In mathematics, an Eells–Kuiper manifold is a compactification of
R
n
by an
n
2
- sphere, 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
R
P
(
2
)
. For
n
≥
4
it is simply-connected and has the integral cohomology structure of the complex projective plane
C
P
2
(
n
=
4
), of the quaternionic projective plane
H
P
2
(
n
=
8
) or of the Cayley projective plane (n = 16).
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
f
:
M
→
R
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:
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
.
