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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.
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.
||The homology of the little disks operad.
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.
||Manifold-theoretic compactifications of configuration spaces.
Selecta Mathematica (new series) Vol 10, No 3 (2004) 391-428.
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.
||A pairing between graphs and trees.