nLab quaternionic projective space

Idea

The quaternionic projective space $\mathbb{H}P^n$ is the space of right (or left) quaternion lines through the origin in $\mathbb{H}^{n+1}$, hence the space of equivalence classes $[q_1, \cdots, q_{n+1}]$ of (n+1)-tuples of quaternions, excluding zero, under the equivalence relation given by right (or left) multiplication with non-zero quaternions

$$\mathbb{H}P^n \;\coloneqq\; \big\{ [q_1, \cdots, q_{n+1}] \big\} \;\coloneqq\; \Big( \big\{ (q_1, \cdots, q_{n+1}) \big\} \setminus \{(0, \cdots, 0)\} \Big) /_{ (q_1, \cdots, q_{n+1}) \sim (q_1 q, \cdots, q_{n+1} q) \vert q \neq 0 }$$

As $n$ ranges, there are natural inclusions

$$\ast = \mathbb{H}P^0 \hookrightarrow \mathbb{H}P^1 \hookrightarrow \mathbb{H}P^2 \hookrightarrow \mathbb{H}P^3 \hookrightarrow \cdots \,.$$

The sequential colimit over this sequence is the infinite quaternionic projective space $\mathbb{H}P^\infty$. This is a model for the classifying space $B Sp(1) \cong B SU(2) \cong B Spin(3)$.

Properties

General

Proposition

Every continuous map $\mathbb{H}P^n\rightarrow\mathbb{H}P^n$ for $n \geq 2$ has a fixed point. This does not hold for $n = 1$ as in this case $\mathbb{H}P^1\cong S^4$ and the antipodal map $S^4\rightarrow S^4, x\mapsto -x$ does not have a fixed point.

(Hatcher 02, page 180)

Cell structure

See at cell structure of projective spaces.

Homology

Proposition

(homology of quaternionic projective space)

The ordinary homology groups of quaternionic projective space $\mathbb{R}P^n$ can be calculated using its CW structure and are given by

(1)$$H^k \big( \mathbb{H}P^n \big) \;=\; \left\{ \array{ \mathbb{Z} &\vert& \; k = 0, 4,\ldots, 4(n-1),4n \\ 1 &\vert& otherwise } \right.$$

As a coset space

As any Grassmannian, quaternion projective space is canonically a coset space, in this case of the quaternion unitary group $Sp(n+1)$ by the central product group Sp(n).Sp(1):

(2)$$\mathbb{H}P^n \;\simeq\; \frac{ Sp(n+1) }{ Sp(n)\cdot Sp(1) }$$

As a quaternion-Kähler symmetric space (Wolf space)

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.

Related concepts

• real projective space

• complex projective space

• octonionic projective space

References

General

• 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

• Topospaces, Homology of quaternionic projective space

• Wikipedia, Quaternionic projective space

In string theory

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

• Michael Atiyah, Edward Witten, p. 75 onwards in $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

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).