(A quick and dirty
description of the course)

Renzo Cavalieri 


MWF 1 – 1:50 pm 


ENGRG E 106 

Textbook 
Basic Topology (Armstrong) 
Springer 


A
PHILOSOPHICAL DISCUSSION
Topology lies at the heart of ANY branch of modern geometry. So let me take a slight
detour and chat a little bit of history first.
I like to think that modern
geometry is born in 1872, when Felix Klein wrote the Erlangen Programme, a mathematical
manifesto that addresses a
philosophical question: what is geometry?
The answer Klein proposes is the following: a geometry is the study of properties of shapes and
spaces  properties that are invariant under a group of transformations. What this means is that, before we make
any statement, we must decide two things:
·
what kind of objects we consider ``shapes and spaces''.
·
what is the group of transformations, i.e. what is our
notion of ``being equal''.
This also means that there is
not a unique geometry, and so yes, it is correct to say that two parallel lines
never meet if you are doing euclidean geometry, but it is not if you are doing
projective geometry. And if you are working projectively over the complex
numbers, then lines are really spheres, but two of them still meet transversally
at one point (huh?).
As you see, geometry is
escaping the boundaries of our ``common sense intuition'', and it will more and
more. So it is our job of mathematicians to try and really understand what is
going on  i.e. to abstract our intuitive geometric notions to a more general
context.
Topology is a fun branch of geometry to
study in order to accomplish such a goal. In fact, in topology, Klein's group
of transformations is simply huge! Two
geometric shapes are considered equal if there are two bijective, continuous functions inverse to each
other between them. This means, for example, that a triangle, a square and a
circle are all the same, because they can be stretched one into another. But a
circle can never be stretched into an infinite straight line, for example. Some
people like to call topology ``rubber geometry'', in the sense that it studies geometric properties of objects
that are made of an extremely stretchable and bendable rubber. Even searching
for properties to study is a chore to begin with! (lengths, areas, angles,
number of sides...all goes topsy turvey if a triangle is the same as a circle).
Properties that remain the same under such a huge flexibility are called topological
invariants.
Now, if this made little sense, don’t get scared, but please read on.
A
MORE DOWN TO EARTH DESCRIPTION OF THE COURSE
Over the course of the semester, I would like to present some basic
concepts in both point set and algebraic topology.
We will start of course, by redefining the notion of continuous
functions. Of course this new notion will coincide with the usual concept you
all know from calculus, but it will be a lot more general. In particular, it
will not depend on any notion of distance on our spaces (so away with epsilon
and deltas, by golly!), but only on the notion of open sets (whatever they might
be).
Once we understand continuity, we will be studying continuous functions
between various spaces, both familiar and unfamiliar. Two spaces will be
considered the same (the technical term will be homeomorphic) if you can find a pair
of continuous inverses between them.
And now comes the crucial question. If somebody hands you two spaces, can
you tell if they are homeomorphic or not? This can be a very difficult problem,
because space that “look” very different might actually be
homeomorpic and viceversa. So we have to be clever, and hunt for properties of
spaces that DO NOT
change when two spaces are homeomorphic. These properties are called topological
invariants.
Some such properties are very down to earth (is the space made up of only one
piece or more pieces, can you “walk” from any point in the space to
any other point in the space). But
the most useful ones are quite sophisticated, and it will take us the rest of
the semester to get familiar with a few of them.
Now, if somebody has already vaguely heard about topology, the topics that
I would like to cover are:
1.
Continuity and Homeomorphisms: what is a topological
space?
2.
The first topological invariants: connectedness and path
connectedness.
3.
A substantially more sophisticated invariant:
compactness.
4.
The topology of compact surfaces, and some more fun
invariants.
5.
A bite of algebraic topology: the fundamental group and
(time permitting) a little theory of covering spaces.