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reference request – Does $pi_k(M)neq 0$ implies $operatorname{ind}(gamma) retort

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reference request – Does $pi_k(M)neq 0$ implies $operatorname{ind}(gamma)

nasty submit from MSE. and sorry if that is an patent query.

Here is a line of proof of Theorem 1.15 from

Brendle, Simon, Ricci rife 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.

Let us moor two
factors $p, q in M$ 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 $operatorname{ind}(gamma) < ok$.

Q: Why $pi_k(M)neq 0 implies operatorname{ind}(gamma) < ok$? Is this a standard reality?

level to: $operatorname{ind}(gamma):=$ Morse index of $gamma$.

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