Cross post from [MSE](https://math.stackexchange.com/q/3744292/272127). and sorry if this is an obvious question.


Here is a line of proof of [Theorem 1.15](https://books.google.com/books?id=GoyDAwAAQBAJ&lpg=PP1&pg=PA8) from


*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 p. (2010). [ZBL1196.53001](https://zbmath.org/?q=an:1196.53001).



> 
> 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 $\gamma$.



Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Very roughly speaking: Geodesics $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=p$ and $\operatorname{ind}(\gamma)<k$ correspond to critical points of Morse index less than $k$ of the energy functional $E:\Omega\_p(M)\to \mathbb{R}$, where $\Omega\_p(M)$ is the space of loops based at $p$.


The assumption that $\pi\_k(M)\cong\pi\_{k-1}(\Omega\_p(M))\neq 0$ implies that $H\_i(\Omega\_p(M))\neq 0$ for some $i\leq k-1$, by the Hurewicz Theorem. Then Morse homology implies there must be a critical point of index less than $k$.


