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# Knots, surfaces, and 4-manifolds: 4-manifolds

Corks, involutions, and Heegaard Floer homology- Abhishek Mallick (Michigan State)

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution of a 3-manifold over any homology ball that it may bound (rather than a particular contractible manifold). We utilize the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification of the integer homology cobordism group which takes into account involutions acting on homology spheres, and prove that this group admits an infinite rank subgroup generated by corks. Using our invariants, we also establish several new families of (strong) corks (This is joint work with Irving Dai and Matthew Hedden). This talk will be complementary to the talk given by Irving Dai (in the "Floer theory and low-dimensional topology" topic group).

A relative invariant of smooth 4-manifolds- Hyunki Min (Georgia Tech)

Abstract: We define a polynomial invariant of a smooth and compact 4-manifold with connected boundary by modifying an invariant of closed 4-manifolds from Heegaard Floer homology. Using this invariant, we show that there exist infinitely many exotic fillings of 3-manifolds with non-vanishing contact invariant. This is a joint work with John Etnyre and Anubhav Mukherjee.

Standardizing Some Low Genus Trisections- Jesse Moeller (University of Nebraska-Lincoln)

Abstract: Trisections are a novel way to study smooth 4-manifold topology reminiscent of Heegaard splittings of a 3-manifolds; a surface, together with families of embedded curves, determines a 4-manifold up to diffeomorphism. Given a specific genus, we would like to know which 4-manifolds have trisection diagrams inhabiting a surface with this genus. For genus one and two, this is known. In this talk, we will introduce a family of seemingly complicated genus three trisection diagrams and demonstrate that they are, in fact, connected sums of well understood diagrams.

Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines - Sümeyra Sakalli (MPIM Bonn)

Abstract: It is known that all minimal complex surfaces of general type have exactly one (Seiberg-Witten) basic class, up to sign. Thus, it is natural to ask if one can construct smooth 4-manifolds with one basic class. First, Fintushel and Stern built simply connected, spin, smooth, nonsymplectic 4-manifolds with one basic class. Next, Fintushel, Park and Stern constructed simply connected, noncomplex, symplectic 4-manifolds with one basic class. Later Akhmedov constructed infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds with nontrivial Seiberg-Witten invariants. Park and Yun also gave a construction of simply connected, nonspin, smooth, nonsymplectic 4-manifolds with one basic class. All these manifolds were obtained via knot surgeries, blow-ups and rational blow-downs.  In this talk, we will first review some main concepts and recent techniques in symplectic 4-manifolds theory. Then we will construct minimal, simply connected and symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines by the so-called star surgeries, and by using complex singularities. We will show that our manifolds have exotic smooth structures and each of them has one basic class. We will also present a completely geometric way of constructing certain configurations of Kodaira’s singularities in the rational elliptic surfaces, without using any monodromy arguments.

Ribbon homology cobordisms- Mike Wong (Louisiana State University)

Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we describe a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies, and some applications. This is joint work with Aliakbar Daemi, Tye Lidman, and Shea Vela-Vick.

## 19 replies on “Knots, surfaces, and 4-manifolds: 4-manifolds”

[…] 4-manifolds: […]

LisaPiccirillosays:

Hi Hyunki, nice talk!
For 4-manifolds $X$ with boundary $Y$, do we know anything about the effect of knot surgery on plain old HF cobordism maps, say $F^+_{\mathfrak{s}}:HF^+(S^3)\to HF^+ (Y,\mathfrak{s’})$? Is there a reason the mixed map is the right thing, other than the conjectural equivalence of the OS polynomial with the SW? Thanks!

hkmin27says:

Hi LIsa. That’s a good point. Actually, Tom Mark showed that the knot surgery formula on a negative version of HF cobordism map $F^-:HF^-(S^3) \rightarrow HF^-(Y)$. We can use any version we want, but to apply the formula, we need a non-vanishing cobordism map. We just found one using the non-vanishing contact invariant. A minus version does not contain a contact invariants and a plus version vanishes because this cobordism is not symplectic (there is no weak filling of $S^3$ other than $B^4$ up to blow up/down.) That’s why we used a mixed map.

Hi Abhishek,

How special are your techniques to involutions? Can you say anything when the maps on the 3-manifolds are periodic with period some other finite number? Or even if the maps are not periodic at all?

Abhishek_Mallicksays:

Hi Maggie,

That is a very interesting question. Unfortunately, our techniques makes use of the fact the diffeomorphisms are involutions in an essential way. Indeed one can try to define a similar invariants for any finite order diffeomorphisms but the invariants ends up being too weak (i.e it results in making some fundamental non-trivial finite order actions on some ‘standard’ chain complexes, locally-trivial) if we keep working on Z_2 coefficient for the Floer homology (as we have done).

Oh okay, that’s interesting. I guess there is some result you are using about Floer homology that is specific to Z_2 coefficients? (I know this comes up often, but don’t know enough about the subject to know where. )

Abhishek_Mallicksays:

Yes, that is right. We are using naturality and diffeomorphism invariance of cobordisms for Heegaard Floer homology in Z_2 coefficient. Although it is believed that it can be extended to other coefficients but I think it is still work in progress.

Hi Sümeyra,

I just want to make sure I understand the basic idea of your argument! Is the idea that you start with a big manifold whose Seiberg-Witten invariants you understand, and then do successive star surgeries to decrease b_2 of the manifold while preserving the Seiberg-Witten invariants? And then at the end try to obtain a small b_2 manifold whose Seiberg-Witten invariants distinguish it from something standard?

SumeyraSakallisays:

Hi Maggie,

Thanks for asking! I start with E(n) manifolds and do blow ups. For E(n) blown up at n points, we know how to compute the Seiberg-Witten basic classes due to Fintushel-Stern. Then I do star surgeries. For the resulting manifold X, I check which basic classes extend from the blown up E(n) to X. In fact, I show that only the top class extends to X. From this fact I show that X is minimal. (So, not all SW classes extend to X after doing star surgeries.)
To show exoticness of my manifolds I again use SW invariants. There are vanishing-nonvanishing theorems (e.g. see Gompf-Stipsicz, Chapter 2). The standard 4-manifolds have trivial SW invariants. But for my manifolds SW invariants are nontrivial (Taubes’ result.) Hence they are not diffeomorphic to each other.
Star surgeries decrease the b2 of the manifold, that’s right! But on this paper I mainly try to control c_1^2 and \chi_h of X. My goal is to construct something between the Noether and half Noether lines.

SumeyraSakallisays:

Hi Abhishek,

In defining the integer homology cobordism group, the 3-manifolds bounding the cobordism are integer homology spheres. I just wondered if we take more general 3-manifolds as the boundary of the cobordism, are there any work related to the corresponding cobordism “group”, parallel to these results?

Same for the strong corks. Is there a notion of generalized Strong corks which does not require Y to be a homology sphere?

Thanks!

Abhishek_Mallicksays:

Hi Sümeyra,

That is an interesting follow-up. One can certainly define such a group for rational homology spheres, here we need to careful about the spin-c structures. Essentially we would need a decoration by a spin-c structure that is fixed by the involution, and same for the spin-c structure on the cobordism as well, and I think parallel of the constructions would still continue to hold.
As per more generalized manifolds is concerned, I think we would run into difficulties on the construction if we allow non-zero b_1 for the 3-manifolds.

For your other question, to my knowledge there is no such generalization that has been studied in the literature. Although it is a very reasonable question to ask.

SumeyraSakallisays:

Hi Hyunki,

In the proof of the main theorem, you attach 2-handles to the convex(?) boundary of [0,1]xY. Could you say a little bit about why you want to have convex boundary rather than concave? Also, how do you insert the Gompf nucleus?
For your construction, is not it possible to compute the b_2 of the manifold? I guess I misunderstood the part saying “controlling the b_2” at the end of your talk.

Thank you!

Anubhavsays:

Hi Sumeyra,

In our construction we need to attach those handles (2-h) along convex end in order extend the symplectic structure to the manifold (which is similar to the idea of Eliashberg and Gompf). We inserted the Gompf nucleus by attaching two two handles to build a cusps neighborhood. We did it in a clever way so that complement remains simply connected and the torus become a symplectic torus with trivial neighbourhood. This part of the proof is availavble on the arxiv version. The only gap in the arxiv version is the last paragraph. And to modify that we had to deal with Osvath-Szabo invariant with twisted co-efficient.

I will let Min to express his thought.

hkmin27says:

Hi Sumeyra.
Thank you for your question. This is because we can attach Weinstein 2-handles on a convex boundary, not concave one. By attaching https://photos.app.goo.gl/VCHd5EAg3TKAnKGC9 these Weinstein 2-handles, we can insert the Gompf nucleus and keeping symplectic structure on the cobordism.
For b_2 numbers, you’re right. We couldn’t compute b_2 numbers of our manifold and it seems large. “controlling b_2” just means whether we can find a small b_2 exotic filling for a given 3-manifold.

SumeyraSakallisays:

Thanks Anubhav and Hyunki for your replies.
Oh that’s right, you attach ‘Weinstein’ 2-handles. I’ve missed that.

SumeyraSakallisays:

Hi Jesse,

What are the 4-manifolds that admit genus 3 trisection diagrams? (May be you mentioned but I missed it, sorry.)

Also, using the Farey triple sounds cool!

Anubhavsays:

Hi Mike,

Nice talk!! I have a question regarding the last theorem of your 2nd talk. Do you know any counter-example if $Y$ is not irreducible? I noticed that you use the fact that it is irreducible to conclude that $W$ retract onto $Y$.

Mike_Wongsays:

Hi Anubhav,

Thank you! So as I mentioned today in office hour, I do not know of any counter-examples if Y is not irreducible. In fact, the theorem also holds for L-spaces, which may not be irreducible.

Hey Jesse,
Cool talk! Quick Q- what inspired you and Roman to create these diagrams coming from Farey triples in this way? Is there some precedent for doing this? Or was the idea that the classification of genus 3 trisections is hard, and these diagrams were for some reason “nice” enough to be able to tackle?
Thanks!