Does trivial fundamental group imply contractible?

Question

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?

Answer 1

No, consider the sphere $S^2$.