On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Let $(m_-,m_+)$ be the pair of multiplicities of an isoparametric hypersurface in the unit sphere $S^{n+1}$ with four distinct principal curvatures -w.r.g., we assume that $m_-\le m_+$. In the present paper we prove that, in the case 4B2 of U. Abresch (Math. Ann. 264 (1983), 283-302) (i.e., where $3m_-=2(m_++1)$), $m_-$ must be either 2 or 4. As a by-product, we prove that the focal manifold $F_-$ of an isoparametric hypersurface is homeomorphic to a $S^{m_+}$ bundle over $S^{m_++m_-}$ if one of the following conditions holds: (1) $m_+>m_->1$ and $m_+=3,5,6$ or $7\pmod{8}$; (2) $m_+>2m_->2$ and $m_+=0\pmod{4}$. This generalizes partial results of Wang (1988) about the topology of Clifford type examples. Consequently, the hypersurface is homeomorphic to an iterated sphere bundle under the above condition.