Professor Alexandru I. Suciu MATH 5121 · Topology 1 Fall 2015

## Course Information

 Course: MATH 5121 · Topology 1 Web site: www.northeastern.edu/suciu/MATH5121/top1.fa15.html Instructor: Prof. Alex Suciu,   < a.suciu@neu.edu > Time and Place: Th 2:30–4:00 pm, in 544 NI and Fr 11:00am–12:30pm in 555 NI Office Hours: Mon & Wed 4:40pm–5:40pm, in 435 LA, or by appointment Prerequisites: MATH 5101 (Analysis 1), MATH 5111 (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, 2007, 2008, 2009, 2011, and 2013. 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 Sept. 25
Munkres 2.16 92 5
Munkres 2.17 101-102 13, 19
Munkres 2.18 112 13
Munkres 2.20 126 3
Munkres 3.23 152 5
Homework 2
Due Oct. 8
Munkres 3.24 158 8, 9, 10
Munkres 3.25 162 5, 6, 7
Homework 3
Due Oct. 22
Munkres 2.22 145 3
Munkres 2.22 supplement 146 5
Munkres 3.26 170-172 1, 5, 8, 12
Homework 4
Due Oct. 30
Munkres 3.29 186 3, 5
Munkres 7.46 289 7
Munkres 9.51 330 3(b)(d)
Munkres 9.58 366 1&6, 8
Homework 5
Due Nov. 14
Munkres 9.52 335 6, 7
Munkres 9.53 341 4, 6(b)
Munkres 9.54 348 8
Munkres 9.58 367 9(a)-(d)
Homework 6
Due Nov. 30
Munkres 11.70 433 1, 2
Hatcher 1.2 53-54 4, 6, 8, 14
Homework 7
Due Dec. 11
Munkres 12.74 454 3, 6
Munkres 12.75 457 3
Munkres 13.79 483 5
Hatcher 1.3 79 9, 10

 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