# Math 472: Introduction to Topology

## Colorado State University, Fall 2022

**Instructor:** Henry Adams
**Email:** henry dot adams at colostate dot edu
**Office:** Weber 120
**Office Hours:** Mondays at 3pm in Clark C362, or by appointment.

**Lectures:** MWF 2:00-2:50pm in Clark C363
**Textbook:** *Basic Topology* by M. A. Armstrong.
This book is freely available as a PDF to CSU students if you login on the CSU library webpage.

**Overview:** This course is an introduction to topology. Topics covered include Euler's theorem, topologies on sets, continuous functions, homeomorphisms, sequences and convergence, metric spaces, compactness, connectedness, surfaces, manifolds, identification spaces, homotopy equivalences, the fundamental group, homotopy groups, Borsuk-Ulam theorem, Brouwer's fixed-point theorem, Morse theory, homology.

**Goals:** Students will become fluent with the main ideas and the language of topology, and will be able to communicate these ideas to others.

**Syllabus:** Here is the course syllabus.

## Videos

The following videos are from Fall 2022.*What is an i-dimensional hole in a space?*

## Notes

Scans of Henry's lecture notes from 2016.## Schedule

Date |
Class Topic |
Remark |

Aug 22 | Introduction and course overview | |

Aug 24 | What is an i-dimensional hole in a space? |
[Video] |

Aug 26 | A spontaneous introduction to knot theory | |

Aug 29 | Recap of metric spaces | |

Aug 31 | Open sets in metric spaces | |

Sept 2 | Topological spaces | |

Sept 5 | Holiday - no class! | |

Sept 7 | Topological spaces; Stereographic projection | Last day to drop or change grading option |

Sept 9 | Continuous functions; Spontaneous introduction to category theory | |

Sept 12 | Continuous functions | |

Sept 14 | Closed sets | |

Sept 16 | The closure, interior, and boundary of a set | Spontaneous discussion on applying to graduate school |

Sept 19 | Homeomorphisms | |

Sept 21 | Homeomorphisms | |

Sept 23 | Alex Elchesen: Category theory | |

Sept 26 | Michael Moy: Compact spaces | |

Sept 28 | Michael Moy: Compact spaces | |

Sept 30 | Alex Elchesen: Category theory; Functors | |

Oct 3 | Homework A, especially the boundary of a set | |

Oct 5 | Connected spaces | |

Oct 7 | Path-connected spaces | |

Oct 10 | Fundamental group | |

Oct 12 | Homotopic maps | |

Oct 14 | Homotopic maps | |

Oct 17 | Homotopy classes of maps form an equivalence relation | |

Oct 19 | Fundamental group | |

Oct 21 | Fundamental group | |

Oct 24 | Michael Moy: Fundamental group: Functoriality, Brouwer's fixed point theorem | |

Oct 26 | Fundamental group: Examples | |

Oct 28 | Fundamental group: Examples | |

Oct 31 | Free groups | |

Nov 2 | Group presentations | Dinner at Henry's at 5pm! |

Nov 4 | Fundamental group and the Projective plane | Video of genus-2 torus |

Nov 7 | Homotopy groups | |

Nov 9 | Evasion Paths in Mobile Sensor Networks | |

Nov 11 | Homotopy equivalences between spaces | |

Nov 14 | Homotopy equivalences between spaces | |

Nov 16 | An introduction to applied topology | |

Nov 18 | An introduction to applied topology | |

Fall Recess, Nov 21-25 | ||

Nov 28 | Homology | |

Nov 30 | Homology | |

Dec 2 | Alex Elchesen: Morse theory, Manifolds | |

Dec 5 | Homology | |

Dec 7 | Brouwer's fixed point theorem, Invariance of domain | |

Math department poster session, Thursday, December 8, 9-11am, LSC Main Ballroom | ||

Dec 9 | Student project |

## Homework

Homework A (LaTeX Source) is due Wednesday, October 5.Homework B (LaTeX Source) is due Wednesday, November 2.

Homework C (LaTeX Source) is due Wednesday, December 7.

Here are the Poster Session Guidelines.