Linear Independence in the Rational Homology Cobordism Group

Golla, Marco; Larson, Kyle P.

doi:10.1017/s1474748019000434

Cited by 3 publications

(1 citation statement)

References 14 publications

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“…It is an interesting problem to understand the properties of these maps. It was first shown by Kim and Livingston [KL14] following the work of Hedden et al [HLR12] that Coker ψ contains a subgroup isomorphic to Z ∞ ⊕ Z ∞ 2 (see also [HKL16,GL19]). In fact, examples from [HLR12, KL14,HKL16] bound topological Q-homology balls.…”

Section: Introductionmentioning

confidence: 99%

Rational cobordisms and integral homology

Aceto¹,

Celoria²,

Park³

2020

Compositio Math.

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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.

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Section: Introductionmentioning

confidence: 99%

Rational cobordisms and integral homology

Aceto¹,

Celoria²,

Park³

2020

Compositio Math.

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More Brieskorn spheres bounding rational balls

Oğuz

2020

Topology and its Applications

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A survey of the homology cobordism group

Şavk

2023

Bull. Amer. Math. Soc.

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In this survey, we present the most recent highlights from the study of the homology cobordism group, with particular emphasis on its long-standing and rich history in the context of smooth manifolds. Further, we list various results on its algebraic structure and discuss its crucial role in the development of low-dimensional topology. Also, we share a series of open problems about the behavior of homology 3 3 -spheres and the structure of Θ Z 3 \Theta _{\mathbb {Z}}^3 . Finally, we briefly discuss the knot concordance group C \mathcal {C} and the rational homology cobordism group Θ Q 3 \Theta _{\mathbb {Q}}^3 , focusing on their algebraic structures, relating them to Θ Z 3 \Theta _{\mathbb {Z}}^3 , and highlighting several open problems. The appendix is a compilation of several constructions and presentations of homology 3 3 -spheres introduced by Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.

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Linear Independence in the Rational Homology Cobordism Group

Cited by 3 publications

References 14 publications

Rational cobordisms and integral homology

Rational cobordisms and integral homology

More Brieskorn spheres bounding rational balls

A survey of the homology cobordism group

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