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.
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