Math 634

Fall 2020

• Syllabus.
• Textbook: Hatcher, Algebraic Topology.
• Lecture 1: Welcome.
  notes, worksheet.
• Lecture 2: What spaces are we interested in?
  notes, worksheet.
• Reading 1 (due 10/4):
  Chapter 0: pages 1–14 (skip “The Homotopy Extension Property”) and also look at the exercises.
  Chapter 1: pages 21–28 (stop at “The Fundamental Group of the Circle”)
• Homework 1 (due 10/5): pdf, tex. Solutions: pdf, tex.
• Lecture 3: Homotopy.
  notes, worksheet.
• Lecture 4: Homotopy equivalence.
  notes, worksheet.
• Lecture 5: The fundamental group.
  notes, worksheet.
• Reading 2 (due 10/11): Finish §1.1.
• Homework 2 (due 10/12):
  Chapter 0 #1, 12, 18; §1.1 #5, 6.
  Solutions: pdf, tex.
• Lecture 6: The fundamental group of the circle, I.
  notes, worksheet.
• Lecture 7: The fundamental group of the circle, II.
  notes, worksheet.
• Lecture 8: First fruits of π₁(S¹).
  notes, worksheet.
• No reading due 10/18.
• Homework 3 (due 10/19): pdf, tex. Solutions: pdf, tex.
• Lecture 9: Induced maps.
  notes, worksheet.
• Lecture 10: π₁(X×Y) and π₁(Sⁿ).
  notes, worksheet.
• Lecture 11: Free products, statement of van Kampen’s theorem.
  notes, worksheet.
• Reading 3 (due 10/25): §1.2.
• Homework 4 (due 10/26): pdf, tex. Solutions: pdf, tex.
• Lecture 12: Proof of van Kampen’s theorem.
  notes, no worksheet.
• Midterm 1 (due 10/30): pdf, tex. Solutions: pdf, tex.
• Lecture 13: Attaching discs; projective space.
  notes, worksheet.
• Lecture 14: Some knots via van Kampen’s theorem.
  notes, no worksheet.
• Reading 4 (due 11/1): §1.3. This is long, but there’s no homework this week.
• Lecture 15: Covering spaces.
  notes, worksheet.
• Lecture 16: Covering spaces, continued.
  notes, no worksheet.
• Lecture 17: Lifting criterion.
  notes, worksheet.
• No reading for 11/8.
• Lecture 18: Deck transformations.
  notes, no worksheet.
• Homework 5 (due 11/11):
  §1.2 #9 (take g = 2 and h = k = 1 if you want).
  §1.3 #8, 12.
  Draw a simply-connected cover of each of the four spaces from Homework 3 #4. Hint: They all have π₁ = ℤ, so the fiber will have to be ℤ.
  Solutions: pdf, tex.
• Lecture 19: Universal covers.
  notes, worksheet.
• Lecture 20: Homology.
  notes, worksheet.
• Reading 5 (due 11/15): Chapter 2; stop at “Exact Sequences and Excision” on page 113.
• Lecture 21: Induced maps.
  notes, worksheet.
• Lecture 22: Relative homology.
  notes, no worksheet.
• Midterm 2 (due 11/20): pdf, tex. Solutions: pdf, tex.
• Lecture 23: Long exact sequence of a pair: statement, examples.
  notes, worksheet.
• Reading 6 (due 11/22): Continue reading Chapter 2; stop at “Excision” on page 119.
• Lecture 24: Long exact sequence of a pair: proof.
  notes, worksheet.
• Lecture 25: Long exact sequence of a pair, continued.
  notes, worksheet.
• Reading 7 (due 11/29): “Cellular Homology” starting on page 137. Some details are important and some are not, so feel free to skim, but be sure to catch Examples 2.36 and 2.42.
• Homework 6 (due 11/30):
  §2.1 #12, 15, 20. Challenge: 21.
  Solutions: pdf, tex.
• Lecture 26: Miscellaneous.
  notes, no worksheet.
• Lecture 27: Homology of surfaces.
  notes, worksheet.
• Lecture 28: Cellular homology.
  notes, no worksheet.
• Final Exam (due 12/11): pdf, tex. Solutions: pdf, tex.