## MAT 530: Topology & Geometry, I

### Announcements

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.

### Handouts

Detailed solutions to the final exam
Notes on Van Kampen's Theorem, with pictures
Detailed solutions to the midterm
Gabriel's solutions to the first problem set, minus the Kuratowski problem, p101 #21

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.

### Course Instructor

Name: Aleksey Zinger     E-mail: azinger@math     Phone: 432-8618
Office: Math Tower 3-117     Office Hours: Mon 130-230 and Wed 3-4

### Daily Schedule and Homework Assignments

 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