Eells–Kuiper manifold

In mathematics, an Eells–Kuiper manifold is a compactification of by a sphere of dimension , where , or . It is named after James Eells and Nicolaas Kuiper.

If , 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 ().

Properties

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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

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References

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  1. ^ Eells, James Jr.; Kuiper, Nicolaas H. (1962), "Manifolds which are like projective planes", Publications Mathématiques de l'IHÉS (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.