Rational cobordisms and integral homology

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We consider the question of when a rational homology -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 -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.
Publisher
London Mathematical Society
Issue Date
2020-09
Language
English
Article Type
Article
Citation

Compositio Mathematica, v.156, no.9, pp.1825 - 1845

ISSN
0010-437X
DOI
10.1112/S0010437X20007320
URI
http://hdl.handle.net/10203/280091
Appears in Collection
MA-Journal Papers(저널논문)
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