Dual submanifolds in rational homology spheres |
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Authors: | FuQuan Fang |
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Institution: | 1.Department of Mathematics,Capital Normal University,Beijing,China |
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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 -) ∈ ? 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 ÉLie 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 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. |
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