This is the webpage of the learning seminar on the L-space conjecture, organized by Allie Embry and myself. The seminar will meet **Fridays at 10 AM CDT** during Fall 2021.

The L-space conjecture connects topological, analytic (geometric), and algebraic properties of 3-manifolds and is currently a very robust and popular area of work for low-dimensional topologists. The conjecture asserts the following.

**The L-space Conjecture** (BGW): Suppose Y is an irreducible rational homology 3-sphere. Then the following statements are equivalent:

(1) Y is a not an L-space (i.e. the Heegaard Floer homology of Y is not “simple”),

(2) π_{1}(Y ) is left-orderable, and

(3) Y admits a taut foliation.

In this seminar, we aim to define and understand this conjecture (including investigating the meanings of L-space, taut foliations, and left-orderability) and present some results that support the conjecture.

**Schedule**

Date | Speaker | Topic |

September 10 | Allie | Introduction to HF (Section 2 in [JH] and Ch 7 in [CG]) |

September 17 | Jacob | Constructing L-spaces via surgery ([OS07]) |

September 24 | Ceren | SFS that are L-spaces (Ch 4 in [CG] and Section 2 in [BGW]) |

October 1 | Joe | Left-orderability of SFS’s π_{1} (Ch 4 in [CG]) |

October 8 | Shiyu | Left-orderability of surgeries on 1-bridge braids ([CGHV] and [Li17]) |

October 15 | Xinghua Gao* | Postponed |

October 22 | Luis | Introduction to contact topology (Sections 2, 3 and 4 in [JE]) |

October 29 | Cass | Foliations (Ch 5 in [CG]) |

November 5 | Siddhi | (3) ⇒ (1) ([OS04]) |

November 19 | Allie | (1) ⇔ (3) for SFS with base S^{2} (Section 3 in [LS07]) |

November 26 | – | Thanksgiving |

December 3 | Xinghua Gao* | PSL2(R)~ representations and left-orderability |

December 6 | Siddhi* | Taut foliations and braid positivity |

**References**

**Papers**

[BGW] Boyer, Gordon, and Watson, “On L-spaces and left-orderable fundamental groups”

[OS04] Ozsv´ath and Szab´o, “Holomorphic disks and genus bounds“

[OS07] Ozsv´ath and Szab´o, “On knot Floer homology and lens space surgeries“

[LS07] Lisca and Stipsicz, “Ozsv´ath–Szab´o invariants and tight contact three–manifolds, III“

**Lecture Notes**

[CG] Jackson Van Dyke’s notes of Orderability and 3-manifolds course given by Cameron Gordon

[JH] Jen Hom’s Lecture notes on Heegaard Floer homology

[JE] John Etnyre’s Lectures on contact geometry in low-dimensional topology