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2014 Stable moduli spaces of high-dimensional manifolds
Søren Galatius, Oscar Randal-Williams
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Acta Math. 212(2): 257-377 (2014). DOI: 10.1007/s11511-014-0112-7

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 Sn×Sn, in the limit g. 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 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 N(P) after stabilisation using certain self-bordisms of P. The stable homology is identified with that of an infinite loop space.

Funding Statement

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.

Note

Dedicated to Ib Madsen on the occasion of his 70th birthday.

Citation

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Søren Galatius. Oscar Randal-Williams. "Stable moduli spaces of high-dimensional manifolds." Acta Math. 212 (2) 257 - 377, 2014. https://doi.org/10.1007/s11511-014-0112-7

Information

Received: 19 September 2012; Revised: 12 November 2013; Published: 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1377.55012
MathSciNet: MR3207759
Digital Object Identifier: 10.1007/s11511-014-0112-7

Subjects:
Primary: 57S05
Secondary: 55P47 , 57R15 , 57R20 , 57R50 , 57R65 , 57R90

Rights: 2014 © Institut Mittag-Leffler

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Vol.212 • No. 2 • 2014
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