Homework Set 6.
To be handed in Friday, November 4th.
Download as pdf
sequence of diagrams for problem 2
Proof of the classification of compact, connected surfaces (to be) discussed in class
Two additional proofs of the Classification Theorem of compact, conected surfaces:
based on "cutting away know parts of the surface"
and Conway's ZIP proof
FYI: The 2016 Nobel prize in physics is "for theoretical discoveries of topological phase transitions and topological phases of matter".
It uses the classiciation of
compact sonnected surfaces, click to read.
uses cinnamon bun, bagel and pretzel to explain – video
This is the animation mentioned in class:
The Klein can be cut along a circle so that the result is a Mobius strip
Summary of what we covered on Friday (Oct 14th)
The connected sum of surfaces and Euler characteristics. The classification theorem.
and Illustration 1, Illustration 2, Illustration 3.
In addition, here is reading on the triangulation of surfaces
the Euler number (these summaries are
Illustrations: A Peano-curve.
Namely, the Hilbert-curve. (An example of a "space filling curve").
Space filling curve art and
More info on Commutative diagrams
Homework 4: download as pdf . Due Friday, Oct 14th.
Proof of the missing part of the Heine-Borel thm
Notes on: compactness
Homework 4: download as pdf . Due Friday, Oct 7th.
Notes on: interior, closure, boundary, the Hausdorff property
Homework 3 due September 30th
Summary for subspace, quotient and product topologies as we discussed Week 2 and 3.
Summary for "cut-and-paste arguments" is included as well.
formal treatment of
"pasting" ("gluing", "identifying")
and cut and paste arguments from Munkres
in case of pairwise identification of edges of polygons
Homework 2 due September 23rd
practice problems (no need to hand them in)
Here are summaries for the quotient topologies:
- from Munkres, discussing the torus case in detail
- from Messer and Straffin, discussing the Mobius case in detail
- Stereographic projection - animation
- Stereographic projection from a
a complex analysis textbook (since they place the unit
sphere differently in the coordinate system as we did in class, the formulae
are different a bit)
The Klein Bottle (and Mobius strip) (animation)
- If you go to Prague, check out this
a Mobius mural
- M.C. Escher's famous Mobius strip
Homework 1 due Friday, September 16th
NOTES FOR WEEK1
practice problems, no need to hand them in.
Also see the exercises within "Notes for week1"
In topology the cube = the sphere
The class meets Tuesdays and Fridays 8:30-10am in Room 105.
My office hours are: Thursdays, 2-3:30pm and by appointment
GRADING: Homework is posted every Friday and is due the next Friday.
You can get 100 pts on Homework, 100 pts on the Midterm and 100 pts on the Final. That is 300 points altogether.
The class is curved with the median placed at around B+. The Midterm date is October 21st, Friday