  Let C_n(M) be the configuration space of n distinct ordered points in M. We
prove that if M is any connected orientable manifold (closed or open), the
homology groups H_i(C_n(M); Q) are representation stable in the sense of
[Church-Farb]. Applying this to the trivial representation, we obtain as a
corollary that the unordered configuration space B_n(M) satisfies classical
homological stability: for each i, H_i(B_n(M); Q) is isomorphic to
H_i(B_{n+1}(M); Q) for n > i. This improves on results of McDuff, Segal, and
others for open manifolds. Applied to closed manifolds, this provides natural
examples where rational homological stability holds even though integral
homological stability fails.
  To prove the main theorem, we introduce the notion of monotonicity for a
sequence of S_n--representations, which is of independent interest.
Monotonicity provides a new mechanism for proving representation stability
using spectral sequences. The key technical point in the main theorem is that
certain sequences of induced representations are monotone.

Algebraic Topology;Geometric Topology;Representation Theory