Abstract
We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S n×S n, in the limit \({g \to \infty}\). Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.
More generally, we study a moduli space \({\mathcal{N}(P)}\) of those null-bordisms of a fixed (2n–1)-dimensional manifold P which are (n–1)-connected relative to P. We determine the homology of \({\mathcal{N}(P)}\) after stabilisation using certain self-bordisms of P. The stable homology is identified with that of an infinite loop space.
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Galatius was partially supported by NSF grants DMS-0805843 and DMS-1105058 and the Clay Mathematics Institute. Randal-Williams was supported by the Herchel Smith Fund. Both authors were supported by ERC Advanced Grant No. 228082, and the Danish National Research Foundation through the Centre for Symmetry and Deformation.
Dedicated to Ib Madsen on the occasion of his 70th birthday.
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Galatius, S., Randal-Williams, O. Stable moduli spaces of high-dimensional manifolds. Acta Math 212, 257–377 (2014). https://doi.org/10.1007/s11511-014-0112-7
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DOI: https://doi.org/10.1007/s11511-014-0112-7