# Math 552 Elementary Geometric and Algebraic Topology: Spring 2011

 Lecture Room: B305 Van Vleck Hall Lecture Time: 11:00–11:50 MWF Lecturer: Jean-Luc Thiffeault Office: 503 Van Vleck Phone: (608)263-4089 Email: Office Hours: Right after class or by appointment.

## Syllabus

See the official syllabus.

## Textbook

The textbook for the class is Essential Topology by Martin D. Crossley.

The version I use is the corrected printintg of 2010. For earlier versions, please consult the erratum, especially for homework problems.

Other books that I will draw from (but which you don't necessarily need):

• W. Fulton, Algebraic Topology: A First Course.
• A. Hatcher, Algebraic Topology.
• C. C. Adams, The Knot Book.

## Prerequisites

Math 551 and 542, or consent of instructor.

## Homework

Each two weeks I will assign homework from the textbook (or other sources) and post it here. The homework will be due two weeks later.

Homework 1: (due Feb. 7) 2.7, 3.1, 3.2, 3.3, 3.4, 3.7, 3.8, 3.9, 4.1, 4.2, 4.5, 4.6, 4.8, 4.9.

Homework 2: (due Feb. 21) 5.1, 5.2, 5.4, 5.5, 5.8, 5.10, 6.1, 6.3, 6.5.

Homework 3: (due Mar. 21)

1. Consider a modified, more physical version of Theorem 6.39 (p. 112), as discussed in class, where the vector field vanishes on the boundary instead of being tangent to it. Try to prove or disprove that there is then a zero of the vector field in the interior of the disk, in addition to all the zeros on the boundary.

2. Following exercice 6.6 in the book, prove that the degree of a composition of maps is the product of the degrees (but do it right: the "solution" at the back of the book is useless).

3. Exercises 7.1, 7.2, 7.4 from Crossley.

Homework 4: (due Apr. 4)

1. Derive a normal form for a compact surface with b boundary components. (Modify as needed the proof given in class for surfaces without boundaries, or just quote the results that carry through without modification.)

2. Exercises 8.1, 8.3, 8.5 from Crossley.

Homework 5: (due May 2)

Exercises from Crossley: 9.1, 9.3, 9.4 (read about the universal coeff. theorem), 9.6 (9.7 in some editions: the question starts with "Take the simplicial torus...").

There will be three midterm exams in class, and no final exam. The final grade will be computed according to:

 Homework 40% Midterm exam 1 20% Midterm exam 2 20% Midterm exam 3 20%

## Exam Dates

The midterm exams will be given during the regular 50 minute lecture period.

 Midterm exam 1 Wednesday February 16, 2011 at 11:00–11:50. Midterm exam 2 Monday March 28, 2011 at 11:00–11:50. Midterm exam 3 Wednesday May 4, 2011 at 11:00–11:50.

## Schedule of Topics

 lecture date sections topic 1 01/19 2 Continuity 2 01/21 3 Topological spaces 3 01/28 3 Continuity in subspace topology 4 01/31 4.1 Connectivity 5 02/04 4.2–4.3 Compact and Hausdorff spaces 6 02/07 5.1 Homeomorphisms 7 02/09 5.2–5.3 Disjoint unions and product spaces 8 02/11 5.4 Quotient spaces 9 02/14 5.4 Quotient spaces (cont'd) – 02/16 – midterm 1 10 02/18 6.1 Homotopy 11 02/21 6.2 Homotopy equivalence 12 02/23 6.3 The circle: Path lifting 13 02/25 6.3 The circle: Homotopy lifting 14 02/28 6.4 Brouwer's fixed-point theorem 15 03/02 6.5; Hatcher 2.2 Vector fields 16 03/04 7.1 Simplicial complexes 17 03/07 7.2 Triangulation of surfaces 18 03/09 7.2–7.3; Fulton C17 Classification of surfaces 19 03/11 8; Fulton C17 Classification of surfaces (end); Homotopy group of the circle 20 03/21 8.1 Homotopy groups 21 03/23 8.1 Homotopy groups (cont'd) 22 03/25 8.2–8.3 Induced homomorphisms; Fundamental group – 03/28 – midterm 2 23 03/30 see notes Mapping class group of the torus 24 04/01 see notes Anosov homeomorphisms 25 04/04 see notes From the torus to the sphere 26 04/06 see notes Topological stirring (with movies!) 27 04/08 8.4 Path connectivity and π0 28 04/11 9.1 Simplicial homology modulo 2 29 04/13 9.1 Computing simplicial homology modulo 2 30 04/15 9.2–9.3 Integral simplicial homology 31 04/18 Adams 1.1–1.2 Knot theory 32 04/20 Adams 1.3–1.4 Reidemeister moves and links 33 04/22 Adams 1.5 Tricolorability 34 04/25 Adams 2.1–2.2 Tabulating knots 35 04/27 Adams 2.3 Tangles and Conway's notation 36 04/29 Adams 6.1 Polynomial invariants 37 05/02 Adams 7.1 DNA and knot theory – 05/04 – midterm 3 – 05/06 – career day

The lecture notes are available in one large PDF file (11.4MB) or a much smaller DjVu file (2.5MB).