Topology (Math 1410)

Course summary

This is an undergraduate course in topology, which studies properties of geometric objects. In this course we will learn

• basic principles of topology (e.g. topological invariants, point-set notions);
• proofs of famous classical results (Heine-Borel, van Kampen, Brouwer fixed point, classification of surfaces);
• connections of topology to other areas of math (free groups, fundamental theorem of algebra) and to the world (robots, hanging pictures, car shades);
• how to calculate topological invariants such as Euler number and fundamental group;
• how to read/write proofs and critique arguments, especially those that are based on pictures/visualization.

Textbook

Armstrong, Basic topology

Topic schedule

The lectures for the course were pre-recorded, and are available on YouTube. Start here.

• Week 1: Introduction. (Armstrong Chapter 1, Sections 1-2)
  • Lecture 1: Euler’s theorem, graphs slides
  • Lecture 2: proof of Euler’s theorem, topological equivalence slides
• Week 2: Topological spaces. (Armstrong Chapter 1)
  • Lecture 3: more topological equivalence, topological spaces slides
  • Lecture 4: metric spaces, continuity, topologies on ℝ slides
  • Lecture 5: surfaces and their classification slides
• Week 3: Point-set topology. (Armstrong Chapter 2-3)
  • Lecture 6: Closed sets and limit points, basis for a topology slides
  • Lecture 7: Product topology, closed and bounded subsets, compactness slides
  • Lecture 8: Properties of compact spaces, Heine-Borel slides
• Week 4: Compactness and connectedness. (Armstrong Chapter 3)
  • Lecture 9: Heine-Borel, compactness of [0,1], connectedness slides
  • Lecture 10: Connectedness, connectedness and continuity slides
  • Lecture 11: Path connectedness, quotient spaces slides
• Week 5: Quotient spaces. (Armstrong Chapter 4)
  • Lecture 12: Quotient spaces, quotient maps, topological groups slides
  • Lecture 13: Quotient maps, identifying quotient spaces, projective space, group actions slides
  • Lecture 14: Cell complexes, orbit spaces slides
• Week 6: Fundamental group. (Armstrong Chapter 5)
  • Lecture 15: more orbit spaces slides
  • Lecture 16: finishing orbit spaces, fundamental group intro, intuitive computations slides
  • Lecture 17: Rigorous definition of the fundamental group slides
• Week 7:
  • Lecture 18: Homotopy and the fundamental group slides
  • Lecture 19: Fundamental group of spheres slides
  • Midterm
• Week 8:
  • Lecture 20: Fundamental group of the circle, functoriality of π_1 slides
  • Lecture 21: Lifting proposition, Brouwer fixed point theorem slides
  • Lecture 22: Homotopy equivalences, fundamental theorem of algebra slides
• Week 9: Triangulations and the fundamental group (Armstrong Chapter 6)
  • Lecture 23: Simplicial complexes, triangulations, edge groups slides
  • Lecture 24: Computing edge groups slides
  • Lecture 25: Presentation for edge groups, free groups and free products slides
• Week 10: van Kampen for triangulated spaces
  • Lecture 26: van Kampen theorem, simplicial maps slides
  • Lecture 27: proof of van Kampen, simplicial approximation slides
  • Lecture 28: Barycentric subdivision, proof of edge group theorem slides
• Week 11: Classification of surfaces. (Armstrong Chapter 7)
  • Lecture 29: Classification of surfaces slides
  • Lecture 30: Triangulating surfaces slides
  • Lecture 31: Combinatorial spheres, surface surgery slides
• Week 12: (Thanksgiving week)
  • Lecture 32: Proof of the classification theorem slides
  • Lecture 33: Surfaces with boundary (video, see slides from Lecture 32)

• Week 13: Reading period

• Week 14: Exam week (submit final project)

Discussion sessions

Discussions sessions happen Monday 5-6pm and Tuesday 12-1pm. You only need to attend one (they are the same). This is an opportunity to get live practice and feedback. It is recommended that you are up-to-date on lectures and the reading before the discussion session.

• Week 2 (9/14): cutting and pasting with surfaces

• Week 3 (9/21): topologies on a set with 3 elements

• Week 4 (9/28): closure and complement on subsets of ℝ

• Week 5 (10/5): Conway-Coxeter Frieze patterns

• Week 6 (10/12): no discussion (university holiday)

• Week 7 (10/19): Midterm review

• Week 8 (10/26): Hexiamonds

• Week 9 (11/2): no discussion (election day)

• Week 10 (11/9): Sprouts

• Week 11 (11/16): 4-color theorem

• Week 12 (11/23): magic car shade

Homework assignments

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9