Professor Alexandru I. Suciu

# MATH 7221 · Topology 2

Fall 2018

## Course Information

 Course: MATH 7221 — Topology 2 Web site: https://web.northeastern.edu/suciu/MATH7221/top2.fa18.html Instructor: Prof. Alex Suciu   Time and Place: Tuesday & Thursday from 5:50pm to 7:20pm, in 509 Lake Hall Office Hours: Tuesday & Thursday 4:40-5:40pm or by appointment, in 505A Lake Hall Prerequisites: MATH 5121 Topology 1, or MATH 4565 Topology (upon request) Textbook: Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2002 Supplement: Topology and Geometry, by Glen Bredon, Springer-Verlag, GTM #139, 2002. 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, 2011, 2014, and 2016. 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 SimpComp package, which runs under GAP, or the SimplicialComplexes package, which is part of Macaulay2.

## Homework Assignments

Homework Due date Section Pages Problems
1 September 20 Homework 1
2 October 9 Hatcher 2.1 132-133 16, 17, 19, 26, 29
Hatcher 2.2 147 Splitting Lemma (give proof with full details)
3 October 23 Hatcher 2.2 156-158 12, 26, 28, 29, 30(a)-(c), 33
4 November 8 Hatcher 1.3 82 32
Hatcher 2.1 132 22
Hatcher 2.2 156-157 16, 17, 20&21, 22&23
5 November 29 Hatcher 2.2 159 43
Hatcher 3.1 205 3, 6, 11
Hatcher 3.A 267 3
Hatcher 3.B 280 1
6 December 11 Hatcher 3.2 228-230 1, 3, 6, 10, 11, 18

## Handouts

 Department of Mathematics Office: 505A Lake Hall Northeastern University Phone: (617) 373-5635 Boston, MA, 02115 Email: a.suciu@northeastern.edu

 Started:  April 29, 2018 Last modified: December 3, 2018 www.northeastern.edu/suciu/MATH7221/top2.fa18.html