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.

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