Eells–Kuiper manifold
In mathematics, an Eells–Kuiper manifold is a compactification of by a sphere of dimension, where, or 16. 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
These manifolds are important in both Morse theory and foliation theory:
Theorem: Let be a connected closed manifold of dimension. Suppose admits a Morse function of class with exactly three singular points. Then is a Eells–Kuiper manifold.
Theorem: 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.
