Some possible topics are below. Feel free to suggest your
own.
Please e-mail me your top three choices (in order) by 28th
March, and then I'll assign topics as soon as possible, as fairly as
possible. The rules I'll use are, in order of precedence: no two
people have the same topic; if possible, everybody gets one of their
three choices; as few people as possible get their third choice; as
many people as possible get their first choice. If there are multiple
ways to do this that are equally good according to those rules, then
I'll choose one of them randomly.
If you change your mind later,
that's OK so long as you consult me and don't choose a topic too
similar to somebody else's.
The books referred to below are:
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1. The Borsuk-Ulam Theorem and the Ham Sandwich Theorem. |
| See, for example, Hatcher pages
32ff.. for the Borsuk-Ulam Theorem and, for
example, Elements
of Algebraic Topology by J.R. Munkres, pages 405-6,
or Topology,
also by J.R. Munkres, pages 358-9 for the Ham Sandwich Theorem. |
|
2. Cayley Complexes |
| See, for example, Hatcher, Pages 77,78. |
|
3. Subgroups of free groups are free |
| See, for example, Hatcher, Pages 83–86. |
| Or Stillwell, Chapter 2. |
|
4. Wirtinger Presentations of Knot Groups
|
| See, for example, Stillwell, pages 144ff. |
| or Hatcher, Ex. 22, page 55. |
| or Rolfsen, pages 56ff. |
| or Chapter 7 of these notes by J. Roberts. |
|
5. The Game of Hex and the Brouwer Fixed Point Theorem
|
| See the paper by David Gale, American Mathematical Monthly, Vol. 86, No. 10, Dec 1979, pages 818–827. |
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6. The Jordan Curve Theorem
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There are lots of references available here.
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7. The Alexander Horned Sphere
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| This is a counterexample to a higher-dimensional analogue of the Jordan Curve Theorem |
| Hatcher, page 170 |
| Rolfsen, page 76ff. |
|
8. Borromean rings and Brunnian links
|
| See Rolfsen, pages 66, 67 |
|
9. The Hawaiian Earring
|
| Hatcher, Example 1.25, page 49; also page 63 |
| Cannon and Conner, The combinatorial structure of the Hawaiian earring group
Topology and its Applications, 106 (2000), pages 225–271
The proof on pages 236–7 that the double of the cone on the Hawaiian Earring is not simply connected is accessible and interesting. |
|
10. The Long Line
|
| Steen and Seebach, Counterexamples in Topology, Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995 |
| Set Theoretic Topology (notes by Greenwood and Cao of the University of Auckland) |
|
11. Topological Dimension and the Menger sponge
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12. Collapsibility is a stronger version of
contractibility for cell complexes. Examples of contractible but
not collapsible spaces are the Dunce
Hat and
the House
With Two Rooms. Zeeman's Conjecture is a conjecture about
collapsibility that is equivalent to the well-known Poincare Conjecture.
|
13. A surface is a topological space which is
locally like 2-dimensional space; i.e., it has an open cover by
open sets that are homeomorphic to open sets in the plane. E.g., a
sphere, a Klein bottle and a torus are all compact surfaces. There
is a classification of these, and a very
nice proof due to Conway
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14. There are higher dimensional analogues of the
fundamental group called homotopy groups. They are
surprisingly complicated, even for simple spaces like
spheres. Doing non-trivial calculations is way beyond this course,
but the basic definitions and properties (such as the fact that
the higher homotopy groups are always abelian, unlike the
fundamental group), and a description, without proof, of the Hopf
Fibration, which is a generator of the second homotopy group of a
3-dimensional sphere, is quite feasible. You could also say a
little about what is known about homotopy groups of spheres. See,
for example, Section 4.1 in Hatcher's book.
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