An L-space is a rational homology 3-sphere with minimal Heegaard Floer
homology. We give the first examples of hyperbolic L-spaces with no symmetries.
In particular, unlike all previously known L-spaces, these manifolds are not
double branched covers of links in S^3. We prove the existence of infinitely
many such examples (in several distinct families) using a mix of hyperbolic
geometry, Floer theory, and verified computer calculations. Of independent
interest is our technique for using interval arithmetic to certify symmetry
groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the
process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus
3 with two distinct lens space fillings. These are the first examples where
multiple Dehn fillings drop the Heegaard genus by more than one, which answers
a question of Gordon.Comment: 19 pages, 2 figures. v2: minor changes to intro. v3: accepted
  version, to appear in Math. Res. Letter

Dunfield, Nathan M.

Hoffman, Neil R.

Licata, Joan E.

English

arXiv

An LL-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic LL-spaces with no symmetries. In particular, unlike all previously known LL spaces, these manifolds are not double branched covers of links in S³. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus 3 with two distinct lens space fillings. These are the first examples where multiple Dehn fillings drop the Heegaard genus by more than one, which answers a question of Gordon.The authors gratefully thank Ken Baker, Francis Bonahon, Craig Hodgson,
Adam Levine, and Jessica Purcell for many helpful conversations. Dunfield
was partially supported by US NSF grant #DMS-1106476 and a Simons
Fellowship; this work was partially done while he was visiting the University
of Melbourne. Hoffman was supported by the ARC Grant DP130103694
and thanks the Mathematical Sciences Institute at the Australian National
University for hosting him during part of this work

The Australian National University

Mathematical Research Letters

Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

Abstract. An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no sym-metries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic ge-ometry, Floer theory, and verified computer calculations. Of independent inter-est is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus 3 with two distinct lens space fillings. These are the first examples where multiple Dehn fillings drop the Heegaard genus by more than one, which answers a question of Gordon. 

Nathan M. Dunfield

Neil R. Hoffman

Joan E. Licata

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Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

Abstract

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The Australian National University

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