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Professor Alexandru I. Suciu
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MTH G221 Topology 2
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Fall 2005
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 Course Information
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Course:
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MTH G221 -- Topology 2
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Web site:
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https://web.northeastern.edu/suciu/G221/top2.fa05.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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Mon. & Th., 10:00 - 11:30 am, in 544 NI
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Office Hours:
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Mon. & Th., 11:30 am - 12:30 pm, or by appointment -- in 441 Lake
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Prerequisites:
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MTH G121 Topology 1
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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, 1997.
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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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Starts with simplicial complexes and simplicial homology groups. Proceeds 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-Hopf fixed-point theorems.
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3. Manifolds and Duality
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Orientation bundle, Poincaré duality, Lefschetz duality, Alexander duality, Euler class, Lefschetz numbers, Gysin sequence, intersection form, and signature.
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For more information, see some older syllabi, from 1998, 1999, 2003, and
2004. 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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Fort the first homework, I recommend using the Simplicial Homology package, which runs under GAP. (The software is already installed on our network.)
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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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Page
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Problems
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1
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September 22
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Hatcher 2.1
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131
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4, 5, 6, 7, 9
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2
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October 3
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Hatcher 2.1
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132
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14, 16, 17, 18, 19
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3
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October 23
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Hatcher 2.1
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133
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26, 29
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Hatcher 2.2
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157-158
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26, 28, 29
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4
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November 7
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Hatcher 2.2
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156-157
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9, 10, 12, 13, 20, 21
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5
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November 28
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Hatcher 3.A
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267
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1
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Hatcher 3.1
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204-206
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3, 6, 11
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Hatcher 3.2
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228
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1, 2
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6
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December 16
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Hatcher 3.3
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257-260
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2, 3, 5, 10, 11, 25, 26
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