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# Floer theory and low-dimensional topology: Heegard Floer homology

Corks, Involutions, and Heegaard Floer Homology- Irving Dai (MIT)

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. As an application, we define a modification of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits an infinitely-generated subgroup of strongly non-extendable corks. We establish several new families of corks and prove that various known examples are strongly non-extendable. This is joint work with Matthew Hedden and Abhishek Mallick. This talk will be complementary to the talk given by Abhishek Mallick (in the "Knots, Surfaces, and 4-Manifolds" topic group), and will discuss the Heegaard Floer theory underlying various detection results.

Knot Floer homology and relative adjunction inequalities- Katherine Raoux (Michigan State University)

Abstract: In this talk, we present a relative adjunction inequality for 4-manifolds with boundary. We begin by constructing generalized Heegaard Floer tau-invariants associated to a knot in a 3-manifold and a nontrivial Floer class. Given a 4-manifold with boundary, the tau-invariant associated to a Floer class provides a lower bound for the genus of a properly embedded surface, provided that the Floer class is in the image of the cobordism map induced by the 4-manifold. We will also discuss several applications to links and contact manifolds. This is joint work with Matt Hedden.

Involutive Heegaard Floer homology and surgeries- Ian Zemke (Princeton University)

Abstract: In this talk, we investigate the behavior of involutive Heegaard Floer homology with respect to surgeries. We prove several surgery exact sequences, and also a mapping cone formula, for involutive Heegaard Floer homology. In this talk, we will describe a few elements of the proof, and also some applications to the homology cobordism group. This project is joint work in progress with Kristen Hendricks, Jen Hom and Matt Stoffregen.

## 20 replies on “Floer theory and low-dimensional topology: Heegard Floer homology”

[…] Heegaard Floer homology: […]

IrvingDaisays:

Hi! I’m Irving and my talk is about some applications of Heegaard Floer homology to the theory of corks. If you have any questions, I am happy to answer them here and/or at the online office hours (June 4 at noon; joint with Abhishek Mallick).

Katherine.Raouxsays:

Hi everyone, I’m Katherine. My talk is about genus bounds and Heegaard Floer tau-invariants. I’m having an offfice hour today (June 3) at 4pm. Come ask me questions or leave a comment here!

Katherine.Raouxsays:

Office hour update: if you are trying to join my office hour right now I’m sorry! My internet is down at the moment and I can’t get on zoom. I’m getting in touch with the organizers about rescheduling hopefully for Friday.

Ian_Zemkesays:

Hi, I’m Ian Zemke. I’m a postdoc at Princeton. My talk is on the surgery exact sequences and mapping cone formulas in involutive Heegaard Floer homology. I had my office hours today, though I’m happy to answer any questions. Feel free to send me an email, or post here!

Hi Katherine, I enjoyed your talk! Can you use your relative tau invariants to find a family of rationally slice knots in three-manifolds $Y \neq S^3$ for which the rational genus grows arbitrarily large?

Katherine.Raouxsays:

Hi Lihn, Thanks for your question! Let’s see, so the most simple way to construct examples that I can think of is this: do $p/q$ surgery on the unknot in $S^3$. The resulting 3-manifold is $L(p,q)$ (or $-L(p,q)$ depending on your sign conventions) and the meridian of the unknot that you surgered becomes a rationally null-homologous knot of order $p$ in $L(p,q)$. Let’s call it $\mu$. Moreover, the knot $\mu$ rationally bounds a disk in $L(p,q)$. Now, what we can do is connect sum with slice knots in $S^3$ to produce lots of rationally slice knots. In particular, I think that if $K\subset S^3$ is any slice knot then $\mu\# K\subset L(p,q)$ is also rationally slice. On the other hand the rational 3-genus should basically add (like the regular 3-genus for knots in $S^3$). You have to be a little careful here about how you connect sum rational Seifert/slice surfaces. I can explain more if you like…

Thinking about tau-invariants… the tau’s won’t necessarily detect sliceness. On the other hand, knot Floer homology does detect rational 3-genus (this is a result of Ni). So I guess you could try to construct knots where all the tau invariants are basically zero but the rational genus is large and then try to prove that they are actually rationally slice somehow.

I hope this answers your question! Let me know if you have more.

LisaPiccirillosays:

Hi Katherine,

Lovely talk! For nullhomologous knots $K, K’$ in a 3-manifold $Y$, there is a nice classical way of obstructing concordance in $Y\times I$; observe that a concordance in $Y\times I$ lifts to a strong concordance between the links $\widetilde{K}, \widetilde{K’}$ in $\widetilde{Y}\times I$, (where tilde’s denote whatever cover you happen to like) and then try to obstruct this strong concordance of links (frequently, linking number is enough). Are there natural examples where where this style of argument isn’t well suited to distinguish a pair of $K$ and $K’$ but your taus are? (I guess a good first place to look is knots that \textit{are} topologically concordant; can you give examples of top-but-not-smooth concordance for not-concordant-to-local nullhomologous knots in any/every $Y\times I$?) Thanks!

Katherine.Raouxsays:

Hi Lisa,

Yeah, looking at knots that are topologically but not smoothly concordant is probably the way to go here. A really simple construction I can think of is take $K\times I\subset Y\times I$ and then on one end do a connect sum with a knot $J\subset S^3$ that is topologically but not smoothly slice and has $\tau(J)\neq 0$. Then we would have a topological but not smooth concordance from $K$ to $K\# J$ in $Y\times I$. And I didn’t mention in my talk, but we have a connect sum formula for our generalized tau-invariants. So, our taus would obstruct a smooth concordance.

Hi Katherine,

Thanks for nice talk! I have a question your comment. What do you mean by saying “the tau’s won’t necessarily detect sliceness”? Even the classical tau invariant detect rationally sliceness of knots, see for instance, Prop 4.8 of https://arxiv.org/pdf/1102.5730.pdf

Katherine.Raouxsays:

Hi Oguz,

Thanks for your question! What I mean by “detect” is that tau(K)=0 does not always imply that K is slice. (Eg. tau of the figure-8 knot is zero.) So knowing the value of tau doesn’t always tell you whether your knot is slice. In particular, if you want to show a knot is slice then you have to do more than show tau=0.

On the other hand, you may hear people say that knot Floer homology “detects” the 3-genus of a knot. This really means that if we can calculate HFK, then we know the 3-genus precisely. (Ni and Wu’s result is the analog for the rational 3-genus.)

Cheers!
–K

Hi Katherine,

Thanks for the reply. Now I understand that you talked about classical knot concordance 🙂 For example, one of the involutive Floer theoretic V_0 invariants detect the smoothly sliceness of the figure-eight knot, https://arxiv.org/abs/1507.00383.

Because figure-eight knot is rationally slice and tau invariant provides an invariant of rational concordance classes of knots as I referred, then tau-invariant of figure-eight knot is zero as expected.

Katherine.Raouxsays:

Whoops, the figure-8 knot was probably not the best example to throw out, since it is rationally slice, as you mention. But what I mean is that there are knots with tau=0 that are not classically slice nor rationally slice.

You are right. I believe that 8_1 is such an example.

Hi Irving,

Thanks for nice talk! Is it possible to extend Floer theoretic invariants h_\tau and its companion to invariants of \Theta_Q^{\tau, spin}?

IrvingDaisays:

Hi Oguz,

It is certainly possible to study \Theta_Q^{\tau, spin} using our invariants, but one has to be careful about how the extension of \tau on the cobordism interacts with the spin structures. The cleanest generalization is to study \Theta_{Z/2}^\tau. In this case, there is always a unique spin structure s on the cobordism, which is necessarily fixed by the extension of \tau over the cobordism. (Since a self-diffeomorphism has to take a spin structure to a spin structure.) Thus the cobordism maps corresponding to s commute with \tau and \iota \circ \tau. This shows h_\tau and h_{\iota \circ \tau} give cobordism invariants.

However, if the cobordism has more than one spin structure, then it may not be true that a given spin structure s is fixed by \tau. In general the correct relation is thus something more like “a Q-homology cobordism equipped with a diffeomorphism f and spin structure s which is fixed by f”. Obviously this is somewhat cumbersome.

One nice set of examples (which we briefly study in our paper) is double branched covers of knots. There is the usual map from the concordance group into \Theta_{Z/2} given by taking double branched covers, but remembering the double covering action gives a map into \Theta_{Z/2}^\tau. We show that there is an infinite linearly independent set of knots whose double branched covers bound contractible manifolds (and so are zero in \Theta_{Z/2}), but if you remember the branching action, then the image in \Theta_{Z/2}^\tau remembers they are independent. (I don’t think this family of knots is otherwise particularly interesting, however.)

Hi Irving,

Thanks for the enlightening comment! I will think about it.

Hi Ian,

Thanks for nice talk! You proved that the integral homology cobordism group is not generated by Seifert fibered homology spheres. I wonder that your examples are coming from (1/n)-surgery 3-manifold of knots?

Ian_Zemkesays:

Hi Oguz,
Our examples are +1 surgery on knots. I think we could also prove the rational surgeries formula as an extension of our work for positive surgeries, though we haven’t written it down yet. Maybe there are more applications if we think about fractional surgeries.

Hi Ian,