Eells–Kuiper manifold

In mathematics, Eells–Kuiper manifold is a compactification of $R^n$ by an $\frac {n}{2}$ - sphere, where n = 2, 4, 8, or 16.

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:

$c=s%2B2$ , and $M^n$ is homeomorphic to the sphere $S^n$ ,

$c=s%2B1$ , and $M^n$ is an Eells—Kuiper manifold, $n=2,4,8$ or $16$ .

See also

References

^ J. Eells, N. Kuiper, Manifolds which are like projective planes — Pub. I.H.E.S., 14 , 1962, pp. 5–46. [1]
^ C. Camacho, B. Scardua, On foliations with Morse singularities. — Proc. Amer. Math. Soc., 136, 2008, pp. 4065–4073. arXiv:math/0611395v1