Math 5520 Algebraic and Geometric
Topology
Instructor:
Mladen Bestvina
Office: JWB
210
Web page:
http://www.math.utah.edu/~bestvina/5520
Office
Hours:
by appointment
Textbooks:
- Allen Hatcher: Algebraic
Topology, hard
copy, or download
from Prof. Hatcher's web page. We will cover only chapters 0 and 1.
- John Stillwell: Geometry of
surfaces, hard
copy
- Richard Schwartz: Mostly
surfaces, hard copy
or download
an unformatted draft.
- (optional) William Thurston: Three-dimensional
geometry and topology, vol 1. hard
copy
Time: MWF
9:40-10:30
Room: LCB 215
The course will cover fundamental group and covering spaces from
Hatcher's book, and basic topology and geometry of surfaces with an
emphasis on hyperbolic geometry. The basic reference is Stillwell's
book. We will also cover selected topics from Schwartz's book.
The grades
will
be based on homework, class presentations, midterms and the final exam.
You can
contact
me by email.
Homework 1, due Jan 25: Hatcher -- read pages 1-4, especially the
discussion of
deformation retractions. Then do #1,2 on p.18 (hint for #2: think of
the torus as an identification space with underlying space the square,
and delete the center).
Also do p.38: #1,7,8. Hint for #7: Consider what happens to the two
paths under the projection S1xI->S1.
Homework 2, due Feb 8:
- read Schwartz, p. 21-72 (the online version). You may skip
sections 3.7, 4.4 and 6.8. We talked about most of this in class and it
should be leisurely reading.
- Hatcher: p.38 #9 (this is the "Ham Sandwich Theorem" -- you may
assume that the volume of the pieces cut by the plane change
continuously, and you may also assume that the sets are convex).
- p.39, #16 a,f
- p.79, #3,4.
Topics to present in class, out of Schwartz's book. Each topic should
be presented by 2 students in one hour.
- Banach-Tarski paradox
- Dehn's Dissection Theorem (or Scissor's Congruence)
- Cauchy's rigidity theorem
- Continued fractions and the modular group (pp. 215-224 in Schwartz)
Homework 3, due Feb 22: Hatcher p.52: #1,2,8,9,17. Hint for #9: What
does C represent in the abelianization?
I am using this
for the lecture on the classification of surfaces.
Homework 4, due March 8. For the following two triangulations carry out
the transformation to a standard form and find out which surfaces they
represent:
1) 123, 234, 345, 451, 512, 136, 246, 356, 416, 526
2) 124, 235, 346, 457, 561, 672, 713, 134, 245, 356, 467, 571, 126, 237
Handwritten notes on the hyperbolic plane, based on Thurston's book Three-dimensional Geometry and Topology.
There are good notes on basic hyperbolic geometry available here. For example, for hyperbolic trig formulas, see lecture 8.
Homework 5, due March 29, is here.