Eells–Kuiper manifold

In mathematics, an Eells–Kuiper manifold is a compactification of \( \mathbb {R} ^{n} \) 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 \( {\mathbb {RP}}^{2} \). For \( n\geq 4 \) it is simply-connected and has the integral cohomology structure of the complex projective plane C \( {\displaystyle \mathbb {CP} ^{2}} \) ( } n=4), of the quaternionic projective plane \( {\mathbb {HP}}^{2} \)( n=8) or of the Cayley projective plane ( \( {\displaystyle n=16} \)).

Properties

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 \( {\displaystyle f\colon M\to \mathbb {R} } \) 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.