Homotopy group

Remarks

Since $I^n / \partial I^n$ is homeomorphic to $S^n$, the elements of $\pi_n(X, x_0)$ are also the pointed homotopy classes of pointed maps $(S^n, s_0) \to (X, x_0)$. For $n = 1$ the definition gives the fundamental group. The set $\pi_0(X, x_0)$ is defined as the set of path components of $X$, with the component of $x_0$ as base point; it is not a group in general.

Example: $\pi_n(\R^m, 0)$ is trivial for all $n \ge 1$. For $f$ as above, $F(s, t) = (1 - t) f(s)$ is a homotopy that keeps $\partial I^n$ at $0$ and ends at the constant map.

Non-example: a map $f \colon I^n \to X$ that does not send all of $\partial I^n$ to $x_0$ does not define an element of $\pi_n(X, x_0)$. Likewise, free homotopy classes of maps $S^1 \to X$ form in general a different set: they correspond to conjugacy classes of $\pi_1(X, x_0)$ when $X$ is path-connected.