Eells–Kuiper manifold
In mathematics, an Eells–Kuiper manifold is a compactification of [math]\displaystyle{ \R^n }[/math] by a sphere of dimension [math]\displaystyle{ n/2 }[/math], where [math]\displaystyle{ n=2,4,8 }[/math], or [math]\displaystyle{ 16 }[/math]. It is named after James Eells and Nicolaas Kuiper. If [math]\displaystyle{ n=2 }[/math], the Eells–Kuiper manifold is diffeomorphic to the real projective plane [math]\displaystyle{ \mathbb{RP}^2 }[/math]. For [math]\displaystyle{ n\ge 4 }[/math] it is simply-connected and has the integral cohomology structure of the complex projective plane [math]\displaystyle{ \mathbb{CP}^2 }[/math] ([math]\displaystyle{ n = 4 }[/math]), of the quaternionic projective plane [math]\displaystyle{ \mathbb{HP}^2 }[/math] ([math]\displaystyle{ n = 8 }[/math]) or of the Cayley projective plane ([math]\displaystyle{ n = 16 }[/math]).
Properties
These manifolds are important in both Morse theory and foliation theory:
Theorem:[1] Let [math]\displaystyle{ M }[/math] be a connected closed manifold (not necessarily orientable) of dimension [math]\displaystyle{ n }[/math]. Suppose [math]\displaystyle{ M }[/math] admits a Morse function [math]\displaystyle{ f\colon M\to \R }[/math] of class [math]\displaystyle{ C^3 }[/math] with exactly three singular points. Then [math]\displaystyle{ M }[/math] is a Eells–Kuiper manifold.
Theorem:[2] Let [math]\displaystyle{ M^n }[/math] be a compact connected manifold and [math]\displaystyle{ F }[/math] a Morse foliation on [math]\displaystyle{ M }[/math]. Suppose the number of centers [math]\displaystyle{ c }[/math] of the foliation [math]\displaystyle{ F }[/math] is more than the number of saddles [math]\displaystyle{ s }[/math]. Then there are two possibilities:
- [math]\displaystyle{ c=s+2 }[/math], and [math]\displaystyle{ M^n }[/math] is homeomorphic to the sphere [math]\displaystyle{ S^n }[/math],
- [math]\displaystyle{ c=s+1 }[/math], and [math]\displaystyle{ M^n }[/math] is an Eells–Kuiper manifold, [math]\displaystyle{ n=2,4,8 }[/math] or [math]\displaystyle{ 16 }[/math].
See also
References
- ↑ "Manifolds which are like projective planes", Publications Mathématiques de l'IHÉS (14): 5–46, 1962, http://www.numdam.org/item?id=PMIHES_1962__14__5_0.
- ↑ Camacho, César; Scárdua, Bruno (2008), "On foliations with Morse singularities", Proceedings of the American Mathematical Society 136 (11): 4065–4073, doi:10.1090/S0002-9939-08-09371-4.
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