Quaternionic projective space: Difference between revisions

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{{Short description|Concept in mathematics}}

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 [[quaternion]]s <math>\mathbb{H}.</math> Quaternionic projective space of dimension ''n'' is usually denoted by

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

In the language of [[Group action (mathematics)|group action]]s, <math>\mathbb{HP}^n</math> is the [[orbit space]] of <math>\mathbb{H}^{n+1}\setminus\{(0,\ldots,0)\}</math> by the action of <math>\mathbb{H}^{\times}</math>, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside <math>\mathbb{H}^{n+1}</math> one may also regard <math>\mathbb{HP}^{n}</math> as the orbit space of <math>S^{4n+3}</math> by the action of <math>\text{Sp}(1)</math>, the group of unit quaternions.<ref>{{cite book |first=Gregory L. |last=Naber, ''|chapter=Physical and Geometrical Motivation |title=Topology, geometry,Geometry and gaugeGauge fields: foundations''|publisher=Springer (|series=Texts in Applied Mathematics |volume=25 |year=2011 |orig-year=1997), p.|isbn=978-1-4419-7254-5 |page=50 |doi=10.1007/978-1-4419-7254-5_0 |chapter-url=https://books.google.com/books?id=MObgBwAAQBAJ&pg=PR1}}</ref> The sphere <math>S^{4n+3}</math> then becomes a [[principal bundle|principal Sp(1)-bundle]] over <math>\mathbb{HP}^n</math>:

:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb{HP}^n.</math>

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===Characteristic classes===

Since <math>\mathbb{HP}^1=S^4</math>, its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial [[Stiefel–Whitney class|Stiefel–Whitney]] and [[Pontryagin class]]es. The total classes are given by the following formulaformulas:

:<math>w(\mathbb{HP}^n) = (1+u)^{n+1}</math>

:<math>p(\mathbb{HP}^n) = (1+v)^{2n+2} (1+4v)^{-1} </math>

where <math>v</math> is the generator of <math>H^4(\mathbb{HP}^n;\Z)</math> and <math>u</math> is its reduction mod 2.<ref>Szczarba,{{cite journal |first=R. H. (1964).|last=Szczarba |title=On tangent bundles of fibre spaces and quotient spaces. |journal=American Journal of Mathematics, |volume=86( |issue=4), 685-697|pages=685–697 |year=1964 |doi=10.2307/2373152 |jstor=2373152 |url=https://www.maths.ed.ac.uk/~v1ranick/papers/szczarba.pdf}}</ref>

==Special cases==

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==Further reading==

*[[{{cite journal |author-link=V. I. Arnol'd]], |first=V.I. |last=Arnol''d |title=Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation'', |journal=Tr. Mat. Inst. Steklova, 1999,|volume=224 Volume|pages=56–6 224,|year=1999 Pages|citeseerx=10.1.1.50.6421 56–67|url=http://mi.mathnet.ru/eng/tm/v224/p56 }} Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.

* {{Citation

| last = Gormley

| first = P.G.

| title = Stereographic projection and the linear fractional group of transformations of quaternions

| journal = [[Proceedings of the Royal Irish Academy, Section A]]

| volume = 51

| pages = 67&ndash;85

| year = 1947

| jstor = 20488472

}}

[[Category:Projective geometry]]