In mathematics, an Eells–Kuiper manifold is a compactification of \R^n by a sphere of dimension n/2, 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 n\ge 4 it is simply-connected and has the integral cohomology structure of the complex projective plane (n = 4), of the quaternionic projective plane (n = 8) or of the Cayley projective plane (n = 16).

Properties

These manifolds are important in both Morse theory and foliation theory: Theorem: ''Let M be a connected closed manifold (not necessarily orientable) of dimension n. Suppose M admits a Morse function of class C^3 with exactly three singular points. Then M is a Eells–Kuiper manifold.'' Theorem: 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: