## Math 821 (Algebraic Topology) Spring 2018

### General Information

Instructor: Jeremy Martin
618 Snow Hall, 864-7114
The best way to contact me is by e-mail:

KU course number: 67998
Syllabus (1/9/18; subject to change)
Meeting times: MWF, 1:00-1:50pm, 564 Snow
Office hours: Wed 10-11am, Fri 2-3pm, or by appointment
Final exam: Monday 5/7/18, 10:30am-1:00pm

Prerequisites: Math 790-791 and Math 820, or permission of instructor. More details in the syllabus.

Blackboard: I will use Blackboard to post solution sets and lecture notes for the use of Math 821 students only. These materials must not be redistributed.

Textbook: Algebraic Topology by Allen Hatcher (Cambridge U. Press, 2002). Available as a free download from the author's website. If you choose to use the downloaded version, do not print out a copy on the department printer! You can also buy a paperback copy for around \$40. See the syllabus for a list of secondary texts that may be helpful.

### Schedule

#### Chapter 0: Basic Geometric Notions

• Wed 1/17: Course logistics; quotient maps and cell complexes
• Fri 1/19: Cell complex examples: polytopes, simplicial complexes, projective spaces, Grassmannians
• Mon 1/22: Homotopy and homotopy equivalence
• Wed 1/24: Two homotopy equivalence criteria; examples; the mapping cylinder
• Fri 1/26: The homotopy extension property for pairs

#### Chapter 1: The Fundamental Group

• Mon 1/29: Paths, path homotopies, and the fundamental group
• Wed 1/31: Induced homomorphisms
• Fri 2/2: $$\pi_1(S^1)=\mathbb{Z}$$ and some consequences (Brouwer)
• Mon 2/5: FTA and Borsuk-Ulam from $$\pi_1(S^1)=\mathbb{Z}$$; intro to Van Kampen's Theorem
• Wed 2/7: Proof of Van Kampen's Theorem
• Fri 2/9: Applications of Van Kampen's Theorem
• Mon 2/12: More applications, notably 2-dimensional cell complexes
• Wed 2/14: Covering spaces: definition, examples
• Fri 2/16: Lifting paths and homotopies
• Mon 2/19: More lifting properties; construction of universal covering space
• Wed 2/21: Universal covering spaces; deck transformations
• Fri 2/23: Deck transformations; covering spaces via group actions and quotients

#### Chapter 2: Homology

• Mon 2/26: Introduction to homology
• Wed 2/8: Simplicial homology
• Fri 3/2: $$\Delta$$-complexes; examples
• Mon 3/5: Singular homology: definition, basic properties
• Wed 3/7: Homotopy invariance of singular homology
• Fri 3/9: Relative homology; exact sequences
• Mon 3/12: Excision; long exact sequence for quotient spaces; homology groups of spheres
• Wed 3/14: Equivalence of simplicial and singular homology
• Fri 3/16: Cellular homology; projective spaces and Grassmannians
• Mon 3/26: Degrees of maps $$S^n\to S^n$$ (theory)
• Wed 3/28: Degrees in practice; examples of cellular homology computations
• Fri 3/30: Homology with coefficients; Euler characteristic
• Mon 4/2: The Mayer-Vietoris sequence
• Wed 4/4: Topology and combinatorics (Stanley-Reisner theory)
• Fri 4/6: The Borsuk-Ulam Theorem

#### Chapter 3: Cohomology

• Mon 4/9: Introduction to cohomology (with motivation from vector calculus)
• Wed 4/11: Pictures of cohomology in low dimension
• Fri 4/13: The cup product
• Mon 4/16: Explicit cup products; Künneth formulas
• Wed 4/18: Cohomology of tori
• Fri 4/20: Proof of the universal coefficient theorem (guest speaker Jacob Hegna)
• Mon 4/23: Poincaré duality I
• Wed 4/25: Poincaré duality II

### Problem Sets

Problem sets are due in class approximately every two weeks. I will post problems on the website at least a week in advance. Problem sets are subject to change before that time, so make sure you have the final version.

• You are encouraged to collaborate with other students, but you must write up the problems by yourself and acknowledge all collaborators. You should not consult outside sources such as the Internet.
• You must submit typed solutions using LaTeX. If possible, figures should be typeset as well, using a package such as TikZ or Ipe (see below). Here is a header file with common macros (1/9/18). LaTeX files posted here will require that header to compile.
• Late homework will not be accepted.
• Solution sets will be posted on Blackboard. They are for the private use of Math 821 students only and must not be redistributed.