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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 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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Related concepts (8)
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved.
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space.
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG. As explained later, this means that classifying spaces represent a set-valued functor on the of topological spaces.
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