# MTH 3105 Topology I

## Course Information

 Course: MTH 3105 (Topology I) Instructor: Alex Suciu Time and Place: Mon. & Wed., 5:30 - 7:00 PM, in 544 NI Office Hours: Mon. & Wed., 4:00 - 5:00 PM, in 441 LA Prerequisites: MTH 3010 (Basics of Analysis) Textbook: Basic Topology, by M. A. Armstrong, Springer-Verlag, UTM, corrected 4th printing, 1994 Grade: Based on problem sets, class projects, and a final exam

## Course Description

This course provides an introduction to Topology, requiring only some elementary background in Analysis. Time permitting, the following topics will be covered:

 Continuity, Compactness and Connectedness Identification Spaces The Fundamental Group Triangulations Surfaces Simplicial Homology Degree and Lefschetz Number Knots and Covering Spaces

The first (shorter) part of the course treats General Topology. The objects of study are metric and topological spaces. The main properties that are studied are connectedness and compactness. We also introduce several constructions of spaces, and study the invariance of various properties under topological equivalence.

The second part of the course treats the basics of Algebraic Topology. It starts with the fundamental group of a space, and methods to compute it. It proceeds with a study of simplicial complexes, and the classification of surfaces. Simplicial homology is then developed. Applications include the Brouwer fixed point theorem, the Euler-Poincaré formula, the Borsuk-Ulam theorem, and the Lefschetz fixed point theorem. (A more thorough treatment of some of these topics may be postponed for Topology II.)

The last part of the course serves as a brief introduction to Geometric Topology. It starts with covering space theory, and the correspondence between coverings of a space and the subgroups of the fundamental group of that space. It ends with a brief excursion into Knot Theory: the fundamental group of a knot complement, Seifert surfaces, and the Alexander polynomial.

Here are some past qualifying exams in Topology, based in large part on the material covered in this course.

## Class Projects

Here is the Surface Calculator.  This Java applet, created by Ivo Nikolov, takes a closed polygon in R2, with edges identified in pairs, and returns a closed surface in standard form, specifying its orientability, genus, and Euler characteristic.  And here is the poster Recognizing Surfaces, based on this class project, and presented at the Poster Session: Connections '99.

## Homework Assignments

Chapter Page Problems
Homework 1 2 31 2, 3, 4
2 35 17
2 36 20
3 50 7, 11, 18
Homework 2 3 55 23, 25
3 63 43
3 72-73 2, 3, 5, 10
Homework 3 4 78 16, 21
4 85 26, 27, 32
5 91 5, 7
Homework 4 5 95 11, 13
5 102 21
5 109 27, 28, 31
Homework 5 6 124 8
6 131 14, 17
6 140 20, 22, 23, 24
Homework 6 7 170 28, 29, 30
8 183 11, 16, 17
8 184 18, 19

## Final Exam

Here is the (take home) final exam, as an Adobe PDF file, or as a TeX DVI file.

 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: alexsuciu@neu.edu Directions