Eells–Kuiper manifold

In mathematics, an Eells–Kuiper manifold is a compactification of $\mathbb {R} ^{n}$ 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 $n\geq 4$ 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 $f\colon M\to \mathbb {R}$ 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 $16$.

See also

• Reeb sphere theorem