Math 525, Topology (Spring 2018)

### Instructor: Pierre Albin

### Office: Illini Hall 237

### Email: palbin [at] illinois .edu

### Lectures: TR 9:30-10:50 Altgeld 347

### Office Hours: W 9:30-11:30, 2:00-3:00

### Web page:
https://math.uiuc.edu/~palbin/Math525.Spring2018/home.html

### Supplementary Texts:

Bredon, *Topology and Geometry*

May, *A concise course in Algebraic Topology *, available on the
author's webpage

### Assignments:
There will be homework each week.You are allowed (and encouraged) to work
with other students while trying to understand the homework problems.
However, the homework that you hand in should be your work alone.
Late homework will not be accepted, but the lowest score will be dropped.

### Exams:
There will be two midterms (in lecture: Feb 22 and Mar 29)
and a final (May 11, 8am, 347 Altgeld).

### Holidays:
This semester we will not have classes on:

**Spring break** March 19-23

### Grading percentages:

Problem sets (30%)

Midterms (35%)

Final Exam (35%)

We will drop the lowest hw score and we will drop the lowest midterm score.

### Description:

This is a first course in Algebraic Topology.

Topology is the study of those properties of a space that are
unchanged by a continuous transformation.
At first this was studied by assigning integers to spaces that
did not change under a reversible continuous map or homeomorphism.
Thus Riemann classified surfaces by the minimum number of simple
closed curves along which to cut in order to obtain a simple
presentation of the surface (namely, twice the `genus' of the
surface), and much later Rado showed that two connected closed
oriented surfaces are homeomorphic if and only if they have the same
genus.
For more complicated spaces we have more sophisticated invariants.
Instead of assigning a number to a space, we might assign a group, a ring,
or a complex of rings to a space in a homeomorphism invariant way.
These richer invariants extend the older numerical ones, e.g., the
group might have its size determined by the numerical invariant as
happens with the fundamental group of a surface and its genus.
In this course we will study some of the earliest algebraic invariants
of topological spaces: the fundamental group, covering spaces, and
homology. Introduced by Poincaré in 1895, these are now
fundamental tools for anyone interested in geometry.