Homological stability for moduli spaces of high dimensional manifolds. I

Abstract

We prove a homological stability theorem for moduli spaces of simply-connected manifolds of dimension $2n>4$, with respect to forming connected sum with $S^n\times S^n$. This is analogous to Harer's stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of $S^n\times S^n$ in a range of degrees.

43 pages, 2 figures; this article supersedes arXiv:1203.6830. v2 is a major revision, proving a stronger statement and combining better with arXiv:1601.00232. v3 has minor revisions and an added remark 7.16, submitted version. v4 reverts an unintended change of title (metadata only). v5 is final accepted version

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• Homological stability for moduli spaces of high dimensional manifolds

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