Homological stability for configuration spaces of manifolds
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- Church, T. Invent. math. (2012) 188: 465. doi:10.1007/s00222-011-0353-4
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Let Cn(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 Hi(Cn(M);ℚ) are representation stable in the sense of Church and Farb (arXiv:1008.1368). Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space Bn(M) satisfies classical homological stability: for each i, Hi(Bn(M);ℚ)≈Hi(Bn+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 Sn-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.