Topology I -- Fall 2013

MTH 430/630 - Topology I. First course in topology. Basics of point-set topology: metric and topological spaces, continuity, connectedness, compactness, products, quotients. Surfaces and simplicial complexes, Euler characteristics. PR: MTH300. 3 hours.

• Time and Place: 3:00 pm - 3:50 pm MWF 516 Smith Hall.
• Instructor: Peter Saveliev (call me Peter)
• Office: Smith Hall 325
• Office Hours: WF 1:30-2:30 and MW 4:00-5:00, or by appointment
• Office Phone: x4639
• E-mail: saveliev@marshall.edu
• Class Web-Page: math02.com
• Prerequisites: Linear algebra
• Text: Applied Topology and Geometry (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.

Lectures

They will appear here as the course progresses.

• Set 1
• Set 2
• Set 3
• Set 4
• Set 5
• Set 6
• Set 7

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 $\star$.

Homework and solutions:

• HW1: Solution 1, Solution 2, Solution 3
• HW2: Solution 1, Solution 3
• HW3: Solution 1, Solution 2, Solution 3
• HW4: Solution 1, Solution 2, Solution 3
• HW5: Solution 1, Solution 2, Solution 3
• HW6: Solution 1, Solution 2, Solution 3
• HW7: Solution 1, Solution 3
• HW8: Solution 1, Solution 2, Solution 3, Solution 4
• HW9: Solution 1, Solution 2, Solution 3, Solution 4
• HW10: Solution 1, Solution 2, Solution 3, Solution 4
• HW11: Solution 1, Solution 2, Solution 3, Solution 4
• HW12: Solution 1, Solution 2, Solution 3, Solution 4

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 1. Introduction to homology

1. Introduction: Topology in real life
2. Homology in Calculus
3. Homology as an equivalence relation
4. Topology of graphs
5. Euler characteristic of graphs
6. Homology groups of graphs
7. Homology maps of graphs

Chapter 2. Point-set topology

1. Continuity as accuracy
2. From continuity to point-set topology
3. Bases of neighborhoods
4. Neighborhoods and topologies
5. Topological spaces
6. Continuous functions
7. Topological equivalence: homeomorphisms
8. Relative topology

Chapter 3. Complexes (algebraization of topology)

1. Discretization of the Euclidean space
2. Cubical chains
3. Chain complex
4. Cubical complexes
5. Oriented chains
6. Euclidization of data
7. Simplicial complexes
8. Simplicial maps and chain maps

$\star$

• Point-set topology: exercises
• Introductory algebraic topology: review exercises

• Rules of proof writing

---

Chapter 4. More topology and complexes

1. Properties of topological spaces
2. Quotients
3. Cell complexes
4. Products
5. Homotopy
6. Singular complexes

Chapter 5. Homology

1. Homology and algebra
2. Homology as a vector space
3. Homology of cubical complexes
4. Examples of homology of cubical complexes
5. Euler-Poincare formula
6. Homology maps
7. Holes and tunnels: Homology in dimension 1
8. Voids: Homology in dimension 2
9. Homology of balls and spheres
10. How to compute Betti numbers
11. How to compute homology

Chapter 6. More...

Notes

Topology I -- Fall 2013 -- midterm: Friday, October 18, includes all up to "Homeomorphisms", excluding "optional".

Topology I -- Fall 2013 -- final exam: Monday, December 9, 3-5 pm, same place.

Related texts:

• Introduction to Topology by Gamelin and Greene
• Topology of Surfaces by Kinsey.

Also

• Point-set topology: course
• Topology 2: course