Rational cobordisms and integral homology

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.

arXiv comment19 pages, final version to appear in Compositio Mathematica

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Cited by in corpus (5)

• More Brieskorn spheres bounding rational balls
• A survey of the homology cobordism group
• Ribbon cobordisms between lens spaces
• The 0-concordance monoid admits an infinite linearly independent set
• Embedding lens spaces in definite 4-manifolds