Ismar Volic

Formality of the little N-disks operad

Introduction Notation, linear orders, weak partitions, and operads CDGA models for operads Real homotopy theory of semi-algebraic sets The Fulton-MacPherson operad The CDGAs of admissible diagrams Cooperad structure on the spaces of (admissible) diagrams Equivalence of the cooperads D and H * (C[ * ]) The Kontsevich configuration space integrals Proofs of the formality theorems Index of notation Bibliography

Finite type knot invariants and the calculus of functors

We associate a Taylor tower supplied by the calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral sequence collapses along this line and that the Taylor tower represents a universal finite type knot invariant.

The rational homology of spaces of long knots in codimension > 2

We determine the rational homology of the space of long knots in R for d 4 . Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree d 1 , which can be obtained as the homology of an explicit graph complex and is in theory completely computable.

A SURVEY OF BOTT–TAUBES INTEGRATION

It is well-known that certain combinations of configuration space integrals defined by Bott and Taubes [11] produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.

Multivariable manifold calculus of functors

Abstract. Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce “multivariable” manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. We construct multivariable Taylor approximations to such functors, classify multivariable homogeneous functors, apply this classification to compute the derivatives of a functor, and show what this gives for the space of link maps. We also relate Taylor approximations in single variable calculus to our multivariable ones.

CONFIGURATION SPACE INTEGRALS AND THE COHOMOLOGY OF THE SPACE OF HOMOTOPY STRING LINKS

Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in

On volume-preserving vector fields and finite-type invariants of knots

\mathbb{R}^n

Configuration Space Integrals: Bridging Physics, Geometry, and Topology of Knots and Links (Commentary on [106], [108], [109])

for

Real homotopy theory of semi-algebraic sets

n>3

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

since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of