Rational cobordisms and integral homology

Aceto, Paolo; Celoria, Daniele; Park, JungHwan

doi:10.1112/S0010437X20007320

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Rational cobordisms and integral homology

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Aceto, Paolo
Max Planck Institute for Mathematics, Max Planck Society;

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Park, JungHwan
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1112/S0010437X20007320
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arXiv:1811.01433.pdf
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Citation

Aceto, P., Celoria, D., & Park, J. (2020). Rational cobordisms and integral homology. Compositio Mathematica, 156(9), 1825-1845. doi:10.1112/S0010437X20007320.

Cite as: https://hdl.handle.net/21.11116/0000-0007-8252-9

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
embeds in 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.