Cross post from [MSE][1]. and sorry if this is an obvious question.

Here is a line of proof of [Theorem 1.15][2] from 

<cite authors="Brendle, Simon">_Brendle, Simon_, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4938-5/hbk). vii, 176&nbsp;p. (2010). [ZBL1196.53001](https://zbmath.org/?q=an:1196.53001).</cite>
> Let us fix two
points $p, q \in M$ such that $d(p, q) = \operatorname{diam}(M, g) > \frac{\pi}{2}$. **Since $\pi_k(M)\neq 0$, there
exists a geodesic $\gamma : [0,1] \to M$ such that $\gamma(0) = \gamma(1) = p$ and $\operatorname{ind}(\gamma) < k$.**

**Q:** Why $\pi_k(M)\neq 0 \implies \operatorname{ind}(\gamma) < k$? Is this a general fact? 

**Note:** $\operatorname{ind}(\gamma):=$ Morse index of $\gamma$.


  [1]: https://math.stackexchange.com/q/3744292/272127
  [2]: https://books.google.com/books?id=GoyDAwAAQBAJ&lpg=PP1&pg=PA8