Math 526, Topology II (Fall 2018)

### Instructor: Pierre Albin

### Office: Illini Hall 237

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

### Lectures: TR 2-3:20 Altgeld 441

### Office Hours: TBA

### Web page:
https://faculty.math.illinois.edu/~palbin/Math526.Fall2018/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.

### Holidays:
Classes begin on August 27 and end on December 12.
There will be no classes on:

**Labor Day**, September 3

**Thanksgiving break**, November 19 - November 23

### Grading percentages:

Problem sets (100%)

### Description:

Poincaré's initial *analysis situs* paper had as its
main objective proving an important duality between the dimensions
of homology groups of complementary degree. The study of duality in
homology naturally led to the definition and study of cohomology
groups. These groups have additional structure over the homology
groups, namely there is a graded multiplication of cohomology classes
that turns the cohomology groups into a ring. We will study this cup
product and establish Poincaré duality.

The other main topic of this course is higher homotopy theory. Whereas
Math 525 studied the homotopy group of pointed maps from the
circle into a topological space (the fundamental group), in this
course we will study homotopy classes of maps from higher
dimensional spheres. These groups are always Abelian, but are almost
always impossible to compute. The techniques invented to study them
are at the center of contemporary algebraic topology.