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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 \(p/q \in {\mathbb{Q}}\).

  • Fact: \(\dim_{{\mathbb{F}}_2} \widehat{\operatorname{HF}}(M) \geq # H_1(M; {\mathbb{Z}})\) 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 \(S^1\times S^2\).

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