By now we seem to have a fairly good idea of what the fundamental group is, and we are able compute it for some interesting spaces such as spheres, the torus, or more generally for products of these spaces. We also saw how covering spaces help us to understand subgroups of the fundamental group. To understand the fundamental group of more complicated constructions, we need to study the group structure that arises from loops in a bit more detail. In Week 7 we will define the wedge product of spaces, which includes the figure-eight, and the concept of a free product of groups. These two concepts are related to each other via the celebrated Seifert-van Kampen Theorem.

**Learning outcomes**

After this week, you should be able to

- construct new spaces using the wedge product
- construct and derive examples of free products of groups
- state the Seifert-van Kampen theorem and use it to compute the fundamental group in some examples

**Tasks and Materials**

- Have a look at the lecture notes when they appear.
- Read pages 43-45 in Hatcher’s book.
- Have a look at the example sheet published this week. The problems should be handed in on Thursday of Week 8

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