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Concept
Quaternionic projective space
Summary
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 Quaternionic projective space of dimension n is usually denoted by
and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line is homeomorphic to the 4-sphere.
Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written
where the are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion c; that is, we identify all the
In the language of group actions, is the orbit space of by the action of , the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside one may also regard as the orbit space of by the action of , the group of unit quaternions. The sphere then becomes a principal Sp(1)-bundle over :
This bundle is sometimes called a (generalized) Hopf fibration.
There is also a construction of by means of two-dimensional complex subspaces of , meaning that lies inside a complex Grassmannian.
The space , defined as the union of all finite 's under inclusion, is the classifying space BS3. The homotopy groups of are given by These groups are known to be very complex and in particular they are non-zero for infinitely many values of . However, we do have that
It follows that rationally, i.e. after localisation of a space, is an Eilenberg–Maclane space . That is (cf. the example K(Z,2)). See rational homotopy theory.
In general, has a cell structure with one cell in each dimension which is a multiple of 4, up to . Accordingly, its cohomology ring is , where is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that has infinite homotopy groups only in dimensions 4 and .
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