Can the following statement be proven without the use of homotopy?
If $(M,g)$ is a closed, orientable Riemannian manifold with even dimension $m$ and positive sectional curvature then $M$ is simply connected.
When I look it up different sources (for example: Petersens book, these lecture notes) all used homotopy classes, but I'm not familiar with this concept.
So, can it be proven not using homotopy classes?