Isoparametric foliations, diffeomorphism groups and exotic smooth structures

In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification problem of isoparametric foliations up to equivalence. In particular, we show that each homotopy $n$-sphere has the "same" isoparametric foliations as the standard sphere $S^n$ has except for $n=4$, reducing the classification problem on homotopy spheres to that on the standard sphere. Moreover, we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere $S^n$ except for $n=5$. Besides, we show that the uniqueness holds on $S^5$ if and only if $\pi_0(Diff(S^4))\simeq\mathbb{Z}_2$, i.e., pseudo-isotopy implies isotopy for diffeomorphisms on $S^4$. At last, some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds.