

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 <a.suciu@neu.edu>

Time and Place:

Tuesday & Thursday from 5:50pm to 7:20pm, in 509 Lake Hall

Office Hours:

Tuesday & Thursday 4:405:40pm 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, SpringerVerlag, 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 interconnected parts:

1. Homology Theory and CWComplexes


We will start with simplicial complexes and simplicial homology, after which we will proceed to singular homology, homological algebra (exact sequences, axioms), MayerVietoris sequence, CWcomplexes 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, BorsukUlam theorem, Brouwer and Lefschetz fixedpoint 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

132133

16, 17, 19, 26, 29

Hatcher 2.2

147

Splitting Lemma (give proof with full details)

3

October 23

Hatcher 2.2

156158

12, 26, 28, 29, 30(a)(c), 33

4

November 8

Hatcher 1.3

82

32

Hatcher 2.1

132

22

Hatcher 2.2

156157

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

228230

1, 3, 6, 10, 11, 18

Handouts
