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Let $X$ be a path-connected topological space with a trivial fundamental group: $$\pi_1(X,x_0)=\{e\}.$$ Does $X$ have to be homotopic to a point?

I know that the converse is true: a contractible space has trivial fundamental group. But what about the converse? Does the fundamental group tells us enough of the space to fix its homotopy type when it is trivial?

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No, consider the sphere $S^2$.

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