# 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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