Is it true that any simply connected compact $n$-dimensional Riemannian manifold with positive isotropic curvature is diffeomorphic to the standard sphere $S^n$? I know that it is true for the case $n=4$ (R. Hamilton proved it in the 1990s), how about $n\geq 5$ ?
I also see that it is true for $n\geq 4$ under a stronger curvature assumption i.e. $(M,g)\times R$ has positive isotropic curvature (S. Brendle proved it in the 2000s).
This is an important open problem, see a discussion of some of the chief difficulties in Sec 1.4 of the book "The Ricci flow in Riemannian Geometry" of Andrews and Hopper.
As an unsolicited remark, I favor the shorter way (equivalent, by Micallef-Moore) of phrasing the question as "Does there exist an exotic sphere with positive isotropic curvature?"
The recent paper of Simon Brendle seems to answer your question "Ricci flow with surgery on manifolds with positive isotropic curvature"
https://arxiv.org/abs/1711.05167
See Corollary 11.3 for details.
If you are interested there is a paper of A. Fraser and J. Wolfson (http://users.math.msu.edu/users/wolfson/fraser-wolfson-final.pdf) where they prove that for a compact Riemannian manifold $(M^n,g)$ of dimension $n\geq 5$, with positive isotropic curvature, the fundamental group does not contain a subgroup isomorphic to the fundamental group of a compact Riemann surface!
The result of Brendle, Corollary 11.3 of https://arxiv.org/abs/1711.05167, needs an additional assumption that the manifold, with dimension n, does not contain non-trivial incompressible (n − 1)-dimensional space forms. A recent preprint on arxiv claims that the additional assumption is not necessary, see https://arxiv.org/abs/1909.12265v1.
