Hatcher-Algebraic-Topology · Problem 1.3.27

The Fundamental Group

Mathematics — Graph Theory · covering-spaces

Problem

For a universal cover p:X~→Xp : \tilde{X} \to X there are two actions of π1(X,x0)\pi_1(X, x_0) on the fiber p−1(x0)p^{-1}(x_0). The first is the action defined on page 69 in which the element of π1(X,x0)\pi_1(X, x_0) determined by a loop γ\gamma sends γ~(1)\tilde{\gamma}(1) to γ~(0)\tilde{\gamma}(0) for each lift γ~\tilde{\gamma} of γ\gamma to X~\tilde{X}, and the second is the action given by restricting deck transformations to the fiber (see Proposition 1.39). Show that these two actions are different when X=S1∨S1X = S^1 \vee S^1 and when X=S1×S1X = S^1 \times S^1 and determine when the two actions are the same. [This is a revised version of the original form of this exercise.]

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