More Brieskorn spheres bounding rational balls

Oğuz, Şavk,

doi:10.1016/j.topol.2020.107400

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…For example, Fintushel and Stern showed that †.2; 3; 7/ bounds a rational homology 4-ball, although it does not bound an integral homology 4-ball [26]. More examples can be found in Akbulut and Larson [4] and Şavk [21]. We show in Section 7.2 that the nonvanishing of ı Z p ;1 .…”

Section: Applicationsmentioning

confidence: 75%

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Equivariant Seiberg–Witten–Floer cohomology

Baraglia,

Hekmati

2024

Algebr. Geom. Topol.

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We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology 3-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of d -invariants d G;c .Y; s/ which are indexed by the group cohomology of G. These invariants satisfy a Frøyshov-type inequality under equivariant cobordisms. Lastly, we consider a variety of applications of these d -invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding 4-manifolds, Nielsen realisation problems for 4-manifolds with boundary and obstructions to equivariant embeddings of 3-manifolds in 4-manifolds.

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Section: Applicationsmentioning

confidence: 75%

“…2; 4n C 1; 12n C 5/ and †.3; 3nC1; 12nC5/ bound rational 4-balls but not integral 4-balls. More examples, †.2; 4nC3; 12nC7/ and †.3; 3n C 2; 12n C 7/ for even n, were constructed by Şavk[21]. Taking p D 2 or 3, the above Brieskorn spheres admit Z p -actions with nonzero delta invariants, as in Example 7.7.…”

mentioning

confidence: 99%

Equivariant Seiberg–Witten–Floer cohomology

Baraglia,

Hekmati

2024

Algebr. Geom. Topol.

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“…It was found in [6] and played a key role in the proof of their main theorem. Later, it was also effectively used by the second author [58,59].…”

Section: Mazur and Poénaru Manifoldsmentioning

confidence: 99%

“…On the other hand, there are rational homology planes that arise from complex normal surface singularities, see Wahl's paper [67]. Brieskorn spheres may bound Mazur and Poénaru manifolds, see [59] and references therein. However, one cannot realize any Brieskorn sphere as a boundary of a homology plane due to Orevkov [46].…”

Section: Introductionmentioning

confidence: 99%

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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“…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]. Thus Corollary 1.7 is a non‐trivial result.…”

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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More Brieskorn spheres bounding rational balls

Cited by 4 publications

References 16 publications

Equivariant Seiberg–Witten–Floer cohomology

Equivariant Seiberg–Witten–Floer cohomology

On homology planes and contractible 4‐manifolds

Brieskorn spheres, cyclic group actions and the Milnor conjecture

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