Algebraic Topology, by Hatcher Professor Alexandru I. Suciu Topology and Geometry, by Bredon

MATH 7221 · Topology 2

Spring 2010

* Course Information

Course:   MTH G221 -- Topology 2
Web site:   http://www.math.neu.edu/~suciu/G221/top2.sp10.html
Instructor:   Prof. Alex Suciu  <a.suciu@neu.edu>
Time and Place:   Mon. and Wed. from 5:50pm to 7:20pm, in 202 Kariotis Hall
Office Hours:   Mon, Wed 2:50pm-3:50pm in 441LA, or by appointment
Prerequisites:   MTH G121 Topology 1 or equivalent
Textbook:   Algebraic Topology by Allen Hatcher, Cambridge University Press, 2002
Supplement: Topology and Geometry, by Glen Bredon, Springer-Verlag, GTM #139, 1997.
Grade:   Based on problem sets, exams, and class participation

* Course Description

This course provides an introduction to the concepts of Algebraic Topology, with an emphasis on homological methods, and with applications to problems in homotopy theory, geometry and combinatorics. It consists of three inter-connected parts:
1. Homology Theory and CW Complexes
We will start with a brief review of homotopy, fundamental group, and covering spaces. Next, we will discuss simplicial complexes and simplicial homology, after which we will proceed to singular homology, homological algebra (exact sequences, axioms), Mayer-Vietoris sequence, CW-complexes and cellular homology, calculation of homology of cellular spaces, and homology with coefficients.
2. Cohomology Theory
Cohomology groups, universal coefficients theorems, Bockstein homomorphism, Künneth formula, cup and cap products, Hopf invariant, Borsuk-Ulam theorem, Brouwer and Lefschetz fixed-point theorems.
3. Manifolds and Duality
Orientability, Poincaré duality, Lefschetz duality, Alexander duality, Euler class, Gysin sequence, intersection form, and signature.
For more information, see some older syllabi, from 1998, 1999, 2003, 2004, 2005, and 2008. You may also want to look at some past qualifying exams in Topology, based in part on the material covered in this course.
As a computational aide, I recommend using the Simplicial Homology package, which runs under GAP, or the Simplicial Complexes package, which is part of Macaulay2.

* Homework Assignments

Homework Due date Section Page Problems
1 February 1 Hatcher 2.1 131-132 4, 5, 6, 9, 11, 12
2 February 15 Hatcher 2.1 132-133 16, 17, 18, 19, 26, 29
3 March 8 Hatcher 2.2 157-158 26, 27, 28, 29, 33, 36
4 March 22 Hatcher 2.2 156-158 9(a)-(c), 10, 12, 13(a), 19, 30
5 March 31 Hatcher 2.2 157-159 20, 21, 22, 23, 40, 41
6 April 12 Hatcher 3.A 267 2, 3
Hatcher 3.B 280 1, 3
Hatcher 3.1 204-206 6, 11(a)
7 April 26 Hatcher 3.2 228 1, 3, 6, 10, 11, 18

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 13, 2009
Last modified:  April 16, 2010
www.math.neu.edu/~suciu/G221/top2.sp10.html