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 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 π1 (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 S2 (Section 3 in [LS07])|
|December 3||Xinghua Gao*||PSL2(R)~ representations and left-orderability|
|December 6||Siddhi*||Taut foliations and braid positivity|
[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“
[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