Homological stability for configuration spaces of manifolds

• 成果类型: Article
• 署名作者: Church, Thomas
• 署名单位: University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0353-4
• 发表日期: 2012
• 页码: 465-504
• 关键词: cohomology

摘要

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);ae) 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 B (n) (M) satisfies classical homological stability: for each i, H (i) (B (n) (M); Q) approximate 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.