On the positive isotropic curvature in higher dimensions 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).
 A: 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?"
A: 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.
A: 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!
A: 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.
