Quaternionic projective space: Difference between revisions

Revision as of 08:05, 4 September 2018

Revision as of 15:13, 20 October 2018

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:<math>[q_0,q_1,\ldots,q_n]</math>

where the <math>q_i</math> 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

:<math>[cq_0,cq_1\ldots,cq_n]</math>.

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==Topology==

===Homotopy theory===

The space <math>\mathbb{HP}^{\infty}</math>, defined as the union of all finite <math>\mathbb{HP}^n</math>'s under inclusion, is the [[classifying space]] ''BS''³. The homotopy groups of <math>\mathbb{HP}^{\infty}</math> are given by <math>\pi_i(\mathbb{HP}^{\infty}) = \pi_i(BS^3) \cong \pi_{i-1}(S^3).</math> These groups are known to be very complex and in particular they are non-zero for infinitely many values of <math>i</math>. However, we do have that

:<math>\pi_i(\mathbb{HP}^\infty) \otimes \Q \cong \begin{cases} \Q & i = 4 \\ 0 & i \neq 4 \end{cases}</math>

It follows that rationally, i.e. after [[localisation of a space]], <math>\mathbb{HP}^\infty</math> is an [[Eilenberg–Maclane space]] <math>K(\Q,4)</math>. That is <math>\mathbb{HP}^{\infty}_{\Q} \simeq K(\Z, 4)_{\Q}.</math> (cf. the example [[K(Z,2)]]). See [[rational homotopy theory]].

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==Differential geometry==

<math>\mathbb{HP}^n</math> carries a natural [[Riemannian metric]] analagousanalogous to the [[Fubini-Study metric]] on <math>\mathbb{CP}^n</math>, with respect to which it is a compact [[quaternion-Kähler symmetric space]] with positive curvature.

Quaternionic projective space can be represented as the coset space

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===Quaternionic projective plane===

The 8-dimensional <math>\mathbb{HP}^{2}</math> has a [[circle action]], by the group of complex scalars of absolute value 1 acting on the other side (so on the right, as the convention for the action of ''c'' above is on the left). Therefore, the [[quotient manifold]]

:<math>\mathbb{HP}^{2}/\mathrm{U}(1)</math>

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==Further reading==

*[[V. I. Arnol'd]], ''Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation'', Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67. Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.

[[Category:Projective geometry]]