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# Investigating the L-space conjecture: Through Floer Homology

From Floer homology to spectral theory, and hyperbolic geometry- Francesco Lin (Columbia)

Abstract: In the first part of the talk, I will review some spectral theory of three manifolds, and discuss its relation with Floer homology, and in particular L-spaces. In the second part of the talk, I will discuss how spectral theory on a hyperbolic three manifold can be understood in terms of natural geometric quantities. This is joint work with M. Lipnowski.

On the monopole Lefschetz number of finite order diffeomorphisms- Jianfeng Lin (UC San Diego)

Abstract: Let K be a knot in an integral homology 3-sphere Y, and Σ the corresponding n-fold cyclic branched cover. Assuming that Σ is a rational homology sphere (which is always the case when n is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of Σ. Our formula is motivated by a Witten-style conjecture relating the two gauge theoretic invariants of homology S1 cross S3s (the Furuta-Ohta invariant and the Casson-Seiberg-Witten invariant). As applications, we give a new obstruction (in terms of the Jones polynomial) for the branched cover of a knot in S3 being an L-space and we define a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers. This is a joint work with Danny Ruberman and Nikolai Saveliev.

L-spaces in instanton Floer homology- Steven Sivek (Imperial)

Abstract: Framed instanton homology $I^\#(Y)$ is a gauge-theoretic invariant which appears to coincide with the hat version of Heegaard Floer homology. Inspired by the notion of a Heegaard Floer L-space, we say that a rational homology sphere Y is an “instanton L-space" if the rank of $I^\#(Y)$ is as small as possible, namely $|H_1(Y)|$. In this talk I’ll summarize what is known about instanton L-spaces, especially those which arise as Dehn surgeries on knots in $S^3$, and what this tells us about fundamental groups, A-polynomials of knots, and the L-space conjecture. Various parts of this are joint with Antonio Alfieri, John Baldwin, Irving Dai, and Raphael Zentner.

## 8 replies on “Investigating the L-space conjecture: Through Floer Homology”

[…] …through Floer homology […]

jianfenglinsays:

Hello everyone, I am Jianfeng Lin. I’m now an assistant professor at UC San Diego. I am interested in gauge theory, Floer Homology and their applications in 3 and 4-manifolds. I hope you find my talk “On monopole Lefschetz number of finite order diffeomorphisms” interesting. Please feel free to ask any question by leaving a comment here or sending me email. I will hold an office hour on June 8 11AM. Everyone, especially graduate students, is welcome !

HannahTurnersays:

Hi Jianfeng! Can you say more about in what sense your results extend to cyclic branched covers of links? Your machinery then gives an invariant Ln of a (oriented) link?

jianfenglinsays:

Hi Hannah:

For branched double covers of links in S3, we actually proved that the monopole Lefschetz number equals the Froyshov invariant plus the Murasugi signature (it was in our paper: On the Froyshov invariant and the monopole Lefschetz number). We expect something similar should hold for links in other 3-manifolds as well…but we didn’t figure that out.

HannahTurnersays:

For Francesco: Hi! Thanks for your talks. Are there any geometric/topological operations on hyperbolic 3-manifolds under which one can understand how this first eigenvalue on coexact 1-forms changes?

francesco.linsays:

Hi Hannah, thank you for your question! That’s a good question – it’s quite hard to understand indeed; for example in the case of Dehn surgery one can in principle provide explicit bounds on how the eigenvalue changes, but the problem is that the estimates one gets are not sharp at all, so not useful for our purposes.

Somehow related: if you take large surgery on a given knot, lambda_1^* tends to zero, so unfortunately our approach cannot show that large surgeries on an L-space knot are L-spaces. That’s a well known fact that I would really like to understand from this perspective!

SiddhiKrishnasays:

Hi Steven! I really enjoyed your talks! I have a question.

You give two examples of non–torus knots with three $SU(2)$ abelian surgeries, the $K_1 = P(-2,3,7)$ and $K_2 = T(4,7;2,1)$, a twisted torus knot.

You mentioned that both these examples admit two lens space surgeries (which, by CGLS, must be consecutive integers), and that the “extra” SU(2) abelian surgery is the average of these surgery coefficients.

Is this always true? That is, if you have a non-torus knot admitting two lens space surgeries, is the average of the surgery coefficients always an SU(2) abelian surgery?

I guess one can check Berge’s (conjectural) list and determine which knots admit two lens space surgeries, and then investigate the above. Is it already known, or is their some expectation either way?

Thanks!

Hi Siddhi, and thanks for your question! I can’t give any reason why that should or should not be true, unfortunately. I also haven’t thought to check Berge’s list, in part because in general it’s very hard to tell when the non-lens-space surgeries are SU(2)-abelian. But it’s a good idea!

The reason I know about these examples is that they’re among Eudave-Muñoz’s complete list of hyperbolic knots with non-integral, toroidal Dehn surgeries: he gave a 4-parameter family of tangles such that one rational filling gives an unknot, while another rational filling at distance 2 from the first has an essential Conway sphere, so taking branched double covers gives a half-integral toroidal surgery on a knot in S^3. He also observed that many of these tangles have two-bridge fillings as well, and I think P(-2,3,7) and T(4,7;2,1) were the only ones known to have two of them. Moreover, these toroidal manifolds happen to fit into Motegi’s family of splicings of torus knot exteriors, which are known to be SU(2)-abelian. (I learned that they belong to this family from a paper by Ni and Zhang.)

I should also mention work of Jianfeng Lin, who proved that SU(2)-abelian surgeries on a knot are constrained by the inequality $\Delta(p/q,r/s) \leq |p|+|r|$ (cf. CGLS). So if you have two lens space surgeries then there’s a relatively small list of other possibilities for SU(2)-abelian slopes, and the average of the lens space slopes is the most natural place to look.