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

# Topology I -- Fall 2013

### From Mathematics Is A Science

**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.

## 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 $\star$.

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

## 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 1. Introduction to homology**

- Introduction: Topology in real life
- Homology in Calculus
- Homology as an equivalence relation
- Topology of graphs
- Euler characteristic of graphs
- Homology groups of graphs
- Homology maps of graphs

**Chapter 2. Point-set topology**

- Continuity as accuracy
- From continuity to point-set topology
- Bases of neighborhoods
- Neighborhoods and topologies
- Topological spaces
- Continuous functions
- Topological equivalence: homeomorphisms
- Relative topology

**Chapter 3. Complexes** (algebraization of topology)

- Discretization of the Euclidean space
- Cubical chains
- Chain complex
- Cubical complexes
- Oriented chains
- Euclidization of data
- Simplicial complexes
- Simplicial maps and chain maps

$\star$

**Chapter 4. More topology and complexes**

**Chapter 5. Homology**

- Homology and algebra
- Homology as a vector space
- Homology of cubical complexes
- Examples of homology of cubical complexes
- Euler-Poincare formula
- Homology maps
- Holes and tunnels: Homology in dimension 1
- Voids: Homology in dimension 2
- Homology of balls and spheres
- How to compute Betti numbers
- How to compute homology

**Chapter 6. More...**

## 3 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:

Also