Let X be a path-connected space and suppose that every map f: S^1 --> X is homotopically trivial but not necessarily by a homotopy leaving the base point x_0 fixed. Show that pi_1(X,x_0) = 0.
Fundamental Groups, Path-Connected Space, Connectivity and Homotopy are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.