Dual submanifolds in rational homology spheres

Abstract

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 _-) ∈ ℕ³ can appear, where n = dimΣ − 1 and m _± = codimm _± − 1. The problem is motivated by several fundamental aspects in differential geometry.

• (i) The theory of isoparametric/Dupin hypersurfaces in the unit sphere Sⁿ⁺¹ initiated by ÉLie Cartan, where m _± are the focal manifolds of the isoparametric/Dupin hypersurface M ⊂ Sⁿ⁺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 S³-bundles over S⁴ homeomorphic but not diffeomorphic to S⁷, where m _± = P _{± × SO(4)} S³, P → S⁴ 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}

• $\frac{n}{m_{+}+m_{-}}$ = 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 $\frac{n}{m_{+}+m_{-}}$ = 2 (for simplicity let us assume 2 6 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 3), which is equivalent to the existence of (m _-−1) linearly independent vector fields on the sphere $\mathbb{S}^{m_ + + m_ - }$ by Adams’ celebrated work. In contrast, infinitely many counterexamples are given if Σ is a rational homology sphere.