Quaternionic projective space: Difference between revisions

Revision as of 10:42, 14 October 2009

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There is also a construction of '''H'''''P''^''n'' by means of two-dimensional complex subspaces of '''C'''^2''n'', meaning that '''H'''''P''^''n'' lies inside a complex [[Grassmannian]].

==Infinite-dimensional quaternionic projective space==

The space <math>\mathbb{HP}^{\infty}</math> is the [[classifying space]] BS³; and, rationally (i.e. after [[localisation of a space]] it is an [[Eilenberg-Maclane space]] K(Z,4) (cf. the example [[K(Z,2)]]). See [[rational homotopy theory]].

==Projective line==

The one-dimensional projective space over '''H''' is called the "projective line" in generalization of the [[complex projective line]]. For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the [[MobiusMöbius group]] to the quaternion context with "linear fractional transformations". See [[inversive ring geometry]] for the uses of the projective line of the arbitrary [[ring (mathematics)|ring]].

==Infinite-dimensional quaternionic projective space==

The space <math>\mathbb{HP}^{\infty}</math> is the [[classifying space]] BS³; and, rationally (i.e. after [[localisation of a space]] it is an [[Eilenberg-Maclane space]] K(Z,4) (cf. the example [[K(Z,2)]]). See [[rational homotopy theory]].

==Quaternionic projective plane==