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 quaternions ${\displaystyle \mathbb {H} .}$ Quaternionic projective space of dimension n is usually denoted by

and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line ${\displaystyle \mathbb {HP} ^{1}}$ is homeomorphic to the 4-sphere.

Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written

where the ${\displaystyle q_{i}}$ 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

In the language of group actions, ${\displaystyle \mathbb {HP} ^{n}}$ is the orbit space of ${\displaystyle \mathbb {H} ^{n+1}\setminus \{(0,\ldots ,0)\}}$ by the action of ${\displaystyle \mathbb {H} ^{\times }}$, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside ${\displaystyle \mathbb {H} ^{n+1}}$ one may also regard ${\displaystyle \mathbb {HP} ^{n}}$ as the orbit space of ${\displaystyle S^{4n+3}}$ by the action of ${\displaystyle {\text{Sp}}(1)}$, the group of unit quaternions. The sphere ${\displaystyle S^{4n+3}}$ then becomes a principal Sp(1)-bundle over ${\displaystyle \mathbb {HP} ^{n}}$:

This bundle is sometimes called a (generalized) Hopf fibration.

There is also a construction of ${\displaystyle \mathbb {HP} ^{n}}$ by means of two-dimensional complex subspaces of ${\displaystyle \mathbb {H} ^{2n}}$, meaning that ${\displaystyle \mathbb {HP} ^{n}}$ lies inside a complex Grassmannian.

The space ${\displaystyle \mathbb {HP} ^{\infty }}$, defined as the union of all finite ${\displaystyle \mathbb {HP} ^{n}}$'s under inclusion, is the classifying space BS³. The homotopy groups of ${\displaystyle \mathbb {HP} ^{\infty }}$ are given by ${\displaystyle \pi _{i}(\mathbb {HP} ^{\infty })=\pi _{i}(BS^{3})\cong \pi _{i-1}(S^{3}).}$ These groups are known to be very complex and in particular they are non-zero for infinitely many values of ${\displaystyle i}$. However, we do have that