---
date: 2021-11-08
tags: [ web/quick-notes ]
---

# 2021-11-08

Tags:
#web/quick-notes

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## 15:05

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

- Two main constructions for 3-manifolds: Dehn surgery and branched cyclic covers
- Idea: $C_n\actson M$, take quotient to get an $n\dash$fold covering map away from a branch locus (usually a knot or link).
- Given a knot $K\embeds S^3$, can produce a canonical cyclic branched cover for any $n$, $\Sigma_n(K)$.
- Dehn surgeries: classified by $p/q \in \QQ$.
- Fact: $\dim_{\FF_2} \hat{\HF}(M) \geq \# H_1(M; \ZZ)$ unless it's infinite, in which case we set the RHS to zero.
  We say $M$ is an $L\dash$space if this is an equality.
  - Note that lens spaces have this property!

- Conjecture: non $L\dash$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 $L\dash$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 $L\dash$spaces?
- Nice trick: quotient by a $C_2$ action to make it a double branched cover $X\to X/C_2$, and find an $n\dash$fold branched cover $\tilde X\to X$.
  Then take an $n\dash$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 $L\dash$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 $L\dash$space: $L\dash$spaces for $S^1\times S^2$.