Eells–Kuiper manifold

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In mathematics, an Eells–Kuiper manifold is a compactification of by an - 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 . For it is simply-connected and has the integral cohomology structure of the complex projective plane (), of the quaternionic projective plane () or of the Cayley projective plane (n = 16).

Properties[edit]

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

Theorem:[1] Let be a connected closed manifold (not necessarily orientable) of dimension . Suppose admits a Morse function of class with exactly three singular points. Then is a Eells–Kuiper manifold.

Theorem:[2] Let be a compact connected manifold and a Morse foliation on . Suppose the number of centers of the foliation is more than the number of saddles . Then there are two possibilities:

  • , and is homeomorphic to the sphere ,
  • , and is an Eells—Kuiper manifold, or .

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, MR 2425748, arXiv:math/0611395Freely accessible, doi:10.1090/S0002-9939-08-09371-4 .