Eells–Kuiper manifold

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

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

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

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

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


 * $$c=s+2$$, and $$M^n$$ is homeomorphic to the sphere $$S^n$$,
 * $$c=s+1$$, and $$M^n$$ is an Eells–Kuiper manifold, $$n=2,4,8$$ or $$16$$.