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A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which admit Riemannian metrics with $\delta$-pinched sectional curvature (i.e. $\delta\le K ≤ 1$).

Are there any known (or conjectured) asymptotics on this number as $\delta\to 0$?

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  • $\begingroup$ What are some examples when $\delta < \frac{1}{4}$? $\endgroup$
    – Deane Yang
    Commented Feb 7 at 0:38
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    $\begingroup$ By Brendle-Schoen, $\delta<\tfrac{1}{4}$ if it's not a sphere and not a symmetric space. So any of the other examples listed in section 2 of this survey of Ziller should have $\delta<\tfrac{1}{4}$. (And some of those are even-dimensional.) $\endgroup$
    – macbeth
    Commented Feb 7 at 2:12
  • $\begingroup$ In the paper of Stefan Peters there is an upper bound of $e^{e^{2n+8}}, $ but there is another assumption on the sectional curvature. $\endgroup$
    – Julian Seipel
    Commented Mar 6 at 14:26
  • $\begingroup$ Thanks! I'll have to study this -- at first glance it's an explicit version for a different version of the Cheeger finiteness theorem than the one I asked about, but maybe there's a way of using it to answer my question. $\endgroup$
    – macbeth
    Commented Mar 6 at 15:58

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