Level:M. Semester:8.

Lecturer: Prof S Bullett (Room 252, Maths Building)
e-mail: s.r.bullett followed by qmul.ac.uk tel: 020 7882 5474

Lecture times: Wednesdays 11.00-1.00 Room G2 Maths, starting 10th January 2007.

From the second week on, there will also be a non-compulsory exercise class/discussion group (time to be arranged, but tentatively Wednesdays 10.00-11.00, before the lecture).

Assessment: 100% final examination (May 2007).

Prerequisites: It helps to have already come across the concepts of "continuity" for maps between topological spaces or metric spaces, and "compactness" for subsets of such spaces. But if there are several in the audience who are unfamiliar with these notions, or lack confidence with them, we could start with a short refresher on them. Familiarity with elementary algebra, for example with the concept of a "group", would also be useful.

SYLLABUS

1. Homotopy
Fundamental groups, covering spaces, homotopy equivalence, functoriality, CW complexes. [approx. 4 weeks]

2. Simplicial Complexes
Triangulation, simplicial homology, examples and applications. [approx. 2 weeks]

3. Singular Homology
Singular homology, homological algebra, relative homology. Mayer-Vietoris sequence and computations. Homology with coefficients. [approx. 3 weeks]

4. Degree and Fixed Point Theorems
Degrees of maps of spheres, vector fields and their zeros, Euler-Poincare-Hopf theorem, Lefschetz fixed point theorem, Borsuk-Ulam theorem and applications. [approx. 2 weeks]

There will be 4 or 5 exercise sheets. Questions from them will be set and marked each week, but the marks will not count towards a candidate's overall score.

BOOKS

Main Text:
Allen Hatcher, Algebraic Topology (Cambridge University Press, paperback, 2002)

Other Books:
M J Greenberg and J Harper, Algebraic Topology: a First Course (Benjamin/Cummings 1981)
J J Rotman, An Introduction to Algebraic Topology (Springer GTM 119, 1988)
J Vick, Homology Theory (2nd ed, Springer GTM 145, 1994)
G E Bredon, Topology and Geometry (Springer GTM 139, 1993)
J R Munkres, Elements of Algebraic Topology (Benjamin/Cummings 1984)
E H Spanier, Algebraic Topology (McGraw Hill 1966)

Comments

Hatcher's book is ideal for this course, and provides plenty of additional reading too for those who want to follow the subject further. The CUP paperback edition is not expensive, but the book is also available on Hatcher's web-page and copies are permitted for non-commercial use.

Of the other books, Greenberg and Harper is probably the closest to the course. Unfortunately it is out of print, but there is a copy in the QM library. Rotman, Vick, and Munkres are standard algebraic topology textbooks that cover most of the topics we shall look at, but they are not quite as easy to read as Greenberg and Harper. Bredon shares our geometric viewpoint, but quickly reaches a much more advanced level than the course. Spanier is the comprehensive text on algebraic topology, but it is very formal in style and difficult to read if one is not already familiar with the basic ideas.

For a very readable account of homotopy, simplicial complexes and simplicial homology, and for background information in point set topology see M A Armstrong, Basic Topology McGraw Hill 1979 (2nd edition, Springer Verlag 1983); for an interesting account of homotopy and homology concentrating on concrete problems in low dimensions, and making use of differential and holomorphic forms (de Rham cohomology), see W Fulton, Algebraic Topology: a First Course (Springer GTM 153, 1995). A book which goes much further in the same direction is R Bott and L W Tu, Differential Forms in Algebraic Topology (Springer GTM 82, 1982).

There are many other books on algebraic topology available in the library, from introductory to very advanced.

SB (last up-dated 18/12/06).

http://www.maths.qmul.ac.uk/~sb