  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.
