Topology, by Munkres

Professor Alexandru I. Suciu

Algebraic Topology, by Hatcher

MTH G121 Topology 1    

Spring 2005

* Course Information

Course:   MTH G121 -- Topology I
Web site:   http://www.math.neu.edu/~suciu/G121/top1.sp05.html
Instructor:   Prof. Alex Suciu   < a.suciu@neu.edu >
Time and Place:   Tue. & Th., 7:30 - 9:00 PM, in 509 Lake
Office Hours:   Wed, Th 11:45-12:50, or by appointment
Prerequisites:   MTH G101 (Analysis 1), MTH G111 (Algebra 1)
Textbooks:   Topology (2nd Edition) by James R. Munkres, Prentice Hall, 2000
Algebraic Topology by Allen Hatcher, Cambridge University Press, 2002
Grade:   Based on problem sets, class participation, and a final exam

* Course Description

This course provides an introduction to the concepts and methods of Topology. It consists of three inter-connected parts.
1. Topological Spaces and Continuous Maps
This part of the course serves as a quick 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 the methods of Algebraic and 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.
3. Simplicial Complexes and Simplicial Homology
This part of the course is a brief introduction to the methods of Combinatorial Topology and Homological Algebra. It starts with simplicial complexes and their realizations, and proceeds to simplicial homology groups, and ways to compute them. We will illustrate these techniques with many concrete examples, and derive various applications.
For more information, including past exams and class projects, see these older syllabi, from 1998, 2001, and 2003. You may also want to look at some past qualifying exams in Topology, based in large part on the material covered in this course.

* Homework Assignments

Homework Chapter Page Problems
1.
Due Jan. 13 		Munkres 2.16 92 4, 5
Munkres 2.18 112 10, 11
Munkres 2.20 126 3
Munkres 2.21 133 1, 2
2.
Due Jan. 25 		Munkres 2.17 101 11, 13
Munkres 2.22 144-145 2, 3
supplement: 1
Munkres 3.23 152 1, 5, 9
3.
Due Feb. 10 		Munkres 3.24 157-158 1, 9, 10
Munkres 3.25 162 5, 6, 7
Munkres 3.26 170-171 1, 8
4.
Due March 8 		Munkres 9.52 334-335 3, 6, 7
Munkres 9.53 341 3, 6(b)
Hatcher 0 18 5, 6(a,b)
5.
Due March 22 		Munkres 9.54 348 8
Munkres 9.55 353 2
Munkres 9.57 359 2, 3
Munkres 9.58 366 7
Hatcher 1.1 38-39 5, 8
6.
Due April 5 		Munkres 11.70 433 1, 2
Munkres 12.74 453 2, 6, 7
Hatcher 1.2 53 4, 6, 8
7.
Due April 21 		Munkres 13.79 483-484 3, 7
Munkres 13.81 492-493 2, 3
Hatcher 1.3 79-80 8, 10, 13, 14

 
Department of Mathematics  Office:  441 Lake Hall  Messages:  (617) 373-2450 
Northeastern University Phone:  (617) 373-4456  Fax:  (617) 373-5658
Boston, MA, 02115  Email:  a.suciu@neu.edu Directions
Home   Started: December 19, 2004
Last modified:  April 12, 2005
URL:  http://www.math.neu.edu/~suciu/G121/top1.sp05.html