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 Sn 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 Sn except for n=5. Besides, we show that the uniqueness holds on S5 if and only if π0(Diff(S4))≃Z2, i.e., pseudo-isotopy implies isotopy for diffeomorphisms on S4. At last, some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds.