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Decoupling Decorations on Moduli Spaces of Manifolds

Part of: Differential topology Homology and cohomology theories Fiber spaces and bundles

Published online by Cambridge University Press:  10 May 2022

LUCIANA BASUALDO BONATTO
LUCIANA BASUALDO BONATTO*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter (550), Woodstock Road, Oxford, OX2 6GG. e-mail: luciana.bonatto@maths.ox.ac.uk
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Abstract

We study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.

MSC classification

Primary: 55R40: Homology of classifying spaces, characteristic classes
Secondary: 55N99: None of the above, but in this section 55R10: Fiber bundles 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 1 , January 2023 , pp. 163 - 198
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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