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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, quaternionic projective space is an extension of the ideas of
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properti ...
and
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 ...
, to the case where coordinates lie in the ring of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s \mathbb. Quaternionic projective space of dimension ''n'' is usually denoted by :\mathbb^n and is a
closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...
of (real) dimension 4''n''. It is a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
for a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
action, in more than one way. The quaternionic projective line \mathbb^1 is homeomorphic to the 4-sphere.


In coordinates

Its direct construction is as a special case of the projective space over a division algebra. The
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
of a point can be written : _0,q_1,\ldots,q_n/math> where the q_i 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 : q_0,cq_1\ldots,cq_n/math>. In the language of
group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
s, \mathbb^n is the
orbit space In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under fun ...
of \mathbb^\setminus\ by the action of \mathbb^, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside \mathbb^ one may also regard \mathbb^ as the orbit space of S^ by the action of \text(1), the group of unit quaternions. The sphere S^ then becomes a principal Sp(1)-bundle over \mathbb^n: :\mathrm(1) \to S^ \to \mathbb^n. This bundle is sometimes called a (generalized) Hopf fibration. There is also a construction of \mathbb^ by means of two-dimensional complex subspaces of \mathbb^, meaning that \mathbb^ lies inside a complex
Grassmannian In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
.


Topology


Homotopy theory

The space \mathbb^, defined as the union of all finite \mathbb^n's under inclusion, is the
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 ...
''BS''3. The homotopy groups of \mathbb^ are given by \pi_i(\mathbb^) = \pi_i(BS^3) \cong \pi_(S^3). These groups are known to be very complex and in particular they are non-zero for infinitely many values of i. However, we do have that :\pi_i(\mathbb^\infty) \otimes \Q \cong \begin \Q & i = 4 \\ 0 & i \neq 4 \end It follows that rationally, i.e. after localisation of a space, \mathbb^\infty is an
Eilenberg–Maclane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
K(\Q,4). That is \mathbb^_ \simeq K(\Z, 4)_. (cf. the example
K(Z,2) K, or k, is the eleventh letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''kay'' (pronounced ), plural ''kays''. The letter ...
). See rational homotopy theory. In general, \mathbb^n has a cell structure with one cell in each dimension which is a multiple of 4, up to 4n. Accordingly, its cohomology ring is \Z v^, where v is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that \mathbb^n has infinite homotopy groups only in dimensions 4 and 4n+3.


Differential geometry

\mathbb^n carries a natural
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
analogous to the Fubini-Study metric on \mathbb^n, with respect to which it is a compact
quaternion-Kähler symmetric space In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature i ...
with positive curvature. Quaternionic projective space can be represented as the coset space :\mathbb^n = \operatorname(n+1)/\operatorname(n)\times\operatorname(1) where \operatorname(n) is the compact
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...
.


Characteristic classes

Since \mathbb^1=S^4, its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial Stiefel–Whitney and
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...
es. The total classes are given by the following formulas: :w(\mathbb^n) = (1+u)^ :p(\mathbb^n) = (1+v)^ (1+4v)^ where v is the generator of H^4(\mathbb^n;\Z) and u is its reduction mod 2.


Special cases


Quaternionic projective line

The one-dimensional projective space over \mathbb is called the "projective line" in generalization of the
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex ...
. For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the
Möbius group Moebius, Mœbius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Friedrich Möbius (art historian) (1928–2024), German art historian and architectural historian * Theodor ...
to the quaternion context with
linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional t ...
s. For the linear fractional transformations of an associative
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
with 1, see
projective line over a ring In mathematics, the projective line over a ring is an extension of the concept of projective line over a field (mathematics), field. Given a ring (mathematics), ring ''A'' (with 1), the projective line P1(''A'') over ''A'' consists of points iden ...
and the homography group GL(2,''A''). From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Defini ...
manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration. Explicit expressions for coordinates for the 4-sphere can be found in the article on the Fubini–Study metric.


Quaternionic projective plane

The 8-dimensional \mathbb^ 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 :\mathbb^/\mathrm(1) may be taken, writing
U(1) In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...
for the
circle group In mathematics, the circle group, denoted by \mathbb T or , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T = \. The circle g ...
. It has been shown that this quotient is the 7-
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, a result of
Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...
from 1996, later rediscovered by
Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
and
Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...
.


References


Further reading

* 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 = 67–85 , year = 1947 , jstor = 20488472 Projective geometry Homogeneous spaces Quaternions