In Week 6 we will first study the relationship of even and odd maps to the property of being a null-homotopy, and prove the celebrated Borsuk-Ulam Theorem. This theorem states that for any map from the sphere to the two-dimensional real space there exist antipodal points where the map has the same value. We then continue to study higher dimensional spheres. Finally, we will look at a deep and important relation between subgroups of the fundamental group and isomorphism classes of coverings.

**Learning outcomes**

After this week, you should be able to

- state, interpret, and prove the Borsuk-Ulam Theorem
- derive the fundamental group of product spaces and higher-dimensional spheres
- understand the relation between covering spaces and subgroups of the fundamental group

**Tasks and Materials**

- Have a look at the lecture notes when they appear.
- Read pages 31 – 32, 35, 41 – 42, 61 in Hatcher’s book.
- Have a look at the example sheet published this week. The problems should be handed in on Thursday of Week 7

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