Topology

Math 70800, Spring 2021, CUNY GC

Monday & Wednesday, 12 - 1:30 pm, online

GC Math     CUNY GC

Announcements:

1. This class meets online. Information about the online classroom has been emailed to the students.


2. (2/4/21) HW 1, Notes on Classification of surfaces and Quotient Spaces has been posted below.


3. (2/24/21) HW 2, Notes on Abelian groups and homomorphisms has been posted below.


4. (3/8/21) HW 3 has been posted below.


5. (4/10/21) HW 5 has been posted below.


6. (4/19/21) Review Sessions (same Zoom link as the class):
   (1) Fundamental Group review: Friday 4/23, 3 - 4:30 pm
   (2) Covering Spaces review: Thursday 4/29, 4 - 5:30 pm
   


7. (5/5/21) HW 6 has been posted below.

Instructor: Abhijit Champanerkar

Office: 4307

Email :

Office Hours: Wednesdays 2 - 3:30 pm online (same link as classroom)

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.

Outline of classes

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

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Week	Topic	Homework	Due
Feb 1	CW complexes and surfaces (Chapter 0)	Surfaces   Quotient Spaces   HW 1	Mon Feb 15th
Feb 8	Delta complexes and Simplicial Homology (pg 104-108)
Feb 15	Singular homology and Homotopy Invariance (pg 108-113)	Note: Monday 2/15 GC closed
Feb 22	Relative Homology (pg 115-118)	HW 2	Mon March 8th
Mar 1	Excision and Applications (pg 119-128)
Mar 8	Mayer-Vietoris Sequence (pg 149-153) & Equivalance (pg 128-131)	HW 3	Mon Mar 29th
Mar 15	Degree and Cellular Homology (pg 134-141)
Mar 22	Applications (pg 166-176)	HW 4	Mon April 12th
Mar 29	No classes	Spring Break
Apr 5	Universal Coefficient Theorem	HW 5	May 3rd
Apr 12	Cohomology
Apr 19	Cup Products
Apr 26	Orientation
May 3	Poincare Duality	HW 6	Friday May 28th
May 10	Products
May 17	Review	Note: 5/17 is last day of classes
May 21	Topology Qualifying Exam	Friday May 21, 10am - 1pm

Useful Links:

1. Previous Qualifying Exams
2. Triangulations of manifolds
3. Finitely generated abelian groups
4. CUNY Geometry and Topology Seminar
5. Geometry and Topology Student Seminar

About online classes

1. Online classroom Information about the online class will be emailed to students.
   
   • Do not share the class zoom information with anybody nor post it online. To attend the class you either have to be registered for the class or need special permission from me.
   • Join 5 minutes ahead of class time so we can start in a timely manner.
   • Join the Zoom link with your name.
   • Mute your mic. You can unmute to ask questions.


2. Pdf scanning: Students may have to handin homeworks/exams as pdf scans of their work. Here is a guide to scan your work to pdf.


3. CUNY policy about the recording of remote classes: Students who participate in this class with their camera on or use a profile image are agreeing to have their video or image recorded solely for the purpose of creating a record for students enrolled in the class to refer to, including those enrolled students who are unable to attend live. If you are unwilling to consent to have your profile or video image recorded, be sure to keep your camera off and do not use a profile image. Likewise, students who un-mute during class and participate orally are agreeing to have their voices recorded. If you are not willing to consent to have your voice recorded during class, you will need to keep your mute button activated and communicate exclusively using the "chat" feature, which allows students to type questions and comments live.


4. Proctoring of assessments: Students are required to keep their cameras on during assesments for this class.


5. Academic integrity and Digital etiquette: Cheating in any form will not be tolerated. Please see CUNY Policy on Academic Integrity. Online cheating will be dealt with severely. Please be aware of proper online class behavior. In the online virtual classroom repeated offenses will result in being barred from participation in the class.