
Math
9500: Introduction to 3Manifolds
Fall 2022
Meets:  Tue/Thu
11:0012:20 in
Wachman Hall, room 617 
Instructor:  David
Futer 
Office:  1026
Wachman Hall 
Office
Hours:  Tue 1:303:00, Wed 10:3012:00, or by
appointment 
Email:  dfuter
at
temple.edu 
Phone:  (215)
2047854 
Course content:
This course is motivated by the homeomorphism problem. Given a pair of
nmanifolds M and N, is there an algorithmic procedure to decide whether
they are homeomorphic? This problem is trivial in dimension 1, fairly
easy in dimension 2, and impossible in all dimensions n ≥
4. Dimension n=3 is the middle ground, where the problem is
solvable but fairly hard.
The solution to the homeomorphism problem in dimension 3 relies on the
Geometrization theorem, posed as a conjecture by Thurston around 1980
and proved by Perelman in 2003. This result says that every 3manifold
can be canonically subdivided into pieces, such that every piece has one
of 8 homogeneous geometries. So we will study how exactly the cutting
procedure works, what the 8 geometries are, and how they can be used to
address the homeomorphism problem.
Surfaces embedded in 3manifolds will play a starring role. We will study
normal surfaces, incompressible surfaces, and Haken hierarchies. We will
also discuss constructions of 3manifolds from mapping classes of
surfaces, via Heegaard splittings and fibrations. Time permitting, we
will discuss some "virtual problems" about covers of 3manifolds.
References:
Prerequisites: Math 806162.
Grading: Grades will be assigned based on homework and a presentation
toward the end of the semester.
Detailed schedule
This will be gradually filled in as the semester progresses.
Day 
Topic 
Reading 
Homework/Note 
8/23  The homeomorphism problem; surfaces  Conway's ZIP proof 
 8/25  Geometric examples, lens spaces  Martelli, p. 303310  Play
with Curved
spaces app
 8/30  Ibundles, connected sums  Martelli, p. 270274 
 9/1  Morse theory, Alexander's theorem  Martelli,
p. 274276  Homework 1, due 9/13
 9/6  Prime vs irreducible manifolds  Martelli, p. 276278 
 9/8  Normal surface theory  Martelli, p. 279282 
 9/13  Prime decomposition  Martelli, p. 282284 
 9/15  Incompressible surfaces  Martelli, p. 287289 
 9/20  Essential surfaces  Martelli, p. 285; 290291 
 9/22  Haken Manifolds  Martelli, p. 294296 
 9/27  Haken hierarchies  Martelli p. 293, 296297; Hatcher
Lemma 3.5 
 9/29  The Loop/Disk Theorem  Martelli, p. 298300; Hatcher
Theorem 3.1; Kent notes Sec. 8 
 10/4  Circle bundles over surfaces with boundary  Martelli,
p. 310312. 
 10/6  Circle bundles over closed surfaces  Martelli,
p. 312313 
 10/11  Seifert fibrations  Martelli,
p. 314318  Homework 2, due 10/20
 10/13  Classification of SFS, Torus decomposition  Martelli,
Section 10.4; Hatcher, Ex. on page 13 
 10/18  Torus (JSJ) Decomposition  Martelli, p. 364366 
 10/20  Geometric structures on surfaces  Martelli,
p. 160162 
 10/25  Threedimensional geometries: S^{3}  Martelli,
p. 370379; Scott, p. 450457  Choose
a presentation topic by next week
 10/27  Euclidean and S^{2}xR geometries  Martelli,
p. 380384 
 11/1  H^{2}xR geometry, start of Nil 
Martelli, p. 384388 
 11/3  Nil geometry  Martelli, p. 388392 
 11/8  SL_{2}R geometry  Martelli, p. 392396;
Scott, p. 462467 
 11/10  Sol geometry  Martelli, p. 396399 
 11/15  H^{3} geometry  Martelli, p. 6268 
 11/17  Commuting isometries of H^{3} 
Martelli, p. 115116 
 11/29  Geometrization  Martelli, p. 400403 
 11/30  Presentations: Rob, Ross  
 12/1  Presentations: Andrew, Brandis, Dipika  

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Last modified: Fri Aug 21 13:41:22 PDT 2009
