2021-11-08

Tags: #web/quick-notes

Refs: ?

15:05

Hannah Turner, GT: Branched Cyclic Covers and L-Spaces

• Two main constructions for 3-manifolds: Dehn surgery and branched cyclic covers

• Idea: $$C_n\curvearrowright M$$, take quotient to get an $$n{\hbox{-}}$$fold covering map away from a branch locus (usually a knot or link).

• Given a knot $$K\hookrightarrow S^3$$, can produce a canonical cyclic branched cover for any $$n$$, $$\Sigma_n(K)$$.

• Dehn surgeries: classified by $$p/q \in {\mathbf{Q}}$$.

• Fact: $$\dim_{{ \mathbf{F} }_2} \widehat{\operatorname{HF}}(M) \geq # H_1(M; {\mathbf{Z}})$$ unless it’s infinite, in which case we set the RHS to zero. We say $$M$$ is an $$L{\hbox{-}}$$space if this is an equality.

• Note that lens spaces have this property!
• Conjecture: non $$L{\hbox{-}}$$space if and only if admits a co-oriented taut foliation (decomposition into surfaces) iff $$\pi_1$$ is left orderable.

Q: push through local system correspondence, what does this say about reps $$\pi_1\to G$$..? Or local systems..?

• We know foliation $$\implies$$ non $$L{\hbox{-}}$$space, the other directions are all wide open.

• Diagrams for knots: boxes with numbers are half-twists, sign prescribes directions.

• Which branched covers of knots are $$L{\hbox{-}}$$spaces?

• Nice trick: quotient by a $$C_2$$ action to make it a double branched cover $$X\to X/C_2$$, and find an $$n{\hbox{-}}$$fold branched cover $$\tilde X\to X$$. Then take an $$n{\hbox{-}}$$fold branched cover $$\tilde{X/C_2} \to X/C_2$$ and then its 2-fold branched cover will be $$\tilde X \to \tilde{X/C_2}$$.

• Weakly quasi-alternating $$K$$: $$\Sigma_2(K)$$ is an $$L{\hbox{-}}$$space.

• There are tools for showing the $$\pi_1$$ you get here are not left-orderable. Showing left-orderability: fewer tools, need representation theory.

• Generalized $$L{\hbox{-}}$$space: $$L{\hbox{-}}$$spaces for $$S^1\times S^2$$.

#web/quick-notes