M6140 Topology Exercises - 7th Week (2020)
1 Fundamental Groups
Exercise 1. Suppose that x ∈ A ⊆ X, where A is a connected component of a topological space X.
Prove that π1(A, x) ∼= π1(X, x).
Exercise 2. Show that the fundamental group at each point of a discrete space is trivial.
Exercise 3. Prove that the fundamental group at each point of an indiscrete space is trivial.
Exercise 4. Show that each star-shaped set is simply connected.
Exercise 5. Prove that if a two-point space is path-connected, then it is simply connected.
Exercise 6. Show that for each topological group G the set of loops in G based at the neutral element
is a group.
Exercise 7. The binary group operation from the previous exercise induces a binary operation on
π1(G, 1). Prove that this binary operation coincides with the binary group operation ∗ on π1(G, 1).
Exercise 8. Show that the fundamental group π1(G, 1) is abelian for each topological group G.
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