A Compactification of the Real Configuration Space as an Operadic Completion

TL;DR

This paper demonstrates that the compactification of configuration spaces in Riemannian manifolds can be understood as an operadic completion, extending previous work on configuration spaces in the plane.

Contribution

It shows that the non-compactified configuration spaces form a partial operad and that their compactification is an operadic completion, generalizing prior results to arbitrary Riemannian manifolds.

Findings

Configuration spaces form a partial operad.

Compactification is described as an operadic completion.

Spectral sequence identified with bar resolution of an operadic module.

Abstract

S. Axelrod and I.M. Singer constructed a compactification of the configuration space of distinct points in a Riemannian manifold V. A similar compactification for the moduli space of configurations of distinct points in the plane (mod the affine group action) was considered by E. Getzler and J.D.S. Jones. They observed that this compactification carries a natural structure of an operad. In the present note we show that (non-compactified) configuration spaces form a partial operad (or a partial module over a partial operad) and that the compactification can be described as an operadic (or modular) completion. This approach immediately gives the operad (or module) structure on the compactification. We also discuss the spectral sequence of the stratification and identify the second term of this spectral sequence to the bar resolution of an operadic module. Our results generalize the work of Getzler, Kimura, Jones, Stasheff, Voronov and others to the case of configurations in a general Riemannian manifold.