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So I understand the mechanics of the fundamental group, but I want to gain a more natural intuition behind it. I imagine the fundamental group $\pi_{1}(X)$ to detect "holes" in a space. For example, it detects the hole in the punctured plane. However, it does not detect all type of holes, specifically the hole at $\mathbb{R}^3-0$. Presumably, $\pi_{2}(\mathbb{R}^3-0)$ could do the job in this case. I was wondering if you guys could give me some more geometrical intuition behind the fundamental group. For example, what types of "holes" can $\pi_{1}(X)$ detect? Let's restrict the discussion to subspaces of $\mathbb{R}^{n}$.

Sorry if this is a repost.

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    The topological spaces $\mathbb{R}^n - \textbf{0}$ and $S^{n-1}$ (the unit $(n-1)$-sphere in $\mathbb{R}^n$) are homotopy equivalent because there is a deformation retraction of $\mathbb{R}^n - \textbf{0}$ onto $S^{n-1}$. In particular, you can also consider the $2$-sphere $S^2$ for the purposes of your question. In some sense, the fundamental group "detects" "two-dimensional holes".2011-12-23

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