Eells–Kuiper manifold

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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[edit]

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[edit]

References[edit]

  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 .