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

MATH 7221 · Topology 2

Fall 2014

* Course Information

Course:   MATH 7221 -- Topology 2
Web site:   http://www.northeastern.edu/suciu/MATH7221/top2.fa14.html
Instructor:   Prof. Alex Suciu  <a.suciu@neu.edu>
Time and Place:   Mon. and Wed. from 7:30pm to 9:00pm, in 509 Lake Hall
Office Hours:   Mon, Wed 4:40-5:40pm, in 435 Lake Hall, or by appointment
Prerequisites:   MATH 5121 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 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 2003, 2004, 2005, 2008, 2010, and 2011. 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 Pages Problems
1 September 17 Hatcher 1.A 86 3
Hatcher 2.1 131-132 6, 9, 23
Hatcher 2.C 184 1
2 September 29 Hatcher 2.1 132-133 16, 17, 18, 19, 26, 29
3 October 22 Hatcher 2.2 156-158 10, 13(a), 26, 28, 29, 33
4 November 10 Hatcher 2.2 156-159 12, 20&21, 22&23, 40
Hatcher 3.A 267 2, 3 (extra credit)
Hatcher 3.B 280 1 (for homology)
5 November 24 Hatcher 3.1 204-205 2&3, 5
Hatcher 3.2 228-230 1, 6, 10, 18
6 December 8 Hatcher 3.3 257-259 2, 3&4, 5, 10, 24, 32&33


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

Home Started:  July 23, 2014
Last modified: December 1, 2014
www.northeastern.edu/suciu/MATH7221/top2.fa14.html