Gauge theory in Tokyo

This is an informal conference planned to be held in the Graduate School of Mathematical Sciences, Room 122 at Komaba campus, the University of Tokyo. 

HOW TO COME

Address: Graduate School of Mathematical Sciences Building, 3 Chome-8-1 Komaba, Meguro City, Tokyo 153-0041

The schedule might be changed depending on the situation. If you want to join, please send me a message to taniguchi.masaki.7m@kyoto-u.ac.jp . We will try to live-stream via zoom. 


30 July

10:00 - 12:00 Nobuo Iida (Tokyo Institute of Technology), Monopoles and Transverse Knots, note

13:30 - 16:30 Mike Miller Eismeier (the University of Vermont), Equivariant instanton homology, and instantons mod 2, note including the second talk, video


31 July

10:00 - 12:00 Jin Miyazawa (RIMS), A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P^2-knots, note, video

13:30 - 16:30 Mike Miller Eismeier (the University of Vermont), Equivariant instanton homology, and instantons mod 2, video


1 August  

10:30 - 12:00  Mike Miller Eismeier (the University of Vermont), Chern--Simons invariants and Cosmetic surgery, video

13:30 - 16:30 Masaki Taniguchi (Kyoto University), Sliceness problems via real Seiberg--Witten theory, note, video


Lunch info 


Title and Abstract


Monopoles and Transverse Knots 


 We develop equivariant Seiberg-Witten Floer theory based on the works of Baraglia and Hekmati,  and introduce a homotopical transverse knot invariant in this context. This framework gives a new slice-torus invariant q_M and several new applications on topology and symplectic/contact geometry. This talk is based on a joint work with Masaki Taniguchi (Kyoto Univ)  arXiv: 2403.15763.



Equivariant instanton homology, and instantons mod 2


Floer's instanton homology, an invariant of homology 3-spheres, was originally introduced to give splitting formulas for Donaldson polynomials of 4-manifolds glued together along some homology 3-sphere. The theory has since been well-developed, with generalizations in multiple directions.   We will focus on two: the introduction of equivariant instanton homology by Austin and Braam, which allows for splitting formulas along rational homology 3-spheres, and the less-studied mod 2 instanton homology, which should give splitting formulas for mod 2 Donaldson invariants. New developments in both of these subjects are related to a certain technique called suspension, which we use to define chain maps for certain obstructed cobordisms, all joint with Ali Daemi and parts with Chris Scaduto.  First I will introduce a finite-dimensional `equivariant chain complex' C-tilde(Y, pi; R) for any integer homology sphere Y and any ring R, which is a dg-module over the algebra C_*^{CW}(SO_3; R), the cellular chains on SO_3. When R = Z/2Z, this algebra is the exterior algebra on a generator in degree 1 and 2; when R = Q this algebra is quasi-isomorphic to the exterior algebra on a generator in degree 3.  Second, in principle, the complex could depend on a choice of perturbation pi, and it is not trivial to show independence of the perturbation. Rather, perturbations are naturally arranged into chambers labeled by signature data. Daemi and I proved that for pi and pi' in `adjacent chambers', the continuation map C-tilde(Y, pi; R) -> C-tilde(Y, pi'; R) is a quasi-isomorphism, so the equivariant quasi-isomorphism type of the complex is independent of perturbation. This uses a technique known as suspension to define chain maps associated with obstructed cobordisms, by truncating and modifying their instanton moduli spaces to forcibly make them smooth, while keeping track of the new boundary components algebraically.  Third, I will focus attention on the case R = Z/2Z. I will explain how to define Froyshov-type invariants using the complex C-tilde(Y; Z/2Z). Then, using the suspension technique applied to cobordisms with b^+(W) > 0, I will explain why the F_2-homology cobordism invariant q_3(Y) satisfies the following inequality: if W: Y -> Y' is a cobordism between F_2-homology spheres with H_1(W; Z) free of 2-torsion, then -b^+(W) <= q_3(Y') - q_3(Y) <= b^-(W).  This can be used to give lower bounds on the surgery number of certain 3-manifolds.


Chern--Simons invariants and Cosmetic surgery


 The Chern--Simons functional has been used extensively to produce useful homology cobordism invariants of 3-manifolds. In this talk, I will instead focus on joint work with Ali Daemi and Tye Lidman on certain diffeomorphism invariants of 3-manifolds. We apply these to the cosmetic surgery conjecture: if K is a non-trivial knot in S^3 and S^3_r(K) is oriented diffeomorphic to S^3_s(K), then r = s. It is known that |r| = |s| and if r > 0 then r in {2, 1, 1/2, 1/3, ...}   We introduce a certain quantitative invariant of the instanton homology of Y, a number ell(Y) in R cup {infty} which is infinity if and only if I_*(Y) = 0, so that if W: Y -> Y' is a negative-definite simply-connected cobordism inducing an injection on instanton homology, we have ell(Y') < ell(Y). By exploiting variations of the surgery triangle, and variations on this principle, we show that this implies that if S^3_r(K) is oriented diffeomorphic to S^3_s(K), then |r| = |s| = 2. That is, we prove that the ell-invariants distinguish S^3_{1/n}(K) and S^3_{-1/n}(K) for all non-trivial knots K.



A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P^2-knots 


When two embeddings of surfaces on a 4-dimensional manifold are given, if they are topologically isotopic but not smoothly isotopic, we call them a pair of exotic surfaces. While there is a great deal of study of exotic surfaces in 4-manifolds, studies of closed exotic surfaces in S^4 are limited. In particular, the existence of orientable exotic surfaces in S^4 remains unknown to date. There are some examples of non-orientable exotic surfaces in S^4, including the first example given by Finashin-Kreck-Viro in 1988, but all such cases have genus greater than or equal to 5.  The difficulty in detecting exotic surfaces in S^4 is to prove that two embeddings of surfaces are not smoothly isotopic. All examples of exotic non-orientable surfaces in S^4 have been detected by proving the 4-manifolds obtained by the double branched covers are exotic. If we attempt to apply this technique to low-genus non-orientable surfaces in S^4, we have to discover exotic small 4-manifolds, which is known to be difficult. We construct an invariant for embedded surfaces in 4-manifolds using Real Seiberg-Witten theory, that is a variant of Seiberg--WItten theory. As an application, we give an infinite family of exotic embeddings into S^4 for the real projective plane.


Sliceness problems via real Seiberg--Witten theory


We prove that the (2n,1)-cable of the figure-eight knot is not smoothly slice when n is odd, by using the real Seiberg-Witten Frøyshov invariant of Konno-Miyazawa-Taniguchi. In order to prove it, we develop an O(2)-equivariant version of the lattice homotopy type, originally introduced by Dai-Sasahira-Stoffregen. This enables us to compute the real Seiberg-Witten Floer homotopy type for a certain class of knots, including even torus knots and arborescent knots. Such computation enables us to give some computations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots. This is joint work with Sunkyung Kang and JungHwan Park. 




Organizer: Masaki Taniguchi (Kyoto University)