In this week we will study the topology of CW complexes in more detail. We will derive important properties that allow to identify when a space is not a CW complex. We will also discuss the real projective plane, and how it relates to the Möbius strip. Using the Seifert-van Kampen Theorem, we will see an alternative way of computing the fundamental group of the projective plane. We will discuss other examples, such as higher-genus surfaces, and argue that the fundamental group is of most use for lower-dimensional spaces. On the algebraic side, we will study presentations of groups and how they can be used to describe fundamental groups.

Learning outcomes

After this week, you should be able to

  • describe the projective plane as a Möbius strip is a cell attached
  • apply the Seifert-van Kampen Theorem to CW complexes
  • explain why some spaces (Hawaiian earring, Warsaw circle) are not CW complexes
  • describe groups using presentations
  • explain why the fundamental group of a CW complex depends only on the 2-skeleton

Tasks and Materials

  • Have a look at the lecture notes when they appear.
  • Read pages 519 – 523, 50 – 52, 97 (first paragraph) in Hatcher’s book.
  • Have a look at the example sheet published this week. The problems should be handed in on Thursday of Week 10
  • An alternative derivation of the Seifert-van Kampen Theorem is sketched in Terence Tao’s blog. The derivation is based on covering theory.