Topology

Math 70800, Spring 2026, CUNY GC

Tuesday & Thursday, 12 - 1:30 pm, GC 5417

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Course Information

Instructor: Abhijit Champanerkar

Office: GC 4307

Email :

Office Hours: (Changed 2/5) Thursdays 2 - 3 pm, Math Lounge and Fridays 2-3 pm on Zoom (please email me for zoom information)

Course description

Books: Algebraic Topology by Allen Hatcher, Cambridge University Press, ISBN 0-521-79540. Free download. We will cover Chapters 2, 3 and some additional topics at the end of each chapter.

Reference books:

• Elements Of Algebraic Topology by Munkres, Westview Press.
• Homology theory. An introduction to algebraic topology by James Vick, Springer-Verlag GTM.

Homework & Exam: Homework will be assigned regularly during the semester and posted here. Instructions about submitting and grading homework will be provided on each homework. We will have a final exam at the end of the semester.

Grade: The grade for this class will be determined on basis of submitted homeworks and the final exam or the qualifying exam.

Outline of classes

Unless otherwise noted, the section numbers and problem numbers refer to the textbook Algebraic Topology by Allen Hatcher.

Week	Topic	Homework & Reading	Due
Jan 26	CW complexes and surfaces (Chapter 0), Covering Spaces (Section 1.3)	Surfaces   Quotient Spaces   HW 1	Mon Feb 9th
Feb 2	Review of Covering Spaces (Section 1.3, pg 56-79)
Feb 9	Covering spaces and Delta complexes (pg 104-108)	Thursday 2/12 No classes  HW 2	Mon Feb 23rd
Feb 16	Simplicial Homology (pg 104-108)	Tuesday 2/17 No classes
Feb 23	Singular homology, Homotopy Invariance, Relative Homology (pg 108-118)
Mar 2	Excision and Applications (pg 119-128)
Mar 9	Mayer-Vietoris Sequence (pg 149-153) & Equivalance (pg 128-131)
Mar 16	Degree and Cellular Homology (pg 134-141)
Mar 23	Applications (pg 166-176)
Mar 30	Universal Coefficient Theorem	Thursday 4/1 No classes
April 6	No classes	Spring Break
Apr 13	Cohomology
Apr 20	Cup Products
Apr 27	Orientation
May 4	Poincare Duality
May 11	Review
May 18	Exam Week

Useful Links

1. Resources on Point-set Topology: Hatcher's notes   Munkres Topology Second edition
2. Previous Qualifying Exams
3. Triangulations of manifolds
4. Finitely generated abelian groups
5. CUNY Geometry and Topology Seminar
6. Geometry and Topology Student Seminar