Is every loop homotopic to a smooth loop?

Question

I'm studying Differential Topology for a lecture, and we're using Guillemin & Pollack as the text for the lectures. In section 1.6, they define 'simply connected manifold' as a manifold $X$ in which every smooth map $f: S^1 \to X$ is homotopic to a constant map. However, the most general definition of fundamental group has as elements equivalence classes of continuous loops. Assuming both definitions are equivalent for manifolds, we would need some result like: every continuous loop in a manifold is homotopic to a smooth loop. Also, Sard's theorem implies that given any smooth map $f: X \to Y$, where $\dim{X} < \dim{Y}$, its image has measure 0. Considering strange continuous functions such as Peano curves, it seems even more confusing for me.

Answer 1

Any continuous path $p : [0, 1] \to \Bbb{R}^n$ is homotopic to the straight line path from $p(0)$ to $p(1)$. Hence any continuous path $q : [0, 1] \to X$ where $X$ is a smooth manifold can be subdivided into a finite sequence of paths $q_i : [t_i, t_{i+1}] \to X$, $i = 1, \ldots k$ that are all smooth. Now by looking at a chart around each $t_i$ with $1 < i < k$ you can adjust the $q_i$ to make their union smooth everywhere.