L-spaces and symplectic fillings

(by Tye Lidman)

Here are two problems pertaining to the famous L-space conjecture. The L-space conjecture predicts that if $Y$ is a prime rational homology sphere, then $Y$ is not an L-space if and only if $\pi_1(Y)$ is left-orderable if and only if $Y$ admits a co-orientable taut foliation.

Ozsvath-Szabo show that if $Y$ admits a co-orientable taut foliation then $Y$ is not an L-space, using topological results of Eliashberg/Etnyre and Eliashberg-Thurston which turn a taut foliation on $Y$ into a symplectic filling of $Y$ with $b^+ > 0$. Therefore, the L-space conjecture predicts that if $Y$ is not an L-space, then $Y$ should admit a symplectic filling of $Y$ with $b^+ > 0$. Currently, it is not clear how to show that non-L-spaces always admit symplectic fillings, even without the assumption on $b^+$.

Problem: Prove that non-L-space admit symplectic fillings.

This problem seems very hard in general, but one can restrict to specific families of non-L-spaces for which the L-space conjecture is still unknown. My favorite family are the boundaries of Mazur manifolds. Recall that a Mazur manifold is a contractible four-manifold built by taking $B^4$ and attaching a single 1-handle and an algebraically cancelling 2-handle with arbitrary framing. The boundary is a homology sphere, and as long as the 4-manifold is not $B^4$ (equivalently the attaching circle for the 2-handle is not isotopic to $pt \times S^1$ in $S^2 \times S^1$) the boundary is a homology sphere which is not an L-space (see for example Conway-Tosun). For these manifolds, one already has a bounding 4-manifold that one could try to augment into a symplectic filling. If the framing is sufficiently negative, then this contractible 4-manifold has a Stein structure. Perhaps in this case, it's easy to augment this to produce $b^+ > 0$.

On the other hand, if $Y$ admits a symplectic filling with $b^+ > 0$, the L-space conjecture would predict that $\pi_1(Y)$ is left-orderable.

Problem: Use the topology of the symplectic filling to order $\pi_1(Y)$.

While it may be hard to see how the 4-dimensional topology would affect $\pi_1(Y)$, when $(Y,\xi)$ admits a Stein filling, a theorem of Giroux says that $(Y,\xi)$ admits an open book decomposition whose monodromy can be expressed as a product of Dehn twists. This gives a more concrete way to see the fundamental group and the problem may become more tractable. There has already been some work on orderability and the fundamental groups of mapping tori, such as work of Perron-Rolfsen and Johnson-Segerman.

Comments

Post a Comment

Popular posts from this blog

L-spaces and F-summands

Problem:  Prove that if $S^3_r(K)$ is not an L-space then its Heegaard Floer homology has a direct $\mathbb{F}$-summand. The L-space conjecture predicts that if an irreducible closed 3-manifold $Y$ is not an L-space then $Y$ admits a coorientable taut foliation. Recently, Lin proved in a short note that if $Y$ admits a coorientable taut foliation then its Heegaard Floer homology $\mathit{HF}^+(Y)$ has a direct $\mathbb{F}$-summand.  For background: recall that $\mathit{HF}^+(Y)$ is a module over $\mathbb{F}[U]$ where $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$. A direct $\mathbb{F}$-summand of $\mathit{HF}^+(Y)$ refers to a direct summand isomorphic as an $\mathbb{F}[U]$-module to $\mathbb{F}[U]/U$. Lin's result therefore gives a potential way to disprove the L-space conjecture: find an irreducible non-L-space whose Heegaard Floer homology does not have a direct $\mathbb{F}$-summand. Alternatively, one could try to prove that Lin's result cannot possibly give a counterexample by sho
Read more

Left-orderability and SU(2)-representations

Problem: Prove that if $Y$ is an irreducible 3-manifold with left-orderable fundamental group then there exists a homomorphism $\pi_1(Y)\to SU(2)$ with nonabelian image. Suppose $Y$ is irreducible with left-orderable fundamental group. Then the L-space conjecture predicts that $Y$ is not an L-space. Together with Kronheimer-Mrowka's conjectured isomorphism $$\widehat{\mathit{HF}}(Y;\mathbb{C}) \cong_{\mathrm{Conj.}} I^\sharp(Y;\mathbb{C})$$ between Heegaard Floer homology and framed instanton homology, this would imply that $Y$ is not an instanton L-space. If $Y$ satisfies a further mild non-degeneracy condition (and perhaps even if $Y$ doesn't) then being an instanton non-L-space implies that there exists a homomorphism $\pi_1(Y)\to SU(2)$ with nonabelian image, per  Baldwin-Sivek . While these conjectures serve as motivation, one can try to prove the stated link between left-orderability and nonabelian $SU(2)$-representations directly. That's the goal of this problem.
Read more

Knot Floer homology and twist coefficients

Question: What information does the knot Floer complex of a fibered knot record about the fractional Dehn twist coefficient of its monodromy? Let $\phi$ be a diffeomorphism of a surface $S$ with connected boundary, which restricts to the identity on the boundary. The fractional Dehn twist coefficient of $\phi$, denoted by $c(\phi)$, is a measure of the twisting near the boundary of the surface in the free isotopy between $\phi$ and its Nielsen--Thurston representative. It was defined by Honda-Kazez-Matic as a quantification of their notion of right-veering surface diffeomorphisms, which in turn came from their study of contact structures on 3-manifolds (a contact structure is tight if and only if all of its supporting open books have right-veering monodromy). It is also closely related to and inspired by Gabai's notion of degeneracy slope . Suppose for example that $\phi$ is freely-isotopic to a pseudo-Anosov map $\phi_0$. The latter fixes a pair of transverse, singular, tran
Read more