Eells–Kuiper manifold

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

Properties

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

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

Theorem:^[2] 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

References

1. Eells, James, Jr.; Kuiper, Nicolaas H. (1962), "Manifolds which are like projective planes", Institut des Hautes Études Scientifiques Publications Mathématiques (14): 5–46, MR 0145544.
2. Camacho, César; Scárdua, Bruno (2008), "On foliations with Morse singularities", Proceedings of the American Mathematical Society 136 (11): 4065–4073, arXiv:math/0611395, doi:10.1090/S0002-9939-08-09371-4, MR 2425748.