknowledgements. We thank J. McClure and J. Smith for their interest in this project, answers to questions, and especially for writing Section 15 of [29]. We thank M. Markl and J. Stasheff for comments on early versions of this work, and M. Kontsevich for helpful conversations.

We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. Using global coordinates we define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities. Mathematics Subject Classification (2000). Primary 55T99.

let [M ] denote the corresponding class in M U G* . Complex projective spaces give a rich collection of examples of G-manifolds. Given a complex representation W of G let P(W ) denote the space of complex one-dimensional subspaces of W with inherited G-action.The starting point in our work is that after inverting Euler classes, M U G * becomes computable by non-equivariant means. That we rely heavily on localization is not surprising because localization techniques have pervaded equivariant topology. For any compact Lie group G let R 0 denote the sub-algebra of M U G * generated by the e V and [P(n⊕ V )] as V ranges over non-trivial irreducible representations. Let S be the multiplicative set in R 0 of non-trivial Euler classes. By abuse, denote the same multiplicative set in M U G * by S. Then the key first result, which we emphasize is true for a large class of groups including p-groups, is the following.Theorem 1.1. Let G be a group such that any proper subgroup is contained in a proper normal subgroup. The inclusion of R 0 into M U G * becomes an isomorphism after inverting S.In other words, we may multiply any class in M U G * by some Euler class to get a class in R 0 modulo the kernel of the localization map S. We are lead to study divisibility by Euler classes as well as the kernel of this localization map. We can do so successfully in the case when the group in question is a torus.Let T be a torus, and let V be a non-trivial irreducible representation of T . Let K(V ) denote the subgroup of T which acts trivially on V . There is a restriction homorphism (of algebras) res T H : M U T * → M U H * for any subgroup H. The restriction of e V to M U K(V ) * is zero, as can be seen using an explicit homotopy. Remarkably, we have the following.

We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. r

We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan [25], which places explicit geometrically meaningful formulae first dating back to Whitehead [28] in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality [20].We build on [24], which uses graph coalgebras to breathe combinatorial life into the category of differential graded Lie coalgebras. We use that framework to construct a new isomorphism of Lie coalgebras η : H * −1 (E(A * (X))) → Hom(π * (X), Q) for X simply connected. Here A * (X) denotes a model for commutative rational-valued cochains on X, and E is isomorphic to the Harrison complex. While the existence of such an isomorphism follows from Quillen's seminal work in rational homotopy theory, giving a direct, explicit isomorphism has benefits in both theory and applications.On the calculational side, we are able to evaluate Hopf invariants on iterated Whitehead products in terms of the "configuration pairing." We can use this, for example, to take the well-known calculation of the rational homotopy groups of a wedge of spheres as a free graded Lie algebra and give a geometric algorithm to determine which element of that algebra a given map would correspond to. On the formal side, we are able to understand the naturality of these maps in the long exact sequence of a fibration. For applications, we can show for example that the rational homotopy groups of homogeneous spaces are detected by classical linking numbers. Ultimately all of these Hopf invariants are essentially generalized linking invariants, as we explain in Section 1.2.We proceed in two steps, first using the classical bar complex to define integer-valued homotopy functionals which coincide with evaluation of the cohomology of ΩX on the looping of a map from S n to X. This was also the starting point of Hain's work [12] using Chen integrals, but our definition of functionals is clearly distinct from his. We establish basic properties and give examples using the classical bar complex. In the second part, we use the Harrison complex on commutative cochains, and thus must switch to rational coefficients. Using our graph coalgebraic presentation, we show that a product-coproduct formula established geometrically in the bar complex descends to the duality predicted by Koszul-Moore theory.Our basic, apparently new, observation is that calculations in bar complexes yield the Hopf invariant formula of Whitehead [28], as well as those of Haefliger, Novikov and Sullivan. This observation could have been made fifty years ago. Our approach incorporates a modern viewpoint by directly using HarrisonAndré-Quillen homology, the standard algebraic bridge from commutative algebras to Lie coalgebras, with the new graphical presentation essential for a self-contained development. One direction we plan to pursue further is the use of Hopf invariants to realize Koszul-Moore duality isomorphisms in general. A second direction we plan to pursue is that of spaces which are not simply connected, where...

We show that the map on components from the space of classical long knots to the nth stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group, and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n − 1) knot invariant. We compute the E 2 -page in total degree zero for the spectral sequence converging to the components of this tower, identifying it as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.

We give the first explicit computations of rational homotopy groups of spaces of "long knots" in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E 1 term is defined in terms of braid Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E 1 term is zero, and make calculations of E 2 in a finite range.