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Spaces of distinct points in a manifold play many, often central, roles in topology. I can name at least three distinct ways in which they arise in my work on knot spaces: compactified configuration spaces are the building blocks for the models I develop; an operad equivalent to the two-cubes operad (whose entries are homotopy equivalent to Euclidean configuration spaces) acts on spaces of Euclidean knots; the cohomology of configuration spaces gives a model for the linear dual of the Lie operad, which governs the behavior of generalized Hopf invariants.

I have become enamored with configuration spaces for their own sake, and my student Matt Miller has been investigating whether they could give tractable homeomorphism invariants, after a result of Longoni and Salvatore which shows they can be sensitive to homeomorphism versus homotopy equivalence.

PS PDF The homology of the little disks operad.
This paper is mostly expository, explaining in terms which are hopefully suitable for beginning graduate students why the homology of the little disks operad is the Poisson operad, a result which is fundamental to my work and that of many others. In the process, we give concrete models for homology and cohomology classes in Euclidean configuration spaces. New results include identifying the homology-cohomology pairing for these spaces, and the cooperad structure dual to the Poisson operad.

PS PDF Manifold-theoretic compactifications of configuration spaces.
Selecta Mathematica (new series) Vol 10, No 3 (2004) 391-428.
My work on knot spaces relies heavily on completions of configuration spaces due originally to Fulton-MacPherson, Axelrod-Singer and Kontsevich. There are now a number of treatments of these completions, in particular by Markl and by Gaiffi in the real setting. For my applications I have needed to use maps between these spaces and show that some of them commute, are homotopy equivalences in some cases, and so forth. This led me to a new, explicit approach to these completions, giving characterizations as subspaces of simple ambient spaces, which is developed in this paper.

PS PDF A pairing between graphs and trees.
The pairing between canonical bases for homology and cohomology of configuration spaces has a pleasant interpretation in terms of graphs and trees. In this paper I give a systematic combinatorial development of this pairing and use it for example to reprove known structure theorems about the Lie and Poisson operads, as well as to give new (to my knowledge) co-operad and other structures, as well as to extend some results to free Lie algebras.