Instructor
Professor Shelly Harvey
Herman Brown 446
shelly at rice dot edu

Course Information
Class meets: MWF 11-11:50am via Zoom
Office Hours: TBA
Webpage: http://math.rice.edu/~shelly/540s21/
All homework and reading assignments can be found on Canvas
Teaching Assistant: TBA

Textbook
James Munkres, Elements of Algebraic Topology, Perseus Books Publishing, Cambridge, Massachusetts, 1984. ISBN: 0-201-62728 (required)

Other useful textbooks
Allen Hatcher, Algebraic Topology; available free online at http://www.math.cornell.edu/~hatcher/AT/ATpage.html
M J Greenberg and J Harper, Algebraic Topology: a First Course (Benjamin/Cummings 1981)

Course objectives and learning outcomes
In this course, the student will study the homology and cohomology of topological spaces. (Co)Homology is a way of associating a sequence of abelian groups to a topological space that are invariant under homeomorphism (and more generally homotopy equivalence). The homology groups of a space are in general easier to compute than the homotopy groups and hence they can be more useful in distinguishing some spaces. Some of the topics we will cover are simplicial and singular (co)homology, relative (co)homology, the Eilenberg-Steenrod axioms, the Universal Coefficient Theorem, some basic homological algebra, the cohomology ring, Poincare-Lefschetz duality, and the Kunneth Theorem. If time permits, we will cover some of the following topics: spectral sequences, the Hurewicz and Whitehead's Theorems (statements only), De Rham cohomology, group cohomology, and Cech cohomology. By the end of the course, the student will master the material covered in the class.

Your grade in the class will be based on the following weights:
 Homework: 40% Midterm: 25% Final Exam: 35%

Homework and Exams
Homeworks will be assigned every Friday and will be due the following Friday by 11:59pm; they will be posted on Canvas. You should turn your assignment in on Gradescope. Homework solutions must be legible. You must show all of your work for full credit. Late homework will receive at most 1/2 credit. The students in Math 540 will be required to do more homework problems than the students in Math 445. The homework is not pledged and you can collaborate with other students in the class. In fact, you are very much encouraged to do so. However, you are not allowed to look up solutions in any written form; in particular, you are not allowed to look up solutions online. Students caught violating this rule will be reported to the Honor Council. You should write up your solutions individually.

Exams
There will be one midterm (the date to be determined) and a final exam. Both exams will be take home exams. Good mathematical exposition will be counted on both exams. The exams are pledged.

Attendance Policy
Attendance is not required. However, you are responsible for all the material and announcements covered in lecture.

Disability Support
Any student with a documented disability seeking academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain as confidential as possible. Students with disabilities will also need to contact the Disability Resource Center.

Collegiality and Respect
The Department of Mathematics supports an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. We expect all participants in mathematics courses (students and faculty alike) to treat each other with courtesy and respect, and to adhere to the mathematics department standards of collegiality, respect, and sensitivity as well as the Rice Student Code of Conduct. If you think you have experienced or witnessed unprofessional or antagonistic behavior, then the matter should be brought to the attention of the instructor and/or department chair. The Ombudsperson is also available as an intermediate, informal option, and contacting them will not necessarily trigger a formal inquiry.