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Professor Alexandru I. Suciu
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MATH 5121 · Topology 1
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Spring 2013
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Course Information
Course Description
This course provides an introduction to the concepts and methods of Topology. It consists of three inter-connected parts.
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1. Topological Spaces and Continuous Maps
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This part of the course serves as a quick introduction to General Topology. The objects of study are topological spaces and continuous maps between them. Key is the notion of homeomorphism, which leads to the study of topological invariants. The main properties that are studied are connectedness, path connectedness, and compactness, as well as their "local" versions. We also introduce several constructions of spaces, including identification spaces.
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2. Fundamental Group and Covering Spaces
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This part of the course is a brief introduction to the methods of Algebraic and Geometric Topology. It starts with Poincaré's definition of the fundamental group of a space, and various methods to compute it, such as the Seifert-van Kampen theorem. It proceeds with the classification of surfaces, and a detailed study of covering spaces. Applications include the Brouwer fixed point theorem, the Borsuk-Ulam theorem, and the Nielsen-Schreier theorem.
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3. Simplicial Complexes and Simplicial Homology
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Time permitting, this part of the course is a brief introduction to the methods of Combinatorial Topology and Homological Algebra. It starts with simplicial complexes and their realizations, and proceeds to simplicial homology groups, and ways to compute them. We will illustrate these techniques with concrete examples, and derive some applications.
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For more information, including past exams and class projects, see these older syllabi, from 1998, 2001, 2003, 2005, 2007, 2008,
2009, and 2011. You may also want to look at some past qualifying exams in Topology, based in large part on the material covered in this course.
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Homework Assignments
Assignment
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Chapter
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Page
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Problems
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Homework 1
Due Jan. 23
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Munkres 2.16
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92
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10
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Munkres 2.17
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101-102
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13, 20
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Munkres 2.18
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112
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13
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Munkres 2.20
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126
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3
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Munkres 3.23
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152
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5
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Homework 2
Due Feb. 6
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Munkres 3.24
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158
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3, 8, 9, 10
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Munkres 3.25
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162
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5, 6
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Homework 3
Due Feb. 21
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Munkres 2.22
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144-145
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2, 3
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Munkres 3.26
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171-172
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2, 5, 9, 12
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Homework 4
Due March 10
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Munkres 3.29
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186
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3, 5
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Munkres 7.46
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289
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7, 9
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Munkres 9.58
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366
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5&6, 8
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Homework 5
Due March 25
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Munkres 9.52
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335
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6, 7
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Munkres 9.53
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341
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3, 4, 6(b)
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Munkres 9.54
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348
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8
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Homework 6
Due April 8
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Munkres 9.58
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367
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9 (a)-(d)
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Munkres 11.70
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433
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1, 2
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Hatcher 1.2
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53
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4, 6, 8
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Homework 7
Due April 22
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Munkres 12.74
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454
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3, 6
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Munkres 12.75
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457
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3
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Munkres 13.79
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483
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3
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Hatcher 1.3
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79
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9, 10
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