Selecta Mathematica

, 10:391

Manifold-theoretic compactifications of configuration spaces

Original Paper

DOI: 10.1007/s00029-004-0381-7

Cite this article as:
Sinha, D.P. Sel. math., New ser. (2004) 10: 391. doi:10.1007/s00029-004-0381-7


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 difieomorphism 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 

Key words.

Configuration spaces compactifications diagonal maps 

Copyright information

© Birkhäuser-Verlag, Basel 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonUSA

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