In mathematics, Eells–Kuiper manifold is a compactification

Compactification

Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...

 of by an - sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, where n = 2, 4, 8, or 16.

If n = 2, the Eells–Kuiper manifold is diffeomorphic

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

 to the real projective plane

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

 . For it is simply-connected

Simply connected space

In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...

 and has the integral cohomology structure of the complex projective plane  (), of the quaternionic projective plane

Quaternionic projective space

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted byand is a closed manifold of dimension 4n...

  () or of the Cayley projective plane (n = 16).

Properties

These manifolds are important in both Morse theory

Morse theory

In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...

 and foliation theory

Foliation

In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....

:

Theorem: Let be a connected

Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...

 closed

Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

 manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 (not necessarily orientable) of dimension . Suppose admits a Morse function  of class with exactly three singular points

Critical point

Critical point may refer to:*Critical point *Critical point *Critical point *Construction point of a ski jumping hill-See also:*Brillouin zone*Percolation thresholds...

. 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 saddle

Saddle

A saddle is a supportive structure for a rider or other load, fastened to an animal's back by a girth. The most common type is the equestrian saddle designed for a horse, but specialized saddles have been created for camels and other creatures...

s . Then there are two possibilities:

• , and is homeomorphic to the sphere ,

• , and is an Eells—Kuiper manifold, or .