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

Text: Hatcher, Algebraic Topology available on the author's webpage

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.