Dual submanifolds in rational homology spheres

Let E be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M＋, M_ C ＋ are called dual to each other if the complement ~ - 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-） E Na can appear, where n ： dime - 1 and m＋ = codimM~ - 1. The problem is motivated by several fundamental aspects in differential geometry. （i） The theory of isoparametric/I）upin hypersurfaces in the unit sphere Sn＋l initiated by ]~lie Cartan, where M=t= are the focal manifolds of the isoparametric/Dupin hypersurface M C Snq-1, and m~= coincide with the multiplicities of principal curvatures of M. （ii） The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres ~, i.e., total spaces of smooth S3-bundles over $4 homeomorphic but not diffeomorphic to S7, where M~ P~ ~so（4） $3, P -＋ $4 the principal SO（4）-bundle of ~ and P~ the singular orbits of a cohomogeneity one SO（4） ~ 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; 1 {1, 2,4, 8}; m＋=m_=1/3n∈ {1,2}; m＋=m_=1/4n∈{1,2}; m＋=m_=1/6n∈{1,2}; n In addition, if E is a homotopy sphere and the ratio n/m＋m-2 （for simplicity let us assume 2 ≤ m_ 〈 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 5（m_） （see the table on Page 1551）, which is equivalent to the existence of （m- - 1） linearly independent vector fields on the sphere Sin＋＋m- by Adams＇ celebrated work. In contrast, infinitely many counterexamples are given if E is a rational homology sphere.

Let Σ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M_+, M_-? Σ are called dual to each other if the complement Σ-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-) ∈ N^3 can appear, where n = dimΣ-1 and m_± = codim M_±-1. The problem is motivated by several fundamental aspects in differential geometry.(i) The theory of isoparametric/Dupin hypersurfaces in the unit sphere S^(n+1) initiated by′Elie Cartan, where M_± are the focal manifolds of the isoparametric/Dupin hypersurface M ? S^(n+1), and m± coincide with the multiplicities of principal curvatures of M.(ii) The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres Σ,i.e., total spaces of smooth S^3-bundles over S^4 homeomorphic but not diffeomorphic to S^7, where M_± =P_±×_(SO(4))S^3, P → S^4 the principal SO(4)-bundle of Σ and P_± the singular orbits of a cohomogeneity one SO(4) × 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 ∈ {1, 2, 4, 8};· m_+ = m_-=1/4n ∈ {1, 2};· m_+ = m_-=1/6n ∈ {1, 2};·n/(m_++m_-)= 1 or 2, and for the latter case, m_+ + m_-is odd if min(m_+, m_-)≥2.In addition, if Σ is a homotopy sphere and the ratio n/(m_++m_-)= 2(for simplicity let us assume 2 m_-< 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 δ(m_-)(see the table on Page 1551), which is equivalent to the existence of(m_--1) linearly independent vector fields on the sphere S^(m_++m_-)by Adams' celebrated work. In contrast, infinitely many counterexamples are given if Σ is a rational homology sphere.