Math 635 Winter 2024

Instructor: Robert Lipshitz

Basic course information

Time: MWF 11:00-11:50 a.m.
Place: Tykeson 260
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: M 2:00-3:00, W 4:00 - 5:00. Subject to change.
Teaching assistant: Philip Thomas
TA office hours: Thursday 4:00-5:00 p.m.
Final exam: per Registrar's final exam schedule

Prerequisites

Math 634.

Description and goals

This course continues where Math 634 left off, studying properties, computations, and applications of singular homology and cohomology, culminating with the proof of Poincaré duality for the cohomology of manifolds.

Particular "learning objects" for the course include:

  • Being able to compute and apply homology and cohomology, including the ring structure on cohomology.
  • Understanding and applying the notion of homology orientations and Poincaré duality.

Policies

Grading

Homework 32%
Providing Peer Feedback on Homework 5%
Midterm 23%
Final exam 40%

 

Reason-Neutral Missed Homework and Exam Policy

The University of Oregon now requires reason-neutral policies for missed work or deadlines. That is, students who miss an exam because of an illness or family crises must be treated the same as students who simply choose not to show up to the exam. So:

  • The lowest homework score will be dropped, to account for illnesses or other temporary obstacles to participating in the course.
  • If you have a conflict with the midterm exam and contact me well in advance of it, you will have the opportunity to take the midterm early, without penalty.
  • If you miss the midterm exam you will be given the opportunity to take an oral midterm exam as a make-up, by the end of week 7.
  • If you miss the final exam and have a passing score in the rest of the course you will receive an Incomplete in the course, and be allowed to take a makeup final exam during the first week of the Spring Quarter to resolve the Incomplete. If you do not take the makeup exam will receive a 0 on the exam.
  • If you miss the final exam and do not have a passing score in the rest of the course, you will fail the course.
  • In all cases of conflicts or missed exams, you should contact me as soon as practically possible to make arrangements.

Continuity Plan in the case of a disruption to grading

Each homework assignment is worth 4 percent of the grade (since one is dropped). If the course is without GTF grader for some period of time, homework sets during that time will not be graded or counted in the grade. For each ungraded homework assignment, the midterm exam will count for 2 percent more and the final exam will count for 2 percent more.

Textbook

The textbook for the class is Algebraic Topology by Allen Hatcher. It is available online or in print. Some other useful textbooks include:

  • Glen Bredon, Topology and Geometry.
  • William Massey, Algebraic Topology: An Introduction and A Basic Course in Algebraic Topology.
  • James Munkres, Elements of Algebraic Topology and Topology.
  • Edwin Spanier, Algebraic Topology.
  • J. Peter May, A Concise Course in Algebraic Topology. Apparently available for download from Peter May's website.
  • Anatoly Fomenko and Dmitry Fuchs, Homotopical Topology. Available for download via SpringerLink, through the library.
  • Jean-Claude Hausmann, Mod Two Homology and Cohomology. Available here and via SpringerLink, through the library.

Students are expected to read the sections in the textbook before coming to class each day. The relevant sections are listed in the syllabus below.

Homework

Homework is due on Friday at 11:59 p.m. most weeks, by upload to Canvas. You may discuss the problems with other students, but must write up your solutions to the problems by yourself. Any resources you use other than the textbook must be cited in your homework. You may not use electronic resources (e.g., Google) other than the textbook and recommended textbook. Failure to follow this policy constitutes cheating; if you are caught cheating on the homework you will receive a 0 for the homework portion of the class and will be reported to the administration.

In addition to being graded by the TA, homework will receive double-blind peer feedback. Each week, you will read and comment on a few solutions written by two other students. You are responsible for submitting your comments on the homework by midnight on Wednesday of the week after the homework is due. More details will be discussed in class.

For homework feedback to be double-blind, please do not include your name on your solutions.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Assignments, handouts, and other resources

Homework

Handouts

Schedule

Sections refer to Hatcher's book. + denotes material beyond that covered in these sections of the textbook.

Updated schedule:

Week Date Topic Textbook pages / sections
1 1/8 Review: modules, free modules, exact sequences, chain complexes, chain maps, etc. +
  1/10 Review: classification of modules over a PID, computing homology. +
  1/12 Review: Hom and tensor product p. 215, +
2 1/15 Martin Luther King day (no class)
  1/17 Snow day; no class.  
  1/19 Simplicial complexes, simplicial homology and cohomology pp. 97 - 107, +
3 1/22 Singular homology and cohomology. Homework 1 due. pp. 108 - 110, +
  1/24 Homotopy invariance pp. 110 - 113, +
  1/26 Long exact sequence for a pair. Homework 2 due. pp. 113 - 118
4 1/29 Excision. pp. 119 - 126
  1/31 More on the proof of excision.  
  2/2 Computations using the formal properties of homology. +
5 2/5 More computations in homology. Universal coefficient theorems. Homework 3 due.  
  2/7 Midterm exam
  2/9 Classical applications (Jordan curve theorem, invariance of domain, etc.). pp. 169 - 176
6 2/12 Mayer-Vietoris theorem, Universal Coefficients Theorems. Homework 4 due. pp. 149 - 153, 190 - 197, 261 - 267
  2/14 More on the Universal Coefficient Theorems  
  2/16 Euler characteristic, degree. pp. 134 - 147
7 2/19 Cellular (co)homology. Homework 5 due.  
  2/21 More cellular (co)homology  
  2/23 More cellular (co)homology  
8 2/26 Geometry of Tor (guest lecture). Homework 6 due. +
  2/28 Simplicial approximation and Lefschetz fixed point theorem pp. 177 - 184
  3/1 Axioms for (co)homology. pp. 160 - 162
9 3/4 Cup product and the cohomology ring. Homework 7 due. pp. 206 - 214
  3/6 More cohomology ring  
  3/8 More cohomology ring.  
10 3/11 Transfer homomorphism and the cohomology of RPn. Homework 8 due. pp. 321 - 322, +
  3/13 Cross product, Künneth theorem. pp. 214 - 219
  3/15 More on Künneth theorem. Homework 9 due. pp. §3B, +
11 (636 Wk 1) 4/1 Poincaré duality: statement, first applications pp. 230 - 233, +
  4/3 Orientations. pp. 233 - 239
  4/5 More on orientations.  

 

Original plan:

Week Date Topic Textbook pages / sections
1 1/8 Review: modules, free modules, exact sequences, chain complexes, chain maps, etc. +
  1/10 Review: classification of modules over a PID, computing homology. +
  1/12 Review: Hom and tensor product p. 215, +
2 1/15 Martin Luther King day (no class)
  1/17 Simplicial complexes, simplicial homology and cohomology pp. 97 - 107, +
  1/19 Singular homology and cohomology. Homework 1 due. pp. 108 - 110, +
3 1/22 Homotopy invariance pp. 110 - 113, +
  1/24 Long exact sequence for a pair pp. 113 - 118
  1/26 Excision. Homework 2 due. pp. 119 - 126
4 1/29 Universal coefficient theorems pp. 190 - 197, 261 - 267
  1/31 Euler characteristic. Degrees of maps. pp. 134 - 137, 146 - 147
  2/2 Mayer-Vietoris theorem. Homework 3 due. pp. 149 - 153
5 2/5 Cellular (co)homology pp. 137 - 146
  2/7 Midterm exam
  2/9 Classical applications (Jordan curve theorem, invariance of domain, etc.). Homework 4 due. pp. 169 - 176
6 2/12 More cellular (co)homology  
  2/14 Simplicial approximation and Lefschetz fixed point theorem pp. 177 - 184
  2/16 Axioms for (co)homology. Homework 5 due. pp. 160 - 162
7 2/19 Cup product and the cohomology ring pp. 206 - 214
  2/21 More cohomology ring  
  2/23 More cohomology ring. Homework 6 due.  
8 2/26 Transfer homomorphism and the cohomology of RPn pp. 321 - 322, +
  2/28 Cross product pp. 214 - 219
  3/1 Künneth theorem. Homework 7 due. pp. §3B, +
9 3/4 More on Künneth theorem  
  3/6 Poincaré duality: statement, first applications pp. 230 - 233, +
  3/8 Orientations. Homework 8 due. pp. 233 - 239
10 3/11 More on orientations, compactly supported cohomology  
  3/13 Proof of Poincaré duality pp. 240 - 252
  3/15 More proof of Poincaré duality. Homework 9 due.