Topology, by Munkres
  Professor Alexandru I. Suciu
  		Algebraic Topology, by Hatcher
MTH G121 · Topology 1
Spring 2008

* Course Information

Course:   MTH G121 · Topology 1
Web site:   www.math.neu.edu/~suciu/G121/top1.sp08.html
Instructor:   Prof. Alex Suciu,   < a.suciu@neu.edu >
Time and Place:   Mon. & Wed., 7:30 - 9:00 PM, in 509 Lake
Office Hours:   Mon. & Wed., 5:30 - 6:30 PM, 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 (also here)
Grade:   Based on problem sets, class participation, and possibly 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
Time permitting, 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 concrete examples, and derive some applications.
For more information, including past exams and class projects, see these older syllabi, from 1998, 2001, 2003, 2005, and 2007. 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

Assignment Chapter Page Problems
Homework 1
Due Jan. 16 		Munkres 2.13 83 3, 5
Munkres 2.16 92 4, 5
Munkres 2.18 112 10, 11
Munkres 2.20 126 3
Homework 2
Due Jan. 23 		Munkres 2.17 101 6, 9, 11, 13, 14
Homework 3
Due Jan. 30 		Munkres 3.23 152 4, 6, 9, 12
Munkres 3.24 158 11
Munkres 3.25 163 9
Homework 4
Due Feb. 11 		Munkres 3.24 158 7(a), 8, 9, 10
Munkres 3.25 162 3, 4, 5
Homework 5
Due Feb. 20 		Munkres 2.22 144-145 2, 3
Munkres 3.23 152 11
Munkres 3.26 170-172 1, 5, 8, 12
Homework 6
Due March 10 		Munkres 9.29 186 1, 3
Munkres 9.51 330 2, 3
Munkres 9.58 366 1, 6, 8
Homework 7
Due March 19 		Munkres 9.52 335 3, 6, 7
Munkres 9.58 366 7
Hatcher 0 18-19 5, 12
Hatcher 1.1 39 13
Homework 8
Due April 2 		Munkres 9.53 341 2, 3, 4, 6(b)
Munkres 9.54 348 8
Munkres 9.55 353 2, 4(a)-(b)
Homework 9
Due April 16 		Munkres 11.70 433 1, 2
Munkres 11.74 453-454 2, 3, 6
Munkres 11.76 457 3
Hatcher 1.2 53 4

 
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 18, 2007
Last modified:  April 8, 2008
www.math.neu.edu/~suciu/G121/top1.sp08.html