M6140 Topology Exercises - 10th Week (2022)
1 Coverings and Fundamental Groups
Exercise 1. Compute π1(S1) by using coverings.
Exercise 2. Compute π1(RPn) by using coverings.
Exercise 3. Compute the fundamental group of the Klein bottle by using coverings.
2 Topological Manifolds
Exercise 4. Show that Sn is an n-dimensional manifold for each n ≥ 1.
Exercise 5. Show that each connected manifold is compactly homogeneous, i.e. for each pair of points
there exists a compactly-supported self-homeomorphism that takes the first point to the second point.
(A support of a homeomorphism is the closure of the set of points that are not fixed points of the
homeomorphism.)
Exercise 6. Show that each connected manifold of dimension at least two is 2-homogeneous, i.e. for
each pair of pairs of points, there exists a self-homeomorphism that takes the first pair to the second
pair.
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