Level 4: Semester 7
Lecturer: Prof S Bullett (Room 252, Mathematics: s.r.bullett@qmul.ac.uk: 020 7882 5474)
Lecture times: Wed 11-1 (Engineering 237a)
beginning Wednesday 25th September 2002.
(Please contact the lecturer if these times are inconvenient. The
first lecture, on 25/9/02, should take place whether or not there is
a tube strike, but will only contain introductory material.)
Lecturer's consultation hours (to be
confirmed): Mon 1.30-2.30 and Wed 1.30-2.30. (Room 252, Maths)
Assessment: 100% final examination (May 2003).
1. Homotopy
Fundamental groups, covering spaces, homotopy equivalence,
functoriality, CW complexes. [4 weeks, including introductory lecture]
2. Simplicial Complexes
Triangulation, simplicial homology, examples and applications. [2 weeks]
3. Singular Homology
Singular homology, homological algebra, relative homology,
Eilenberg-Steenrod axioms. [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. [1 week]
5. Cohomology and its Applications
Singular cohomology, duality, products, applications.[2 weeks]
Main Text:
Allen Hatcher, Algebraic Topology (Cambridge University Press 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)
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: http://www.math.cornell.edu/~hatcher 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 23/9/02
http://www.maths.qmul.ac.uk/~sb