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SØREN GALATIUS (Stanford, Copenhagen): Homological stability for moduli spaces of manifolds (12 June 2012)


The connected sum of g copies of S^n x S^n can be viewed as a higher-dimensional analogue of the orientable 2-manifold of genus g. Denoting this manifold by W_g, we consider the space BDiff(W_g,D^{2n}), classifying smooth fiber bundles with fiber W_g. In recent joint work with Oscar Randal-Williams, we prove homological stability for these spaces (for n > 2): The k'th integral homology group is independent of g, as long as k < (g-3)/2. For n=1, the statement is equivalent to Harer's homological stability for mapping class groups of surfaces, so our theorem can be viewed as a higher-dimensional version of Harer's.

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