# Math 531

## Algebraic Topology

## Professor Michael Sullivan

### Spring 2016

The idea in algebraic topology is to associate to a topological space
an algebraic object such as a group or ring. If two topological spaces
are mapped to non isomorphic algebraic objects then they cannot be
homeomorphic. Thus algebraic topology becomes a major tool for the
study of topological spaces, especially manifolds and CW-complexes.
The main tools are the following.

- The fundamental group of a space
*X* with base point *x* is a group (often nonabelian).
It is denoted Π_{1}(*X,x*).
- The higher homotopy groups of (
*X,x*) are abelian groups denoted by Π_{n}(X,x)
for each integer *n* > 1.
- The
*n*-th homology group of a space *X* is an abelian group. There is one for each
non-negative integer *n*. These are computed using coefficients from some other group *G*,
usually the integers. They are denoted H_{n}(*X,G*).
- The n-th cohomology group of a space
*X* is an abelian group. There is one for each
non-negative integer *n*. These are computed using coefficients from some other group *G*,
usually the integers. They are denoted H^{n}(*X,G*).
- There is a natural homomorphism from the homotopy groups to the homology groups over the integers.
For
*n*=1 it is just abelianization.
- The homology and cohomology groups of a space
*X* can be combined to form the cohomology ring of *X*.

Algebraic topology has applications in dynamical systems (including differential equations), mathematical physics,
image processing and computer science. It has also evolved into an independent area of work within algebra.

As you might have summarized algebraic topology is a demanding topic. But this is an introductory course and I
realize most students will be doing or beginning to do research in other fields. Thus, while the material is
harder than that in 530, the amount the of homework is less and any tests will be take home. There will be
an emphasis on connecting the geometric or visual understanding of spaces with the algebraic properties of
their groups.

Syllabus

**Instructor:** Prof. Sullivan. Neckers 385.

**Coordinates:** T&Th 2:00-3:15, Engineering A 210. CRN=27364.

**Material:**We will finish some material on the fundamental group, Π_{1}, from the end of Munkres' textbook *Topology*.
Then we will do a little on the higher homotopy groups, Π_{n}, using
Hatcher's textbook *Algebraic Topology* (free online).
Most of the course will be on homology using Munkres' textbook *Elements of Algebraic Topology*. (This is the only book you will need to buy.)
We will touch on cohomology toward the end.

**Grades:** Homework will be collected about every other week. There will be a takehome final. Grades will be 50% homework, 50% final.

**Prerequisites:** A basic topology course such as Math 530 and some knowledge of group theory and linear algebra at the undergrad level.

**Handouts**

Tietze Transformations
Connected Sums of 3-Manifolds
Knot Groups