Image: Introduction to Algebraic Topology

Math 74, Spring 2005

Introduction to Algebraic Topology

General Information Homework Assignments Class Material

General Information

Textbook Scheduled Lectures Syllabus
X hours Examinations Homework Policy
Grades Honor Principle Disabilities


Elementary Topology, A First Course: Textbook in Problems by O. Viro, O. Ivanov, N. Netsvetaev, and V. Kharlamov.
Available for download from here. A hardcopy can be purchased from the Copy Center (Thayer Hall, 2nd floor).

Secondary textbook: Algebraic Topology by Alan Hatcher, Cambridge University Press, 2002. Also available from here.

Scheduled Lectures
Contact Information

Instructor Alexander Shumakovitch
Classroom 215 Silsby Hall
Lecture (MWF) 10:00 – 11:05
X-hour Th 12:00 – 12:50
Office 402 Bradley Hall
Office Hours TuF 6:00 – 7:00pm;
Th 1:00 – 2:00pm
or by appt.
Phone 646-1614
E-mail BlitzMail

Tentative Syllabus

  • Fundamental Group and Covering Spaces: homotopy; properties of path multiplication; definition of fundamental group; theorems of path lifting; universal coverings and calculations of fundamental groups.
  • Fundamental Group and Mappings: induced homomorphisms and their applications (winding number, Borsuk-Ulam Theorem); retraction and fixed points; homotopy equivalence; covering spaces via fundamental groups; hierarchy of coverings.
  • Cellular Spaces: examples of cellular spaces; fundamental group of a cellular space; Seifert-van Kampen Theorem; one-dimensional homology and cohomology.
If time permits, we can venture into the following subject as well:
  • Manifolds and Classification of Surfaces: locally Euclidean spaces and manifolds; isotopy; classification of one-dimensional manifolds; triangulation and handle decomposition; topological classification of compact surfaces.


There will be one take-home midterm and the take-home final exam. They are scheduled as follows:

Exam Date given Date due
Midterm April 27, Wednesday May 2, Monday
Final June 3, Friday June 7, Tuesday

You will have the whole final examination period to work on the final take-home exam, though it should not require more than a few hours if you know the material. You are expected to work alone on the exams. You may use any printed matter (or your class notes) of your choice but you may not consult one another or other humans. The honor principle applies.

If you have a legitimate conflict with the exams dates and times, please contact the instructor as soon as possible, do not wait until shortly before the exam.

Note. Exams will not be given early to accommodate travel plans.


We will meet on X-hours (almost) every week to discuss homework problems. Each time a random student will be chosen to present his/her solutions to selected problems. These presentations will count towards the final grade. Please note that you can always decline an opportunity to present your solutions if you feel uncomfortable about it.

Homework Policy

  • Homework sets will be assigned for each class but are to be handed in weekly. Assignments for Wednesday and Friday on a given week as well as the one for next Monday are to be submitted in class next Friday. There will also be appropriate adjustments for holidays. You can check the exact due dates on the Homework Assignments web page.
  • Besides usual written assignments, additional practice problems will be listed for some classes. You are strongly encouraged to solve these problems, but please do not hand in solutions to them. Their sole purpose is to help you better prepare for the exams.
  • Late homeworks will not be accepted. Unexcused late and missing papers count zero.
  • More details are posted on the Homework Assignments web page.


The course grade will be based upon the scores for the homework (including presentations), midterm and final exams.

In-class presentations of homework 20 points
Homework 100 points
Midterm Exam 100 points
Final Exam
130 points
Total 350 points

The Honor Principle

  • On Exams: No help given or received. You cannot consult any other human (students and professors included, but not limited to).
  • On Homework: Collaboration is permitted and encouraged, but NO COPYING.   Discussions of the general ideas of the class with instructors, tutors, fellow students and others are desirable. However, each student is expected to complete his or her assignments individually and independently.


Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see their instructor as soon as possible. Also, they should stop by the Academic Skills Center in Collis Center to register for support services.