I was thinking about the spaces which are homotopy equivalent to $\mathbb{R}^2$ minus two points and I managed to confuse myself.
I know that this space is a deformation retract of wedge sum of two circles and the fundamental group of this space is the free group on two letters $\mathbb{Z} \ast \mathbb{Z}$. However, if we first remove one point from $\mathbb{R}^2$, we know that it is a deformation retract of a circle $S^1$. Then, we remove the second point from the circle $S^1$, we get a space that is homotopic to an open interval of $\mathbb{R}$. Since the fundamental group is a homotopy invariant, I would expect that these two spaces ($\mathbb{R}^2$ minus two points and an open interval) have the same fundamental group. But I know that they do not.
What am I missing?