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Rational cobordisms and integral homology

Part of: Low-dimensional topology Topological manifolds

Published online by Cambridge University Press:  29 October 2020

Paolo Aceto ,
Daniele Celoria  and
JungHwan Park
Paolo Aceto
Affiliation:
Mathematical Institute, University of Oxford, Oxford, UKpaoloaceto@gmail.com
Daniele Celoria
Affiliation:
Mathematical Institute, University of Oxford, Oxford, UKdaniele.celoria@maths.ox.ac.uk
JungHwan Park
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USAjunghwan.park@math.gatech.edu
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Abstract

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

Keywords

lens spacesthree-dimensional homology cobordism groups

MSC classification

Primary: 57N13: Topology of $E^4$, $4$-manifolds 57M27: Invariants of knots and 3-manifolds 57N70: Cobordism and concordance 57M25: Knots and links in $S^3$
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 9 , September 2020 , pp. 1825 - 1845
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Copyright
Copyright © The Author(s) 2020

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