A survey of the homology cobordism group

Şavk, Oğuz

doi:10.1090/bull/1806

Cited by 3 publications

(3 citation statements)

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“…Since there are several examples of homology planes that are known to be non‐contractible [2, 38], we curiously ask the following question. Compare with [60, Problem G]. Question Does a Kirby–Ramanujam sphere bound a non‐contractible homology plane?…”

Section: Further Directionsmentioning

confidence: 99%

“…Therefore, they also provide examples of Kirby–Ramanujam spheres. Since our all Kirby–Ramanujam spheres bound contractible 4‐manifolds, they are homology cobordant to the 3‐sphere

S^3

; and therefore, they represent the trivial element in the homology cobordism group

\Theta ^3_\mathbb {Z}

, see survey articles [35, 60].…”

Section: Introductionmentioning

confidence: 99%

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On homology planes and contractible 4‐manifolds

Aguilar,

Şavk

2024

Bulletin of London Math Soc

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We call a non‐trivial homology sphere a Kirby–Ramanujam sphere if it bounds both a homology plane and a Mazur or Poénaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur manifold and it has remained a single example since then. By tracing their initial step, we provide the first additional examples and we present three infinite families of Kirby–Ramanujam spheres. Also, we show that one of our families of Kirby–Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Since this family of Kirby–Ramanujam spheres bound contractible 4‐manifolds, they lie in the class of the trivial element in the homology cobordism group; however, both splice components are separately linearly independent in that group.

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Section: Further Directionsmentioning

confidence: 99%

“…Therefore, they also provide examples of Kirby–Ramanujam spheres. Since our all Kirby–Ramanujam spheres bound contractible 4‐manifolds, they are homology cobordant to the 3‐sphere

S^3

; and therefore, they represent the trivial element in the homology cobordism group

\Theta ^3_\mathbb {Z}

, see survey articles [35, 60].…”

Section: Introductionmentioning

confidence: 99%

On homology planes and contractible 4‐manifolds

Aguilar,

Şavk

2024

Bulletin of London Math Soc

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“…The

r=3

case of the above result was proven by Anvari–Hambleton [2], under the assumption that the bounding manifold is contractible. We note that there is a conjecture that Brieskorn spheres with

r &gt; 3

cannot bound contractible manifolds (see, for example, [40, Problem I]). On the other hand, there are many examples of Brieskorn spheres which bound rational homology balls, but not integer homology balls [1, 14, 39].…”

Section: Introductionmentioning

confidence: 99%

Brieskorn spheres, cyclic group actions and the Milnor conjecture

Baraglia,

Hekmati

2024

Journal of Topology

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In this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants defined by the first author satisfy for torus knots, whenever is a prime not dividing . Since is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere does not extend smoothly to any homology 4‐ball bounding . In the case of a non‐free cyclic group action of prime order, we prove that if the rank of is greater than times the rank of , then the ‐action on does not extend smoothly to any homology 4‐ball bounding . Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres.

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