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Fundamental Groups, Path-Connected Space, Connectivity and Homotopy

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.

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Solution:

Some standard results:
1) Suppose X is a space, and x0 is a point of X. We define 1(X, x0) as homotopy classes of maps f: [0, 1]  X, such that f (0) = f (1) = x0.

In other-words, homotopy means that one views as equivalent two maps which lie on a 1-parameter family of functions ft [0,1]  X which satisfy the boundary conditions ft(0) = ft(1) = x0.

So, we call 1(X, x0) as fundamental group of X, if X is path connected.

2) We say a space is "simply connected" if its fundamental group is trivial.

3) Covering space: A map p: X  Y is a covering map if around each point y in Y there exists a neighborhood Ny of b, so that ...

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