Department of Mathematcs, The Graduate Center, City University of New York (CUNY)

## MATH 70700: Topology 1, Fall 2015

Course time and room:   TTh 12-1:30pm in GC 6417
Instructor:   Ilya Kofman
GC Office:   4307
Office hours:   TTh 1:30-2:30pm
Email:   ikofmanmath.csi.cuny.edu
Website:   http://www.math.csi.cuny.edu/~ikofman/

## Course outline

• (2 weeks) Review point-set topology
Reading: Notes by J.P. May and Notes by A. Hatcher.

• (2 weeks) Homotopy and cell complexes
Reading: Chapter 0 of Hatcher's book.

• (1 week) Classification of surfaces
Reading: For the "standard" classification proof, see Gallier and Xu's A Guide to the Classification Theorem for Compact Surfaces. See also Conway's ZIP Proof. See also Putnam's note.

• (6 weeks) Fundamental group, covering spaces, K(G,1) spaces
Reading: Chapter 1 of Hatcher's book.

• (3 weeks) Introduction to homology
Reading: Chapter 2.1 of Hatcher's book.

## Lessons

• Aug 27.  Review point-set topology
Reading:  Sections 1,2 of Notes by J.P. May and Chapter 1 of Notes by A. Hatcher. For solved problems, see Schaum's Outlines General Topology. For more background, see e.g. the text by Munkres.
Homework:  Problem Set 1.

• Sept 1.  Review point-set topology
Reading:  Sections 3,4 of Notes by J.P. May and Chapter 2 of Notes by A. Hatcher.
Homework:  Problems 1-7 on p.28 (Chapter 2) of Notes by A. Hatcher.

• Sept 3.  Review point-set topology
Reading:  Sections 5,6 of Notes by J.P. May and Chapter 3 of Notes by A. Hatcher. Also, read about the compact-open topology on p.529-532 in the Appendix to Hatcher's book.
Homework:  Problem Set 2. Note that after Tues Sept 8, our next class is Thurs Sept 17, so you have two weeks for this assigment.

• Sept 8.  Review point-set topology
Homework:  Problems 2, 3, 4a on p.52 (Chapter 4) of Notes by A. Hatcher.

• Sept 17.  Homotopy
Homework:  Problems 5, 6ab, 10 on p.18 (Chapter 0) of Hatcher's book.

• Sept 24-25.  CW-complexes
Reading:  pp. 5-17 of Chapter 0 of Hatcher's book. Also, read pp. 519-525 in the Appendix to Hatcher's book.
Homework:  Problems 14, 17, 20, 21, 23 on p.19 (Chapter 0) of Hatcher's book.

• Sept 29.  Classification of surfaces
Reading:  Hatcher's book has a brief treatment of surfaces at the end of section 1.2 (pages 50-52) in the context of 2--dimensional cell complexes. For the "standard" classification proof, see Gallier and Xu's A Guide to the Classification Theorem for Compact Surfaces. See also Conway's ZIP Proof. See also Putnam's note.

• Oct 1.  Seifert surfaces, intro to fundamental group
Homework:  Classify the spanning surfaces on any of these exams by D. Bayer. Also, practice identifying surfaces from surface symbols (example).

• Oct 6-8.  Fundamental group
Reading:  pp. 25-33 of Chapter 1 of Hatcher's book. See also an application of Sperner's Lemma in the NY Times.
Homework:  Problems 1, 3, 5, 6, 8 on p.38 (Section 1.1) of Hatcher's book.

• Oct 13-15.  Induced homomorphisms and Van Kampen Theorem
Reading:  pp. 34-37, 40-52 of Chapter 1 of Hatcher's book.
Homework:  Problems 16-18 on p.39 (Section 1.1) and 4, 7-14, 16 on p.53 (Section 1.2) of Hatcher's book.

• Oct 20-22.  Covering spaces
Reading:  pp. 56-63 of Chapter 1 of Hatcher's book.

• Oct 27-29.  Covering spaces
Reading:  pp. 63-70 of Chapter 1 of Hatcher's book. See also universal covering space of the double torus.
Homework:  Problems 4, 8-12 on p.79 (Section 1.3) of Hatcher's book.

• Nov 3-5.  Covering spaces
Reading:  pp. 70-78 of Chapter 1 of Hatcher's book.
Homework:  Problems 18-21, 24, 27, 29, 30 on p.80 (Section 1.3) of Hatcher's book.

• Nov 10-12.  Covering spaces and K(G,1) spaces
Reading:  pp. 87-96 (Section 1.B) of Hatcher's book.

• Nov 17-19.  Simplicial homology
Reading:  pp. 97-107 of Chapter 2 of Hatcher's book.
Homework:  Problems 3-6 on p.131 (Section 2.1) of Hatcher's book.

• Nov 24, Dec 1-3.  Singular homology, homology and fundamental group
Reading:  pp. 108-116 and 166-168 (2.A) of Chapter 2 of Hatcher's book.
Homework:  Problems 11-13 on p.132 (Section 2.1) of Hatcher's book.

• Dec 8-10.  Equivalence of singular and simplicial homology
Reading:  pp. 117-131 of Chapter 2 of Hatcher's book.
Homework:  Problems 15, 17, 22, 29 on pp.132-3 (Section 2.1) of Hatcher's book.