Английская Википедия:Eells–Kuiper manifold

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In mathematics, an Eells–Kuiper manifold is a compactification of <math>\R^n</math> by a sphere of dimension <math>n/2</math>, where <math>n=2,4,8</math>, or <math>16</math>. It is named after James Eells and Nicolaas Kuiper.

If <math>n=2</math>, the Eells–Kuiper manifold is diffeomorphic to the real projective plane <math>\mathbb{RP}^2</math>. For <math>n\ge 4</math> it is simply-connected and has the integral cohomology structure of the complex projective plane <math>\mathbb{CP}^2</math> (<math>n = 4</math>), of the quaternionic projective plane <math>\mathbb{HP}^2</math> (<math>n = 8</math>) or of the Cayley projective plane (<math>n = 16</math>).

Properties

These manifolds are important in both Morse theory and foliation theory:

Theorem: Let <math>M</math> be a connected closed manifold (not necessarily orientable) of dimension <math>n</math>. Suppose <math>M</math> admits a Morse function <math>f\colon M\to \R</math> of class <math>C^3</math> with exactly three singular points. Then <math>M</math> is a Eells–Kuiper manifold.

Theorem: Let <math>M^n</math> be a compact connected manifold and <math>F</math> a Morse foliation on <math>M</math>. Suppose the number of centers <math>c</math> of the foliation <math>F</math> is more than the number of saddles <math>s</math>. Then there are two possibilities:

• <math>c=s+2</math>, and <math>M^n</math> is homeomorphic to the sphere <math>S^n</math>,
• <math>c=s+1</math>, and <math>M^n</math> is an Eells–Kuiper manifold, <math>n=2,4,8</math> or <math>16</math>.

See also

• Reeb sphere theorem