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\textbf{Math 468: Topology} \\
Spring 1999 \\
J. Palmieri
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Consider these three-dimensional figures:
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If, for each shape, you count the numbers $v$, $e$, and $f$ of
vertices, edges, and polygons (a.k.a.~faces), and compute $v - e + f$,
you get 2 for each one.  This is not just a coincidence: each of these
shapes can be deformed continuously into each of the others, and in
fact into a sphere, and a famous theorem of Euler's says that for any
such shape, $v-e+f=2$.

If, on the other hand, you consider this shape (a brick with a
square hole drilled through it):
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\vspace{-0.2in}
then for it, $v-e+f = 0$.  This shape cannot be continuously deformed
into a sphere, which is why you get a different number.

Counting things like vertices, edges, and faces and then considering
expressions like $v-e+f$ is part of \emph{combinatorial topology},
which is the focus of this course.

We will start with Euler's formula, then talk about some fundamental
topological ideas like compactness.  We will move on to vector fields,
the Brouwer fixed point theorem, winding numbers, and the Poincar\'e
index theorem.  Next comes the Jordan curve theorem: every simple
closed curve in the plane divides the plane into two pieces.  This is
notorious for being easy to state, ``obviously'' true, but difficult
to prove.  We will also discuss the classification of surfaces, and
(time permitting) homology, Betti numbers, Euler's formula revisited,
and the Poincar\'e index theorem revisited.

\textsc{Prerequisites}: The students should have a basic background in
advanced calculus, passing familiarity with differential equations,
and it wouldn't hurt to know a bit of group theory.

\textsc{Textbook}: \emph{A Combinatorial Introduction to Topology} by
Michael Henle (the paperback edition of which costs around \$8.00, the
last time I checked).

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