Gluck surgery along a 2-sphere in a 4-manifold is realized by surgery along a projective plane.

Katanaga, Atsuko; Saeki, Osamu; Teragaito, Masakazu; Yamada, Yuichi

doi:10.1307/mmj/1030132479

Cited by 9 publications

(7 citation statements)

References 12 publications

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“…RP 2 ) that is diffeomorphic to the normal disk bundle (cf. [KSTY99]). The condition of admitting a properly transnormal function provides Σ 4 with a further restriction during its construction through these surgeries, that is the exterior of the embedding (i.e., S 4 −N (S 2 ) or respectively S 4 − N (RP 2 )) is also a tubular neighborhood of an embedding of S 2 (resp.…”

Section: Furthermentioning

confidence: 99%

Isoparametric functions and exotic spheres

Tang

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Dedicated to Professor Banghe Li on his 70th birthday.Abstract. The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the GromollMeyer sphere. In particular, as one of main results we prove there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an S 3 -invariant isoparametric function on Sp(2), we construct a properly transnormalbut not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.

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Section: Furthermentioning

confidence: 99%

Isoparametric functions and exotic spheres

Tang

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In this paper, we use trisection diagrams to give an entirely new proof of the following theorem that relates these surgeries, proved by Katanaga, Saeki, Teragaito, and Yamada [11]. This is made possible by recent work on trisection diagrams of complements of surfaces in 4-manifolds; the existence of a purely trisection-diagrammatic proof of this theorem answers a question of Kim and Miller [12].…”

Section: Introductionmentioning

confidence: 93%

Trisection diagrams and twists of 4-manifolds

Naylor¹

2022

Comptes Rendus. Mathématique

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A theorem of Katanaga, Saeki, Teragaito, and Yamada relates Gluck and Price twists of 4-manifolds. Using trisection diagrams, we give a purely diagrammatic proof of this theorem, and answer a question of Kim and Miller.Résumé. Un théorème de Katanaga, Saeki, Teragaito, et Yamada établit une connexion entre des torsions de Gluck et Price. On donne une nouvelle démonstration de ce théorème en utilisant des diagrammes de trisection, et on répond à une question de Kim et Miller.

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“…Let Σ G K (X) be the 4manifold obtained by the Gluck twist along K, where K is a 2-knot in X. Then, from [KSTY99], we see that for a P 2 -knot S = K#P ± , Σ S (X) ∼ = Σ G K (X) holds, where P ± is an unknotted P 2 -knot with normal Euler number ±2 in X. So, for a 2-knot K satisfying Σ G K (S 4 ) ∼ = S 4 such as a twist spun 2-knot, we have Σ S (S 4 ) ∼ = S 4 .…”

Section: The Price Twistmentioning

confidence: 99%

Trisections obtained by trivially regluing surface-knots

Isoshima¹

2022

Preprint

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Let S be a P 2 -knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted P 2 -knot with normal Euler number ±2 in a closed 4-manifold X with trisection T . Then, we show that the trisection of X obtained by the trivial gluing relative trisections of ν(S) and X − ν(S) is isotopic to a stabilization of T . It should be noted that this result is not obvious since we need boundary-stabilizations introduced by Kim and Miller to construct a relative trisection of X − ν(S). As a corollary, we prove the conjecture that is a 4-dimensional analogue of Waldhausen's theorem on Heegaard splittings for the case of the trisection.

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Gluck surgery along a 2-sphere in a 4-manifold is realized by surgery along a projective plane.

Cited by 9 publications

References 12 publications

Isoparametric functions and exotic spheres

Isoparametric functions and exotic spheres

Trisection diagrams and twists of 4-manifolds

Trisections obtained by trivially regluing surface-knots

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