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MAT 530: Topology & Geometry, I
Stony Brook Fall 2005 |
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The textbook for MAT 531 next term will be Frank Warner's Foundations of Differentiable Manifolds and Lie Groups, GTM 94. It should be available at the Stony Brook bookstore by the beginning of the semester. It is available on amazon at a discount, with free shipping. Currently, and likely through the end of December, it is also available directly from Springer at an even higher discount. Please do not wait weeks into the semester to acquire the book.
We should be able to cover Chapters 1,2,4,5 and perhaps 6.
Detailed solutions to:
Here are the homework assignments, the course summary, the midterm, the final exam, and the notes on the Fundamental Group and Covering Spaces from last year, prepared by Vladlen Timorin.
Name:
Aleksey Zinger
E-mail: azinger@math
Phone: 432-8618
Office: Math Tower 3-117
Office Hours: Mon 130-230 and Wed 3-4
| Date | Topic | Read | Problem Set |
| Aug 29 | Topological Spaces | 1-4,12-14 | p39 #5; p83 #4,7,8 p92 #3,4,8,10; p100 #7,9,11,21 p111 #2,3,6,12 |
| Aug 31 | Induced Topologies | 5,15-17 | |
| Sep 2 | Continuous Functions | 18,19 | |
| Sep 5 | No Class | ||
| Sep 7 | Product and Quotient Topologies | 19,22 | p118 #6,7; p144 #3,4 p126 #3,5,8; p133 #2,4,6,7 |
| Sep 9 | Metric Spaces | 20,21 | |
| Sep 12 | Connectedness and Path-Connectedness | 23,24 | p152 #8,10 p157 #1,2,3,8,12; p162 #2 p170 #1,4,8; p177 #1,5 |
| Sep 14 | Components and Path-Components | 9,10,25 | |
| Sep 16 | Compactness | 26,27 | |
| Sep 19 | Compactness for Metric/Order Topologies | 27,28 | p67 #7; p72 #8 p181 #1,3,7 p186 #4,6; p235 #1,5 |
| Sep 21 | One-Point Compactification | 29 | |
| Sep 23 | Tychonoff Theorem | 11,37 | |
| Sep 26 | Countability and Separation Axioms | 30,31 | p194 #5,6; p199 #2,4,9 p205 #1,4; p213 #5,7 p218 3,9; p223 #2,3; p227 #5 |
| Sep 28 | Urysohn Lemma | 32,33 | |
| Sep 30 | Tietze Extension Theorem | 35,36 | |
| Oct 7 | Urysohn Metrization Theorem | 34 | |
| Oct 3,5 | No Class | ||
| Oct 10 | Locally Finite Collections | 39,40 | p248 #5,6; p252 #3,4 p260 #2,7; p262 #2 |
| Oct 12 | Nagata-Smirnov Metrization Theorem | 40,41 | |
| Oct 14 | Smirnov Metrization Theorem | 41,42 | |
| Oct 17 | Function Spaces | 43,44 | p271 #7,10; p274 #2 p280 #2; p288 #3,4 p292 #1; p299 #7 |
| Oct 19 | Compactness in Function Spaces | 45,46 | |
| Oct 21 | Ascoli's Theorem | 46,47 | |
| Oct 24 | Baire Spaces | 48,49 | |
| 10/26 | Review | ||
| 10/28 | Midterm | ||
| Oct 31 | Path Homotopies | 51 | p330 #3; p335 #4,5,6 p341 #3,5; p348 #4,6 |
| Nov 2 | The Fundamental Group | 52,53 | |
| Nov 4 | The Fundamental Group of S1 | 53,54 | |
| Nov 7 | Applications | 55 | p353 #2,4; p359 #3,4 p366 #7,9; p370 #4 |
| Nov 9 | More Applications | 56,57 | |
| Nov 11 | Properties of the Fundamental Group | 58,59 | |
| Nov 14 | Fundamental Groups of Some Spaces | 60 | p375 #2,4,5 p483 #2,3,5 p493 #2ab,4,5 |
| Nov 16 | Equivalence of Covering Spaces | 79,80 | |
| Nov 18 | Deck Transformations | 80,81 | |
| Nov 21 | Construction of Covering Spaces | 81 | |
| Nov 23 | Construction of Covering Spaces (cont'd) | 82 | p433 #1,2; p438 #2,5 p441 #1,3; p445 #1,2 |
| 11/25 | No Class | ||
| Nov 28 | van Kampen's Theorem | 67-70 | |
| Nov 30 | Applications | 71,72,73 | |
| Dec 2 | Surfaces and Decorated Polygons | 73,74 | p454 #3,4,5; p457 #4 p462 #2; p470 #1; p476 #5 |
| Dec 5 | Abelianization of Fundamental Group | 75,76 | |
| Dec 7 | Equivalence Operations | 76,77 | |
| Dec 9 | Classification of Surfaces | 77,78 | |
| 12/12 | Review | ||
| 12/19 | Final Exam, 2-4:30pm | ||