---
date: 2022-04-05 23:42
modification date: Wednesday 6th April 2022 17:02:17
title: "Three-manifolds MOC"
aliases: [Three-manifold, Three manifold, "3-manifolds", "3-manifold"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #geomtop #geomtop/3-manifolds  #MOC 
- Refs:
	- Books:
		- A. Candel and L. Conlon, _[Foliations I](https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/q1ojeg/alma99800497612205899)_ (Chapters 1-3) #resources/books 
		- A. Candel and L. Conlon, _[Foliations II](https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/q1ojeg/alma99800497512205899)_ (Chapters 8-11) #resources/books 
		- D. Calegari, _[Foliations and the geometry of 3-manifolds](https://math.uchicago.edu/~dannyc/books/foliations/foliations.html)_ (Chapters 4-5) #resources/books 
		-   A. Hatcher, _[Notes on basic 3-manifold topology](https://pi.math.cornell.edu/~hatcher/3M/3M.pdf)_ #resources/books 
		-   J. Schultens, _[Introduction to 3-manifolds](https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/gpjosq/alma99759787612205899) #resources/books 
		-   B. Martelli, _[An introduction to geometric topology](https://people.dm.unipi.it/martelli/geometric_topology.html)_ #resources/books 
	- [Lectures by Dunfield](https://nmd.pages.math.illinois.edu/classes/2021/595A/index.html#notes): #resources/notes/lectures #projects/to-read 
		1. [ ]  **Aug 23.** [Intro and course overview](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Aug23.pdf).
		2. [ ]  **Aug 25.** [Definitions of foliations and contact structures.](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Aug25.pdf) Got halfway though page 9, also mentioned co-orientability on page 10.
		3. [ ]  **Aug 27.** [Examples of foliations and contact structures.](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Aug27.pdf) Did through page 14, with a very brief preview of 15-16.
		4. [ ]  **Aug 30.** [Holonomy, gluing, and foliating the 3-sphere.](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Aug30.pdf) Did through page 20, and also 21 very quickly.
		5. [ ]  **Sept 1.** [Reeb stability](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept1.pdf). Covered everything.
		6. [ ]  **Sept 3.** [Limits of leaves and the proof of Reeb stability](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept3.pdf). Covered everything.
		7. [ ]  **Sept 8.** [Foliating all 3-manifolds I](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept8.pdf). Ended halfway through page 35. See [Etnyre's lecture notes](https://arxiv.org/abs/math/0409402) for more on the contact story.
		8. [ ]  **Sept 10.** [Foliating all 3-manifolds II](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept10.pdf). Did through the top half of 39; added more detailed account of handle decompositions.
		9. [ ]  **Sept 13.** [Dehn surgery on links](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept13.pdf). One reference is Chapter 9 of Rolfsen's classic [Knots and links](https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/gpjosq/alma99824785712205899). For Lickorish's theorem, see e.g. Section 6.5 of Martelli. Covered everything.
		10. [ ]  **Sept 15.** [Incompressible surfaces in 3-manifolds](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept15.pdf). Here is an [old lecture](https://ensemble.illinois.edu/app/plugin/embed.aspx?playlistEmbed=true&isNewPluginEmbed=true&idn_playlist=509edf4d-f426-48e6-af50-8a8ba59a570d&idn_init=False&idn_sig=G9MK7zPBNEmvLMnbnMdmf1XbnHQ%3D&destinationID=Td-eUCb05kivUIqLpZpXDQ&contentID=iQXEzqKnfkmNSdyeWLbMLA&searchString=dunfield&pageIndex=1&pageSize=10) on the Virtual Haken Conjecture. Covered everything.
		11. [ ]  **Sept 17.** [Taut foliations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept17.pdf). Through page 54.
		12. [ ]  **Sept 20.** [Properties of taut foliations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept20.pdf). Covered everything.
		13. [ ]  **Sept 22.** [More on taut foliations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept22.pdf). Everything but the theorem at the bottom of the last page.
		14. [ ]  **Sept 24.** [Universal covers of taut foliations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept24.pdf). Covered everything.
		15. [ ]  **Sept 27.** [Thurston's Universal Circle](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept27.pdf). Did through 73.
		16. [ ]  **Sept 29.** [More on Thurston's Universal Circle](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Sept29.pdf). Covered everything.
		17. [ ]  **Oct 1.** [The L-space conjecture I](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct1.pdf). Did through 85.
		18. [ ]  **Oct 4.** [The L-space conjecture II](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct4.pdf). Covered all but the theorem on the last page.
		19. [ ]  **Oct 6.** [The Thurston norm](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct6.pdf). Through page 95.
		20. [ ]  **Oct 8.** [The Thurston norm and foliations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct8.pdf).r Covered everything.
		21. [ ]  **Oct 11.** [Essential laminations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct11.pdf). Covered everything.
		22. [ ]  **Oct 13.** [Branched surfaces](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct13.pdf). Covered everything.
		23. [ ]  **Oct 15.** [Triangulations to foliations](https://nmd.pages.math.illinois.edu/classes/2021/595A/notes/Oct15.pdf). The end.
	- [More lectures by Dunfield](https://nmd.pages.math.illinois.edu/classes/2021/595B/index.html)
		1. [ ]  Oct 18. [Introduction](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture1.pdf). Some references:
		    - [ ]   Peter Scott, [The geometries of 3-manifolds](http://www.math.lsa.umich.edu/~pscott/8geoms.pdf).
		    - [ ]   Bruno Martelli, [An Introduction to Geometric Topology](https://arxiv.org/abs/1610.02592).
		    - [ ]   Greg Kuperberg, [Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization](https://arxiv.org/abs/1508.06720).
		    - [ ]   Culler, Dunfield, Goerner, Weeks, et. al. [SnapPy, a computer program for studying the geometry and topology of 3-manifolds](http://snappy.computop.org/).
		2. [ ]  Oct 20. [Basic examples](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture2.pdf). References for basic hyperbolic geometry include Scott and Martelli above as well as
		    - [ ]   Bonahon, [Low-Dimensional Geometry](https://i-share-uiu.primo.exlibrisgroup.com/permalink/01CARLI_UIU/gpjosq/alma99800517512205899) (the most elementary of all these sources).
		    - [ ]   Thurston, [Three-Dimensional Geometry and Topology](https://press.princeton.edu/books/hardcover/9780691083049/three-dimensional-geometry-and-topology-volume-1).
		3. [ ]  Oct 22. [Geometry of Cusps](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture3.pdf). Did 1-4.
		4. [ ]  Oct 25. [From triangulations to hyperbolic structures.](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture4.pdf) Did 1-4.
		    - [ ]   Jeff Weeks, [Computation of Hyperbolic Structures in Knot Theory](https://arxiv.org/abs/math/0309407).
		5. [ ]  Oct 27. [Thurston's gluing equations](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture5.pdf). Did 1-4.
		6. [ ]  Oct 29. [Hyperboloid model](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture6.pdf). Did 1-4.
		7. [ ]  Nov 1. [Canonical cell decompositions](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture7.pdf). Did 1-4.
		    - [ ]   Jeff Weeks, [Convex hulls and isometries of cusped hyperbolic 3-manifolds.](https://www.sciencedirect.com/science/article/pii/0166864193900329) Topology Appl. **52** (1993), no. 2, 127–149.
		8. [ ]  Nov 3. [More on canonical cell decompositions](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture8.pdf). Did 1 to middle of 4.
		9. [ ]  Nov 5. [Finding the canonical decomposition](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture9.pdf).
		10. [ ]  Nov 8. [Canonical decompositions in 3D; closed hyperbolic manifolds](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture10.pdf).
		11. [ ]  Nov 10. [More on closed manifolds](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture11.pdf).
		12. [ ]  Nov 12. [Hyperbolic Dehn filling](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture12.pdf). Through example on page 4.
		13. [ ]  Nov 15. [More on Hyperbolic Dehn filling](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture13.pdf). Did 1-4.
		14. [ ]  Nov 17. [Volumes of hyperbolic 3-manifolds](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture14.pdf). Did everything.
		15. [ ]  Nov 19. **Office hour instead of class.** Come by my office if you want topic ideas or references for your final paper, or want to discuss the material so far.
		16. [ ]  Nov 29. [Certifying solutions to gluing equations](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture15.pdf).
		17. [ ]  Dec 1. [The HIKMOT method](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture16.pdf).
		    - [ ]   Zgliczynski, [Notes on Krawczyk's test.](https://ww2.ii.uj.edu.pl/~zgliczyn/cap07/krawczyk.pdf)
		    - [ ]   HIKMOT, [Verified computations for hyperbolic 3-manifolds](https://arxiv.org/abs/1310.3410).
		18. [ ]  Dec 3. [Applications of verified hyperbolic structures](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture17.pdf).
		    - [ ]   Dunfield, Hoffman, Licata, [Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling.](https://arxiv.org/abs/1407.7827)
		    - [ ]   Dunfield, [Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds](https://arxiv.org/abs/1904.04628), Section 6.
		19. [ ]  Dec 5. [Proof by parameter space](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture18.pdf).
		20. [ ]  Dec 7. [Machine learning for fun and profit](https://nmd.pages.math.illinois.edu/classes/2021/595B/notes/Lecture19.pdf).
		
		-   Davies, Juhász, Lackenby, Tomasev, [The signature and cusp geometry of hyperbolic knots](https://arxiv.org/abs/2111.15323).
		-   [Nature article](https://doi.org/10.1038/s41586-021-04086-x).
- Links:
	- [Geometrization](Unsorted/Geometrization.md)
	- [Hilbert symbol](Unsorted/Hilbert%20symbol.md)
	- [Chern-Simons invariant](Chern-Simons%20invariant.md)
	- [Dehn invariant](Dehn%20invariant.md)

---

# Three-manifold

>Take a look at Machlachlan and Reid's book "The Arithmetic of Hyperbolic 3-Manifolds". #resources 

> Since finite volume hyperbolic structures are unique whenever an $n$-manifold ($n\geq 3$) has them, any invariants of the hyperbolic structure are invariants of the manifold. [Hyperbolic manifolds](Hyperbolic%20manifolds) are $K(\pi,1)$ spaces, so they're not just diffeo/homeomorphism invariants, but invariants of the homotopy-type.

- [Rohklin invariant](Rohklin%20invariant) : a $\ZZ/2$ invariant $r$ for $\ZHS^3$
- [Casson invariant](Casson%20invariant.md) : a $\ZZ$ invariant $c$ for $\ZHS^3$ where $c\mod 2 = r$.