Place: University of Tsukuba(Shizen D-building D509)
Date：2024/Mar/27-March/29
Keywords:
[Abstract]
Corks were originally defined and introduced by Akbulut.
It is known as a fundamental theorem that for every exotic pair of simply connected closed 4-dimensional manifolds,
there exists a cork related to the maniflds. In other words, corks are believed to govern the entire differential structure of 4-dimensional smooth manifolds.
However, research in this area has not progressed significantly for a long time.
Recently, there has been a growing body of research on corks, leading to a better understanding of the differential structures of 4-dimensional manifolds.
The purpose of this symposium is to cover the basics of corks as well as the recent developments in contemporary cork theory.
To Institute Natural Sciences D-Building D509
by bus
(1) Get off at Tsukuba station
(2) Get on a bus of Kanto Tstsudo Bus(Uni Tsukuba circling(clockwise))(Nr.6 platform) Time table
(3) Get off the Daiichi-Eria-Mae station
(4) After that YouTube video
on foot.
(*) You can walk on foot for about 46 minutes from Tsukuba center to D-building.
Registration Resitration Form
Registration is ended. If you want to attend this workshop, then please email Tange (tange_at_math.tsukuba.ac.jp (_at_=@))
Confirmed speakes
Sungkyung Kang (University of Oxford)
Hokuto Konno(University of Tokyo)*
Roberto Ladu(Max Planck Institute for Mathematics)
Hokuto Konno From families Seiberg-Witten theory to corks
We shall explain an approach to corks and related topics from families Seiberg-Witten theory.
For example, we show that families Seiberg-Witten theory is capable of detecting (strong) corks, and we also give the first example of "1-parameter version of cork" by a related technique.
If time permits, we mention detection of exotic surfaces related to corks. This is joint work with Abhishek Mallick and Masaki Taniguchi.
Makoto Sakuma Three and four dimensional equivariant genera of strongly invertible knots,
and the equivariant cobordism group
A knot $K$ in the 3-sphere is strongly invertible if there is an orientation-preserving smooth involution $h$ of the 3-sphere
such that $h(K)=K$ and $h$ reverses the orientation of $K$.
In my paper “On strongly invertible knots” published in 1986,
I introduced the notion of a direction on a strongy invertoble knot
and defined the equivariant cobordism group of directed strongly invertible knots.
I then observed that the natural homomorphism from the equivariant cobordism group to the cobordism group has a nontrivial kernel,
by using a certain polynomial invariant based on the Kojima-Yamasaki eta-invarinat.
To my pleasant surprise, the strongly invertible knots recently attract attention of various researchers,
in particular, significant progresses have been made on the 4-dimensional equivariant genera
of strongly invertible knots by using modern technology.
On the other hand, in my recent joint works with Mikami Hirasawa and Ryota Hiura,
we studied 3-dimensional equivariant genera of directed strongly invertible knots, by using classical techniques.
In this talk, I quickly recall basic facts on strongly invertible knots,
and then give a survey of recent developments.
(Please understand that I cannot mention to the technical part using modern technology
by the lack of my knowledge and ability. )
Masaki Taniguchi Corks, Instantons, and Chern-Simons functional
(joint work with Abhishek Mallick (Rutgers University), Irving Dai (University of Texas at Austin) and Antonio Alfieri (CRM-ISM))
We present a variant of involutive instanton Floer theory that obstructs the existence of diffeomorphisms on certain 4-manifolds with boundaries.
Our research yields strong corks which survive after stabilization by $n{\mathbb C}P^2$ or $-n{\mathbb C}P^2$ and new strong corks obtained as a family of surgeries along slice knots, and establishes topological constraints concerning non-orientable surfaces with boundaries.
Moreover, we prove that any smooth self-homology cobordism of the boundary of the Akbulut cork is not simply connected if there is a diffeomorphism extending involutions on the boundaries.
This is joint work with Abhishek Mallick, Irving Dai, and Antonio Alfieri.
Kouki Sato Applications of involutive Heegaard Floer theory to corks (survey)
In this talk, we survey Dai-Hedden-Mallick's paper ``Corks, involutions
and Heegaard Floer homology", which constructs an involutive Heegaard
Floer complex for a homology 3-sphere with involution and apply it to
the study of corks. First, we mention fundamental theories established
by Hendricks-Manolescu-Zemke and Juhasz-Thurston-Zemke, and then we
state the defiintions of involutive Floer complexes and an equivalence
relation among them. Next, we show some properties and practical
examples of the Floer complexes, and explain how to apply them to the
study of corks.
Natsuya Takahashi Relative trisections for corks and exotic 4-manifolds
An important conjecture in trisection theory is that “if two 4-manifolds
are exotic, then they have the same trisection genus”. In this talk, we
present an exotic pair of 4-manifolds with boundary whose trisection
genera are both 4. Among the known supporting evidence for the
conjecture, our exotic pair has the smallest trisection genus. Our
trisections of the exotic pair are derived from two minimal genus
relative trisections of the Akbulut cork.
Yohei Wakamaki Stabilizations of some small simply-connected closed 4-manifolds
An oriented 4-manifold is said to be almost completely
decomposable (ACD) if it becomes diffeomorphic to $\#a{\mathbb C}P^2 \#
(-b){\mathbb C}P^2$ after a single ${\mathbb C}P^2$-stabilization
(i.e., taking a connected sum with ${\mathbb C}P^2$ for some integer
$a,b \geq 0$. It is a conjecture of Mandelbaum that any simply-connected
closed 4-manifold is ACD for at least one orientation. While many
examples are known to be ACD, those with small second Betti numbers
$b_2$ are quite rare. In this talk, we give new examples of ACD
4-manifolds with $b_2 = 7,8,9$, and $10$. We also discuss a similar
result for $S^2\times S^2$-stabilization.
Kouichi Yasui Genus functions of 4-manifolds and their applications
Typical applications of corks are constructions of exotic 4-manifolds, and their smooth structures are often distinguished by their genus functions. Here, the genus function of a smooth oriented 4-manifold is the function that maps a second homology class to the minimal genus of an embedded closed oriented surface representing the class. In this talk, I will briefly review genus functions, and give applications to exotic smooth structures and properties of smooth 4-manifolds such as the geometric simple connectivity and the mod 2 simple type conjecture. I will also discuss a limitation of genus functions and its applications.
Roberto Ladu On the universal cork conjecture.
I will discuss some recent results on the universal cork conjecture including a sufficient condition for a cork to be not universal and the non-existence of a universal cork in the case of manifolds with boundary.
Sungkyung Kang
(Talk 1) Bordered Floer homology and the invariant splitting principle
We introduce bordered Floer theory and its involutive version, as well as their applications to knot complements.
We will sketch the proof that invariant splittings of CFK and those of CFD correspond to each other under the Lipshitz-Ozsvath-Thurston correspondence, via invariant splitting principle, which is an ongoing work with Gary Guth.
Sungkyung Kang
(Talk 2) One is not enough, with a view towards two
Using the invariant splitting principle, we construct an infinite family of exotic pairs of contractible 4-manifolds which survive one stabilization. We argue that some of them are potential candidates for surviving two stabilizations.
Motoo Tange Review for Cork
Corks were introduced by Akbulut. This talk is a brief review of cork and its related topics.
26th 9:30- (D509)
Tange's talk (Cork and $S^2\times S^2$ stabilization)
Taniguchi's talk (TBA)
Cork short course(1pm-2pm)
(3pm-)
Short talks or discussion
This workshop is partilly supported by
JSPS KAKENHI Grants-in-Aid for Scientific Research C「Some solutions related to manifolds and handle decomposition, and Dehn surgery」(Organizer Motoo Tange、Nr. 21K03216)
Organizers：Tetsuya Abe（Ritsumeikan University）, Motoo Tange（University of Tsukuba）
