This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Topology II -- Spring 2014

### From Mathematics Is A Science

**MTH 631 - Topology II.** First course in algebraic topology. Homotopy, fundamental group, simplicial homology. PR: MTH550 and MTh630. 3 hours.

- Time and Place: 12:30 pm - 1:45 pm TR 263 Smith Hall.
- Instructor: Peter Saveliev (call me Peter)
- Office: Smith Hall 325
- Office Hours: MW 2:30 - 4:30, or by appointment
- Office Phone: x4639
- E-mail: saveliev@marshall.edu
- Class Web-Page: math02.com
- Prerequisites: Topology I -- Fall 2013
- Text: Topology Illustrated (online draft), specific chapters linked below
- Goals: Introduction to point-set and algebraic topology
- Grade Breakdown:
- homework and quizzes: 40%
- midterm: 25%
- final exam: 35%

- Letter Grades: A: 90-100, B: 80-89, C: 70-79, D: 60-69, F: <60

See also Course policy.

## 1 Lectures

They will appear here as the course progresses.

A better organized version of the content of the lectures appears in one of the chapters below within a week. The most current material is marked with $\star$.

## 2 Chapters

This is the text that will be followed. The chapters will be updated or rewritten, sometimes significantly, as the course progresses. Read them.

**Chapter 4. Topological spaces**

**Chapter 5. Maps**

- Homotopy
- Cell maps
- Homology theory $\star$
- Euler and Lefschetz numbers
- Homology of parametric complexes

**Chapter 6. Advanced topics**

- Short overview
- Singular complexes
- Homology as a group
- Properties of homology groups
- Spaces vs subspaces: relative homology
- Exact sequences
- Cochain complexes and cohomology
- Homology vs homology of the complement: Alexander duality
- Homology vs cohomology in manifolds: Poincare duality
- Cup product and cap product

- Appendices:

- Commutative diagrams
- Computing homology
- Fundamental group
- Homology and algebra
- Homology as a vector space
- More about manifolds
- Mobius band

## 3 Notes

- Topology II -- Spring 2014 -- midterm: March.
- Topology II -- Spring 2014 -- final exam: Tuesday, May 6, 12:45-2:45. Note: If the two-hour time allowance results in a conflict in exam times, it is the student’s responsibility to notify the professor of the later course and to reschedule the later exam.

Related texts: