Professor Alexandru I. Suciu MATH 4565 · Topology Spring 2016

## Course Information

 Course: MATH 4565 · Topology Web site: www.northeastern.edu/suciu/MATH4565/utop.sp16.html Instructor: Prof. Alex Suciu,   < a.suciu@neu.edu > Time and Place: Mon & Wed 2:50–4:30 pm, in 235 Forsyth Office Hours: Mon 4:40pm–5:40pm and Wed 1:40pm–2:40pm in 435 LA, or by appointment Textbook: Topology (2nd Edition) by James R. Munkres, Prentice Hall, 2000 Grade: Based on problem sets (50%), midterm exam (20%), and final exam (30%). It is expected that you will work on the problem sets together; however, they must be written up separately.

## Course Description

This course provides an introduction to the concepts and methods of Topology. It consists of two inter-connected parts.
 1. Topological Spaces and Continuous Maps This part of the course serves as an introduction to General Topology. The objects of study are topological spaces and continuous maps between them. Key is the notion of homeomorphism, which leads to the study of topological invariants. The main properties that are studied are connectedness, path connectedness, and compactness, as well as their "local" versions. We also introduce several constructions of spaces, including identification spaces. 2. Fundamental Group and Covering Spaces This part of the course is a brief introduction to Geometric Topology. It starts with Poincaré's definition of the fundamental group of a space, and various methods to compute it, such as the Seifert-van Kampen theorem. It proceeds with the classification of surfaces, and a detailed study of covering spaces. Applications include the Brouwer fixed point theorem, the Borsuk-Ulam theorem, and the Nielsen-Schreier theorem.

## Homework Assignments

Assignment Chapter Page Problems
Homework 1
Due Jan. 27
Munkres 2.13 83 3
Munkres 2.16 91-92 1, 4, 5
Munkres 2.18 111-112 3, 10
Homework 2
Due Feb. 3
Munkres 2.17 100-102 2&3, 6, 11&12, 13, 14
Munkres 2.20 126 3
Homework 3
Due Feb. 17
Munkres 3.23 152 4, 6, 9
Munkres 3.24 158 8, 9, 10
Homework 4
Due Feb 29
Munkres 3.24 158 2, 3
Munkres 3.25 162 4, 5(a), 6, 7
Homework 5
Due March 30
Munkres 2.22 145 3
Munkres 2.22 supplement 146 4, 5(a)-(c)
Munkres 3.26 172 12, 13(a)-(b)
Munkres 9.51 330 3(c)(d)
Homework 6
Due April 11
Munkres 9.52 335 3, 6
Munkres 9.53 341 3, 4, 6(b)
Munkres 9.58 366 5&6

## Exams

 Department of Mathematics Office: 435 Lake Hall Messages: (617) 373-2450 Northeastern University Phone: (617) 373-3899 Fax: (617) 373-5658 Boston, MA, 02115 Email: a.suciu@neu.edu Directions