Algebraic Topology I Blok 1 2006
Home page for
Teaching Assistant Morten
Poulsen
Latest news
- Wed Sep 27 15:15:35 CEST 2006
- Please fill out the evaluation form
some time between September 30th and October 8. The
result of the
evaluation.
- Fri Sep 1 10:24:19 CEST 2006
- Look at Morten's
home-page for
solutions to some of the exercises.
- Thu Jul 13 09:58:21 CEST 2006
- The course will start Tuesday 29th August at 9.15 in Aud
6. The course will be taught in English.
Textbook
Our textbook will be
Allen Hatcher: Algebraic Topology. ISBN-13: 9780521795401.
Cambridge University Press.
The book is available from Universitetsbogladen
for DKK 335 (minus student discount) or you may download it
directly from the author Allen Hatcher who
also maintains a site with updates.
Notes
The textbook will be supplemented by the following
notes:
- Homotopy theory (pdf)
- Morten Poulsen: The fundamental groups of spheres (pdf)
- Classification of covering maps (pdf)
- From singular chains to Alexander duality (
pdf)
Course Plan and Goals
The plan is to read Chp 0 - 2 of Hatcher's book. The main
topics will be the fundamental group and simplicial and
singular homology.
The aim of the first part is the classification of covering spaces
and the aim of the second part is the
Lefschetz fixed point theorem.
Platon
tells us that true knowledge is achieved only when the student
in his/her own words formulates his/her understanding.
Introduction (1 week) and the fundamental group (3
weeks)
| Week |
Hatcher |
Topics |
Exercises |
| 1 |
Chp 0 |
Abstract homotopy theory
Homotopy, deformation retraction, CW-complex.
The surface M2
as a CW-complex.
Homotopy extension property (HEP) |
0.2, 0.3, 0.9, 0.10, 0.12
In 0.3 it's a good idea to do point b) first.
|
| 2 |
1.1 - 1.16 |
Paths, homotopy of paths, loops.
The fundamental group of a space.
Covering spaces (p 56).
Unique path lifting and unique HLP for covering spaces
(1.7, 1.30).
The fundamental group of the circle.
The fundamental theorem of algebra.
The fundamental group as a functor.
Brouwer's fixed point theorem.
|
0.17, 0.20,
1.1.5, 1.1.6
|
| 3 |
1.14
1.17 - 1.29 |
Borsuk-
Ulam theorem.
The fundamental group of the n-sphere.
Free coproducts of groups and the van Kampen
theorem.
The fundamental group of a CW-complex.
Torus knot complements.
|
1.1.10, 1.1.16 (e) and (f),
1.2.4, 1.2.7
|
| 4 |
1.30 - 1.44 |
A lifting criterion
The universal covering space
Classification of covering maps
Deck transformations.
Cayley graphs and complexes |
1.2.9, 1.2.20,
1.3.4, 1.3.9, 1.3.30
|
If you want a grade for this course:
1.1.18, 1.2.14,
1.3.13
Deadline: 29.09.06 |
Simplicial, singular, and cellular homology (5 weeks)
| Week |
Hatcher |
Topics |
Exercises |
| 5 |
2.1 - 2.12 |
 -complexes (revision
1) (
revision 2)
Simplicial homology
Singular homology of general spaces
Homotopy invariance
Relative homology groups |
2.1.1, 2.1.4, 2.1.9,
Additional exercises: 2.1.1
|
| 6 |
2.13 - 2.21 |
Excision
The Mayer-
Vietoris sequence
Homology of quotient spaces
The equivalence of simplicial and singular homology |
2.1.11, 2.1.16, 2.1.18,
2.1.29 (hint: use 2.25), 2.1.30
Bonus exercise: 2.1.14 |
| 7 |
2.22 - 2.26
2.28 - 2.33 |
The degree of a self-map of a sphere
Cellular homology
Homology of compact surfaces
Computation of degree from local degrees |
2.2.2, 2.2.4, 2.2.9.(a),
2.2.9.(d), 2.2.9 (the rest), 2.2.13
|
| |
|
No teaching 16 - 20 Oct (Fall Break) |
|
| 8 |
Chp 2 |
Euler characteristic
Homology with coefficients
Homology of projective spaces
Moore spaces |
2.2.20, 2.2.21, 2.2.22, 2.2.23, 2.2.32,
2.2.36 |
| 9 |
2C |
Applications of homology:
Jordan curve theorem (background
information)
Alexander horned
sphere
Lefschetz fixed point theorem |
2C.8, 2C.9
From
Additional exercises: 2.2.4, 2.2.5
(hint: 2.2.7)
If you don't want a grade: 2C.2 |
If you want a grade for this course:
2.1.8,
2.2.11,
2.2.14, 2.2.28
Deadline: November 17 |
Prerequisites
You should know basic general topology: Topological spaces,
continuous maps, (locally) connected topological space,
(locally) compact topological space, quotient space, manifold.
You may use my notes
(based on Munkres' book) as a reference. You should also know
very basic algebra: Group, ring, vector space, module. You can
get an idea of the required prerequisites by leafing through
Hatcher's book.
Credits
The course is worth 7.5 ECTS. To earn these points you are
expected to hand in a number of exercises, to give a number of
small presentations in class, and in general to contribute
actively to running of the course.
Online Books
J.P. May: A
concise course in algebraic topology
Jesper M. Møller: General
Topology
Hans-Bjørn Foxby: Homological
Algebra
Anders Thorup: Kommutativ
Algebra and Kommutativ
Algebra II (Danish).
Andrew Baker: Introduction
to Galois Theory.
Abstract
Algebra On Line
Books
Bott and Tu: Differential forms in algebraic topology.
Bredon: Geometry and Topology.
Dold: Lectures on Algebraic Topology.
Greenberg and Harper: Algebraic Topology.
Massey: A basic course in algebraic topology.
Jiri Matousek. Using the Borsuk-Ulam Theorem; Lectures on
Topological Methods in Combinatorics and Geometry (Springer
2002).
Rotman: An introduction to algebraic topology.
Spanier: Algebraic topology.
Switzer: Algebraic topology - homology and homotopy.
Whitehead. Elements of homotopy theory.
George K. Francis: A topological Picture Book.
Links
Hopf
Topology Archive
Algebraic Topology
Discussion List
Algebraic Topology in the Mathematics
ArXiv
RECOGNIZING
SURFACES
The Knot Plot Site
Electronic
Geometric Models
Topology Atlas (General
Topology)
Maple
computes homology (look under Mathematics - Topology)
Klein
Bottles for sale!
The Optiverse
On the
number of subgroups..
Jesper's home-page.
Jesper Michael
Møller
Last modified: Wed Nov 8 10:06:30 CET 2006