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Professor Alexandru I. Suciu
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MATH 7221 · Topology 2
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Fall 2018
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 Course Information
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Course:
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MATH 7221 — Topology 2
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Web site:
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https://web.northeastern.edu/suciu/MATH7221/top2.fa18.html
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Instructor:
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Prof. Alex Suciu <a.suciu@neu.edu>
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Time and Place:
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Tuesday & Thursday from 5:50pm to 7:20pm, in 509 Lake Hall
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Office Hours:
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Tuesday & Thursday 4:40-5:40pm or by appointment, in 505A Lake Hall
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Prerequisites:
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MATH 5121 Topology 1, or MATH 4565 Topology (upon request)
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Textbook:
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Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2002
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Supplement:
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Topology and Geometry, by Glen Bredon, Springer-Verlag, GTM #139, 2002.
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Grade:
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Based on problem sets, exams, and class participation
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Course Description
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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:
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1. Homology Theory and CW-Complexes
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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.
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2. Cohomology Theory
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Cohomology groups, universal coefficients theorems, Bockstein homomorphism, Künneth formula, cup and cap products, Hopf invariant, Borsuk-Ulam theorem, Brouwer and Lefschetz fixed-point theorems.
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3. Manifolds and Duality
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Orientability, Poincaré duality,
Lefschetz duality,
Alexander duality,
Euler class,
Gysin sequence,
intersection form, and
signature.
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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.
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As a computational aide, I recommend using the SimpComp package, which runs under GAP, or the SimplicialComplexes package, which is part of Macaulay2.
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Homework Assignments
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Homework
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Due date
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Section
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Pages
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Problems
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1
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September 20
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Homework 1
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2
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October 9
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Hatcher 2.1
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132-133
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16, 17, 19, 26, 29
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Hatcher 2.2
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147
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Splitting Lemma (give proof with full details)
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3
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October 23
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Hatcher 2.2
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156-158
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12, 26, 28, 29, 30(a)-(c), 33
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4
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November 8
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Hatcher 1.3
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82
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32
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Hatcher 2.1
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132
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22
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Hatcher 2.2
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156-157
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16, 17, 20&21, 22&23
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5
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November 29
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Hatcher 2.2
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159
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43
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Hatcher 3.1
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205
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3, 6, 11
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Hatcher 3.A
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267
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3
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Hatcher 3.B
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280
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1
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6
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December 11
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Hatcher 3.2
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228-230
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1, 3, 6, 10, 11, 18
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Handouts
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a.suciu@northeastern.edu
