Dual submanifolds in rational homology spheres

Let Sigma be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M (+),M (-) aS, Sigma are called dual to each other if the complement Sigma - M (+) strongly homotopy retracts onto M (-) or vice-versa. In this paper, we are concerned with the basic problem of which integral triples (n;M (+),M (-)) a a"center dot(3) can appear, where n = dim Sigma - 1 and m (+/-) = codimm (+/-) - 1. The problem is motivated by several fundamental aspects in differential geometry. The theory of isoparametric/Dupin hypersurfaces in the unit sphere S (n+1) initiated by AeLie Cartan, where m (+/-) are the focal manifolds of the isoparametric/Dupin hypersurface M aS, S (n+1), and m (+/-) coincide with the multiplicities of principal curvatures of M. The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres Sigma, i.e., total spaces of smooth S-3-bundles over S-4 homeomorphic but not diffeomorphic to S-7, where m (+/-) = P (+/- x SO(4)) S-3, P -> S-4 the principal SO(4)-bundle of Sigma and P (+/-) the singular orbits of a cohomogeneity one SO(4) x SO(3)-action on P which are both of codimension 2. Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true: m (+) = m (-) = n m (+) = m (-) = 1/3n a {1, 2, 4, 8} m (+) = m (-) = 1/4n a {1, 2} m (+) = m (-) = 1/6n a {1, 2} = 1 or 2, and for the latter case, m (+) + m (-) is odd if min(m (+),m (-)) ae<yen> 2. In addition, if Sigma is a homotopy sphere and the ratio = 2 (for simplicity let us assume 2 6 m (-) ae m (+)), we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either (m (+),m (-)) = (5, 4) or m (+) +m (-) +1 is divisible by the integer delta(m (-)) (see the table on page 3), which is equivalent to the existence of (m (-)-1) linearly independent vector fields on the sphere by Adams' celebrated work. In contrast, infinitely many counterexamples are given if Sigma is a rational homology sphere.