On L-spaces and left-orderable fundamental groups

By S. Boyer, Cameron McA. Gordon, Liam Watson. 2011

222 citations recorded by the source corpus.

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in $$\text{ Homeo}_+(S^1)$$.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/146e44e89b87d69311f668232a9d21bf5a19026e

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Citations in the corpus

Foliations, orders, representations, L-spaces and graph manifolds (2014, background): “e a compact, connected orientable 3-manifold with torus boundary and f: ∂R→ ∂Nt to be a gluing map. Set Wt(f) = R∪f Nt We call Wt(f) an Nt-ﬁlling of R. A striking property of N2-ﬁllings was proved in [BGW]. Proposition 7.2. ([BGW]) Let Rbe a compact, connected, orientable 3-manifold with torus bound”
A survey of Heegaard Floer homology (2013, background): “ivalent. (1) Y is an L-space, (2) π1(Y ) is not left-orderable, (3) Y carries no taut foliation. 10 ANDRAS JUH´ ASZ´ The conjecture that (1) and (2) are equivalent is due to Boyer, Gordon, and Watson [6]. Ozsv´ath and Szabo´ [71] proved that (1) implies (3). Now we state another conjecture, original”
Bordered Floer homology for manifolds with torus boundary via immersed curves (2016, methodology): “We note that, in light of the L-space conjecture [7], the L-space gluing theorem makes some striking predictions about the behaviour of foliations on manifolds with torus boundary, as well as the behaviour of left-orderes on the fundamental groups of these manifolds.”
Grid Homology for Knots and Links (2015, background): “For other connections between Heegaard Floer homology and the fundamental group, see [15].”
Floer simple manifolds and L-space intervals (2015, unknown): “One could also regard the classification by Jankins, Neumann [24], and Naimi [33] of Seifert fibered spaces over S2 satisfying (1), together with the classification in the present article’s Theorem 5.1 of Seifert fibered L-spaces over S2, as an alternative proof that (1) = (3).”