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) π1(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. 


September 10AllieIntroduction to HF (Section 2 in [JH] and Ch 7 in [CG])
September 17JacobConstructing L-spaces via surgery ([OS07])
September 24CerenSFS that are L-spaces (Ch 4 in [CG] and Section 2 in [BGW])
October 1JoeLeft-orderability of SFS’s π1 (Ch 4 in [CG])
October 8ShiyuLeft-orderability of surgeries on 1-bridge braids ([CGHV] and [Li17])
October 15Xinghua Gao*Postponed
October 22LuisIntroduction to contact topology (Sections 2, 3 and 4 in [JE])
October 29CassFoliations (Ch 5 in [CG])
November 5Siddhi(3) ⇒ (1) ([OS04])
November 19Allie(1) ⇔ (3) for SFS with base S2 (Section 3 in [LS07])
November 26Thanksgiving
December 3Xinghua Gao*PSL2(R)~ representations and left-orderability
December 6Siddhi*Taut foliations and braid positivity
* the department’s topology seminar


[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