Eells–Kuiper manifold explained

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 RP ^{2} . For n\ge 4 it is simply-connected and has the integral cohomology structure of the complex projective plane CP ^{2} (n = 4 ), of the quaternionic projective plane HP ^{2} (n = 8 ) or of the Cayley projective plane (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 f\colon M\to \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 .

See also

• Reeb sphere theorem

Notes and References

1. .
2. .