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Tautological rings for high-dimensional manifolds

Part of: Fiber spaces and bundles Differential topology

Published online by Cambridge University Press:  13 March 2017

Søren Galatius ,
Ilya Grigoriev  and
Oscar Randal-Williams
Søren Galatius
Affiliation:
Department of Mathematics, Stanford University, Stanford CA, 94305, USA email galatius@stanford.edu
Ilya Grigoriev
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA email ilyagr@math.uchicago.edu
Oscar Randal-Williams
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK email o.randal-williams@dpmms.cam.ac.uk
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Abstract

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^{\ast }(M)$ of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $S^{n}\times S^{n}$ for $n$ odd.

Keywords

tautological ringcharacteristic class

MSC classification

Primary: 57R20: Characteristic classes and numbers 55R40: Homology of classifying spaces, characteristic classes 55R35: Classifying spaces of groups and $H$-spaces 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 851 - 866
Copyright
© The Authors 2017 

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