On L-spaces and left-orderable fundamental groups

Abstract

Examples suggest that there is a correspondence between L-spaces and 3-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-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 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group of the 2-fold 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 Homeo_+(S^1).

30 pages, 4 figures, uses colour

References in corpus (7)

• Floer homology and knot complements
• Bordered Heegaard Floer homology: Invariance and pairing
• Bimodules in bordered Heegaard Floer homology
• On L-spaces and non left-orderable 3-manifold groups
• Knot doubling operators and bordered Heegaard Floer homology
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Cited by in corpus (9)

• Non-left-orderable double branched coverings
• Graph manifolds Z-homology 3-spheres and taut foliations
• Left-orderable fundamental group and Dehn surgery on the knot $5_2$
• Left-orderable fundamental group and Dehn surgery on twist knots
• Strong L-spaces and left orderability
• Heegaard Floer homology, L-spaces and smoothing order on links II
• Left-orderable, non-L-space surgeries on knots
• Low Dimensional Topology and Ordering Groups
• Genus one open books with non-left-orderable fundamental group