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 $n\geq 4$ 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 $16$ .

References

↑ 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 .
↑ 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 .

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[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 .

Eells–Kuiper manifold

Properties

See also

References