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Professor Alexandru I. Suciu
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MTH G121 Topology 1
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Spring 2007
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Course Information
Course:
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MTH G121 -- Topology I
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Web site:
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www.northeastern.edu/suciu/G121/top1.sp07.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. & Wed., 7:30 - 9:00 PM, in 509 Lake
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Office Hours:
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Mon. & Wed., 5:00 - 6:00 PM, or by appointment
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Prerequisites:
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MTH G101 (Analysis 1), MTH G111 (Algebra 1)
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Textbooks:
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Topology (2nd Edition) by James R. Munkres, Prentice Hall, 2000
Algebraic Topology by Allen Hatcher, Cambridge University Press, 2002
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Grade:
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Based on problem sets, class participation, and a final exam
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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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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 many concrete examples, and derive various applications.
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For more information, including past exams and class projects, see these older syllabi, from 1998, 2001, 2003, and 2005. 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. 17
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Munkres 2.13
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83
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3, 4
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Munkres 2.16
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92
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4, 5
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Munkres 2.18
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112
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10, 11
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Munkres 2.20
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126
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3
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Homework 2
Due Jan. 29
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Munkres 2.17
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101
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11, 13, 14
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Munkres 3.26
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170-171
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1, 5, 7, 8
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Homework 3
Due Feb. 5
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Munkres 3.23
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152
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6, 9, 12
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Munkres 3.24
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157-158
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1, 3, 11
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Homework 4
Due Feb. 14
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Munkres 3.24
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158
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8, 9, 10
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Munkres 3.25
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162-163
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5, 6, 7
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Homework 5
Due March 12
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Munkres 2.22
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144-145
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2, 3 supplement: 1
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Munkres 9.51
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330
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3
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Munkres 9.52
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334-335
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3, 6, 7
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Homework 6
Due March 28
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Munkres 9.53
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341
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3, 4
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Munkres 9.54
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348
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8
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Munkres 9.55
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353
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2, 4(a)-(d)
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Munkres 9.58
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366
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7
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Homework 7
Due April 18
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Munkres 11.70
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433
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1, 2
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Munkres 11.74
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453-454
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3, 6
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Munkres 11.76
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457
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3
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Hatcher 1.2
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53
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4
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Homework 8
Due April 27
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Munkres 13.79
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483
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3, 4, 5(b)
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Munkres 13.81
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492-493
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2
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Hatcher 1.3
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79-80
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9, 14
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