Lectures: Tue, Th 9am-10:20am, room 380D
Professor: Eleny Ionel, office 383L, ionel "at" math.stanford.edu
Office Hours: TBA and by appointment.
Course Description: This course is a graduate level course on algebraic topology, the first quarter of the Math 215 sequence. Topics covered include: fundamental group and covering spaces, basics of homotopy theory, homology and cohomology (simplicial, singular, cellular), products, introduction to topological manifolds, orientations, Poincare duality.
Prerequisites: Point set topology and abstract algebra, at the level of Math 144 and 120. (Undergraduates require instructor permission to enroll.) Best taken along with Math 210A which develops the algebraic theory while our focus is primarily on the topological applications.
Familiarity with some basic algebraic notions such as group actions, tensor products (of vector spaces) and modules while not strictly necessary, is certainly helpful as we go through them very fast. Similarly, prior exposure to some basic topics covered in undergraduate topology and geometry classes (eg Math 147) while not strictly necessary, could be quite helpful eg for intuition and motivation.
Textbook: Allen Hatcher, Algebraic Topology, available online
here. We will cover roughly Chapters 1-3 (with Chapter 0 as needed). We may cover some topics in a different order than in Hatcher, eg we start introducing categorial perspectives much earlier (but not as early as in some other references).
Students are expected to read the relevant sections of the textbook (listed below) and any handouts (posted on Canvas) before coming to class each day.
Other recommended references (for alternative perspectives and exposition) 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.
- Peter May, A Concise Course in Algebraic Topology, available from Peter May's website.
- Tammo tom Dieck, Algebraic Topology.
- Anatoly Fomenko and Dmitry Fuchs, Homotopical Topology.
Tentative Schedule: (which may be adjusted as the quarter goes on)
- Week 1: Fundamental group and induced homomorphisms; Fundamental group of circle and applications, eg Brouwer Fixed Point Thm and Borsuk Uhlam Thm (§1.1 and p. 1-14 in Hatcher; see also Basic notions handout).
- Week 2:
Van Kampen Thm; Van Kampen Thm and CW complexes (§1.2, §1.A and p 14-17 in Hatcher; see also Homotopy extension property handout)
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Week 3: Covering spaces, action on fibers, universal cover, deck transformations, classification of covering spaces (§1.3; see also Groups actions handout)
- Week 4: Simplicial homology; Singular homology; Homotopy equivalence (§2.1),
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Week 5: Excision and Mayer-Vietoris sequences; Refinement and barycentric subdivision; Computing homology and cellular homology (§2.2);
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Week 6:
Geometric applications: degree, Jordan Thm and Brouwer Thm, simplicial approximation and Lefschetz Fixed Point Thm (§2.C); Homology and the fundamental group (§2.A)
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Week 7:
Axiomatic approach (§2.3) Note: NO CLASSES Nov 3 (Democracy Day)
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Week 8:
Cohomology Groups; Universal Coefficients Thm; Ext and Tor (§3.1).
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Week 9:
Cup product and the cohomology ring; Kunneth formula (§3.2; see also Modules and tensor products handout)
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Week 10:
Topological manifolds, orientations, fundamental class; Poincare duality (§3.3) and Alexander-Lefschetz-Poincare Duality
Course Policies: Please see https://goto.stanford.edu/mathcoursepolicies for important course policies on exam conflicts, academic accommodations, AI guidance, and taking exams. It is your responsibility to thoroughly read this information.
Homework Policy: Weekly homework assignments given out on Wednesday, and due the following Wednesday at 11:59PM on Gradescope (unless otherwise noted). No late submissions will be accepted under any circumstances. (This is as much a courtesy to the grader as an incentive to stay current with the course and not fall behind.) To accommodate situations such as a serious illness or anything else that may arise (even if it is an obstruction known in advance due to your schedule), your lowest homework score will be dropped at the end of the quarter.
You are encouraged to attend office hours and to form study groups to discuss and work on homework together, ideally after thinking about it on your own. However, you must write up your own solutions
individually and in your own words, and indicate the names of any collaborators or sources of outside help that you received. Copying solutions from another student or from other sources (such as AI generated) and then submitting it for credit will be considered a violation of the Stanford Honor Code.
Exams: There will be two in-person, proctored exams: an evening midterm and final exam.
Grading: Homework 15%, Midterm 35% and Final 50%.
Important dates:
- Add/Drop Deadline: Friday, October 9, 5:00pm
- Midterm Exam: Thursday, Oct 29 (tentative).
- Final Exam: Wednesday, December 9, 8:30am-11:30am.
Academic Integrity:
The Honor Code articulates Stanford University's expectations of students and faculty in establishing and maintaining the highest standards in academic work. Its purpose is to uphold a culture of academic honesty. Students will support this culture of academic honesty by neither giving nor accepting unpermitted academic aid in any work that serves as a component of grading or evaluation, including assignments, examinations, and research. Examples of conduct that have been regarded as being in violation of the Honor Code (and are most relevant for this course) include copying from another student's work (or other sources such as the internet, AI generated etc) or allowing another student to copy from your own work; plagiarism; representing as one's own work the work of another. Please visit the OCS website for more information on the Honor Code, and please see link above for the Math Department AI Policy.
Responsibility for understanding: You are expected to understand your submitted work in homework and exams and to be able to verbally explain to your instructor (upon request, and without advance notice) any work you submit. When you apply for jobs, you will be interviewed by professionals who instantly recognize a lack of real understanding due to over-reliance on AI; we hold you to the same standard. Do not submit what you do not understand. The response
"I don't understand what I was writing but I know it works'' is never acceptable when asked to explain a submission.