Differential Topology 19

Knotted surfaces in 4-manifolds and their surgeries

This work shop was over.
The organizers thank to the speakers and people coming to the workshop.(Group photo)

Japanese version

Place : Ritsumeikan University(Tokyo Campus) (Tower 8F 4, 5)
Address : 1-7-12 Marunouchi Chiyoda-ku Tokyo 100-0005 Japan
Date: 2019/3/11-13

Subject: Ribbon surface, Cork, Gluck twist, 4-dimensional Light bulb theorem,

Registration form

Speakers

• Akito Kawauchi(Osaka City University)
• Kouki Sato(The University of Tokyo)
• Hironobu Naoe(Tohoku University)
• Mizuki Fukuda(Tohoku University)
• Mark Hughes (Brigham Young University)
• Kouichi Yasui(Osaka University)
• Yuichi Yamada(The University of Electro-Communications)
• Tetsuya Abe(Ritumeikan University)
• Motoo Tange(University of Tsukuba)

Schedule

Title and Abstract

• 3/11(Mon.)
• Motoo Tange(14 : 20-15 : 20)
  4-dimentional light bulb theorem by Gabai I
  Abstract
• Yuichi Yamada(15 : 40-16 : 40)
  4-dimentional light bulb theorem by Gabai II
  Abstract
• 3/12(Tue.)
• Hironobu Naoe(10 : 00-11 : 00)
  2-knots in shadows of 4-manifolds
  Abstract
• Mizuki Fukuda(11 : 20-12 : 20)
  Branched twist spins does not change by Gluck twists
  Abstract
• Akio Kawauchi(14 : 20-15 : 20)
  Ribbon surface-knot and stable-ribbon surface-knot
  Abstract
• Kouki Sato(15 : 40-16 : 40)
  Unit surfaces in CP² and Gluck twists
  Abstract
• Mark Hughes(17 : 00-18 : 00)
  Describing surfaces and isotopies in 4-manifolds via banded unlinks
  Abstract

Banquet(Rakuzo Utage)


• 3/13(Wed)
• Kouichi Yasui(10 : 00-11 : 00)
  Minimal genus functions and smooth structures of 4-manifolds
  Abstract
• Tetsuya Abe(11 : 20-12 : 20)
  Annulus twists via 3-dimensional light-bulb technique
  Abstract

Title and Abstract

• Tetsuya Abe Annulus twists via 3-dimensional light-bulb technique
  We reinterpret annulus twists using 3-dimensional light-bulb technique.
  
  
• Mizuki Fukuda: Branched twist spins does not change by Gluck twists
  A branched twist spin is a 2-knot, that is an embedded 2-sphere in the 4-sphere, and it is characterized by a 1-knot and two coprime integers. For a 2-knot, the dual is obtained from the 2-knot by the Gluck twist. It is known by Pao that the 4-sphere does not change by the Gluck twist along any branched twist spin. However, the knot type of the dual of the branched twist spin was not known. In this talk, we show the dual of the branched twist spin is also a branched twist spin and determine the knot type of the dual.
  
  
• Mark Hughes: Describing surfaces and isotopies in 4-manifolds via banded unlinks
  There are a number of well-establigd ways to represent knotted surfaces and isotopies between them in S⁴, including motion pictures with movie moves, or broken surface diagrams with Roseman moves. In this talk I will discuss another method of representing surfaces in 4-space via banded unlink diagrams, which can also be used to describe surfaces in an arbitrary oriented 4-manifold X. I will present a set of moves which are sufficient to relate any two banded unlink presentations of isotopic surfaces in X, which generalizes a theorem in S⁴ due to Swenton. As an application of this theorem we prove that bridge trisections of surfaces in 4-manifolds are unique up to perturbations. This is joint work with Seungwon Kim and Maggie Miller.
  
  
• Akio Kawauchi: Ribbon surface-knot and stable-ribbon surface-knot
  After reporting speaker's recent results on a ribbon surface-knot, a meaning of a stable-ribbon surface-knot is explained.
  
  
• Hironobu Naoe: 2-knots in shadows of 4-manifolds
  A shadow is a description of a 4-manifold by using a 2-dimensional simple polyhedron. We apply this notion to describing knotted surfaces in 4-manifolds. We also talk about Gluck twists on shadows and some examples. This is partially joint work with Mizuki Fukuda.
  
  
• Kouki Sato: Unit surfaces in CP² and Gluck twists
  A genus g unit surface is a closed oriented genus g surface in CP² which intersects CP¹ at exactly a single point. In their preprint, Hughes, Kim and Miller claimed that all genus g unit surfaces are isotopic. Although they discovered a gap in their proof of the claim later, if the claim is true, then we can prove that any 4-manifold obtained by Gluck twist on S⁴ is diffeomorphic to S⁴.In this talk, we review the idea of their proof and the gap discovered later.
  
  
• Motoo Tange: 4-dimensional Light bulb theorem by Gabai I
  It is well-known that any knot in S²×S¹ with a single transverse point in S²× {p} is isotopic to {p}×S¹. This is called the light bulb theorem. Gabai proved that in 4-dimension similar statement: if any 2-sphere S in S²×S² homologous to {p}×S² and has a transverse sphere S²×[q} which meets one point with S then S is isotopic to {p}×S². In this talk, we give a overview of this theorem.
  
  
• Yuich Yamada: 4-dimensional Light bulb theorem by Gabai II
  We give an outline of Gabai's proof of 4-dimensional light bulb theorem. The method “Tubed surface”. Starting with Small's theorem (homotopy implies regular homotopy). to avoid finger moves and Whitney moves, the original surface is connected to parallel spheres of the transverse sphere by tubes. A tubed surface is a kind of a surface with information (curves) of the tubes.
  [Reference]
  David Gabai, "The 4-Dimensional Light Bulb Theorem", Arxiv GT1705.09989.
  
  
• Kouichi Yasui: Minimal genus functions and smooth structures of 4-manifolds
  The minimal genus function of a 4-manifold is a map that sends a second homology class to the minimal genus of surfaces representing the class. We discuss several applications of the functions to smooth structures, such as 3-manifolds bounding exotic 4-manifolds, non-existence of surgeries generating exotic 4-manifolds, and geometrically simply connected 4-manifolds.
  
  

Participants

• Yoshinori Asano (Tohoku University)
• Yoshihiro Fukumoto(Ritsumeikan University)
• Ayaka Ise(Tokyo Institute of Technology)
• Kazuhiro Kawamura(University of Tsukuba)
• Min Hoon Kim(Korea Institute for Advanced Study)
• Takao Matumoto(Osaka City University)
• Naoyuki Monden(Okayama University
• Osamu Saeki (Kyusyu University)
• Taketo Sano (The University of Tsukuba)
• Yoshihisa Sato(Kyusyu Technology University)
• Xiaobing Sheng(University of Tokyo)
• Kokoro Tanaka (Tokyo University)
• Akira Yasuhara(Waseda University)

Organizers

Tetsuya Abe(Ritsumeikan University), Motoo Tange(University of Tsukuba)

Seminars