Professor Alexandru I. Suciu

MATH 7221 · Topology 2

Fall 2014

Course Information

 Course: MATH 7221 -- Topology 2 Web site: https://web.northeastern.edu/suciu/MATH7221/top2.fa14.html Instructor: Prof. Alex Suciu   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