Topology III

MATH3281

Lecturer : Dirk Schuetz

Term : Michaelmas 2011

Lectures :
  • Thursday 10:00 in CM107
  • Friday 14:00 in CG218

Problems classes:

  • Monday 31.10 at 17:00 in CG91
  • Thursday 24.11 at 13:00 in PH30

Literature

The following is a list of books on which the lecture is based. Although we will not follow a book strictly, the material can be found in them and they may sometimes offer a different approach.
  • M.A.Armstrong, Basic Topology. Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983.
  • W.Fulton, Algebraic topology. A first course. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995.
  • J.R.Munkres, Topology: a first course. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975.
  • W.S.Massey, Algebraic topology: an introduction. Springer-Verlag, New York-Heidelberg, 1977.

Handouts

Here is a handout on quaternions, including details on why RP3 is SO(3).

Assignments

Homework Date Hand in Solutions
Problem Set 1 pdf 13.10.11 - pdf
Problem Set 2 pdf 27.10.11 04.11.11 pdf
Problem Set 3 pdf 10.11.11 - pdf
Problem Set 4 pdf 24.11.11 02.12.11 pdf

Lecture Outline

Date Outline
09.12. In this lecture we show that the fundamental group of the circle is the integers, and use this to prove the Brouwer Fixed Point Theorem for the 2-disc.
08.12. In this lecture we define simply connected spaces, give some examples of such spaces, and approach the calculation of the fundamental group of the circle.
02.12. In this lecture we define the fundamental group, give some examples and study some of its properties.
01.12. In this lecture we give an example of an 'irrational flow' on the torus, introduce the notion of homotopy and study some of its properties.
25.11. In this lecture we show that RP3 is SO(3), and show that SO(n) is connected.
24.11. In this lecture we give examples of orbits and orbit spaces, including real projective space and lens spaces.
18.11. In this lecture we discuss actions of topological groups on topological spaces, give examples and look at orbits and orbit spaces.
17.11. In this lecture we show that the 3-sphere is a topological group with quaternion multiplication. We also look at components of topological groups and show that SO(n) and O(n) are compact.
11.11. In this lecture we discuss the glueing lemma, define topological groups and give various examples, including matrix groups and quaternions.
10.11. In this lecture we will define quotient spaces and identification maps, give various examples and discuss their properties.
04.11. In this lecture we will deal more with components, and define the notion of path-connectedness. We will give examples to highlight the differences between path-connected and connected.
03.11. In this lecture we will see more properties of connectedness, in particular behaviour under continuous functions and products.
28.10. In this lecture we will see more properties of the product topology. In particular we will show that the product of compact spaces is compact. We may also introduce the concept of connectedness.
27.10. In this lecture we will prove the Lebesgue Lemma and introduce the product topology on a cartesian product of topological spaces.
21.10. In this lecture we will prove the Heine-Borel Theorem, use it to get important compact spaces, and show the Bolzano Weierstrass Theorem.
20.10. In this lecture we define compactness, including basic properties of compact spaces and the Heine-Borel Theorem.
14.10. In this lecture we look at properties of continuous functions, define homeomorphisms, and give criteria equivalent to continuity.
13.10. In this lecture we look at properties of closed sets, define a basis of a topology and continuous functions.
07.10. In this lecture we show that a metric space gives rise to a topology, define the induced topology and closed sets, and discuss some of their properties.
06.10. In this lecture we will define topological spaces and metric spaces, and give various examples of them.

Last modified: 08.12.2011