The quaternionic projective space is the space of right (or left) quaternion lines through the origin in , hence the space of equivalence classes of (n+1)-tuples of quaternions excluding zero, under the equivalence relation given by right (or left) multiplication with non-zero quaternions
Every continuous map for has a fixed point. This does not hold for as in this case and the antipodal map does not have a fixed point.
(homology of quaternionic projective space)
The ordinary homology groups of quaternionic projective space can be calculated using its CW structure and are given by
As any Grassmannian, quaternion projective space is canonically a coset space, in this case of the quaternion unitary group by the central product group Sp(n).Sp(1):
By the coset space-realization (2), quaternion projective space is naturally a quaternion-Kähler manifold which is also a symmetric space. As such it is an example of a Wolf space.
Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
Pierre Conner, Edwin Floyd, Section I.4 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
See also
M-theory on the 8-manifold HP2, hence on a quaternion-Kähler manifold of dimension 8 with holonomy Sp(2).Sp(1), is considered in
and argued to be dual to M-theory on G₂-manifolds in three different ways, which in turn is argued to lead to a a possible proof of confinement in the resulting 4d effective field theory (see there for more).
Last revised on July 18, 2024 at 12:47:43. See the history of this page for a list of all contributions to it.