# Eells–Kuiper manifold

In mathematics, an Eells–Kuiper manifold is a compactification of ${\displaystyle R^{n}}$ by an ${\displaystyle {\frac {n}{2}}}$ - sphere, 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 ${\displaystyle RP(2)}$. For ${\displaystyle n\geq 4}$ it is simply-connected and has the integral cohomology structure of the complex projective plane ${\displaystyle CP^{2}}$ (${\displaystyle n=4}$), of the quaternionic projective plane ${\displaystyle HP^{2}}$ (${\displaystyle n=8}$) or of the Cayley projective plane (n = 16).

## Properties

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

Theorem:[1] Let ${\displaystyle M}$ be a connected closed manifold (not necessarily orientable) of dimension ${\displaystyle n}$. Suppose ${\displaystyle M}$ admits a Morse function ${\displaystyle f:M\to R}$ of class ${\displaystyle C^{3}}$ with exactly three singular points. Then ${\displaystyle M}$ is a Eells–Kuiper manifold.

Theorem:[2] Let ${\displaystyle M^{n}}$ be a compact connected manifold and ${\displaystyle F}$ a Morse foliation on ${\displaystyle M}$. Suppose the number of centers ${\displaystyle c}$ of the foliation ${\displaystyle F}$ is more than the number of saddles ${\displaystyle s}$. Then there are two possibilities:

• ${\displaystyle c=s+2}$, and ${\displaystyle M^{n}}$ is homeomorphic to the sphere ${\displaystyle S^{n}}$,
• ${\displaystyle c=s+1}$, and ${\displaystyle M^{n}}$ is an Eells—Kuiper manifold, ${\displaystyle n=2,4,8}$ or ${\displaystyle 16}$.