Eells–Kuiper manifold

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

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

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 ${\displaystyle f\colon M\to \mathbb {R} }$ 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 ${\displaystyle 16}$.

See also

• Reeb sphere theorem