The space <math>\mathbb{HP}^{\infty}</math> is the [[classifying space]] BS<sup>3</sup>;. 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, rationallywe do have that <math>\pi_i(\mathbb{HP}^\infty) \otimes \mathbb{Q} \cong \mathbb{Q}</math> if <math>i = 4 </math> and <math>\pi_i(\mathbb{HP}^\infty) \otimes \mathbb{Q} = 0 </math> if <math>i \neq 4 </math>. It follows that rationally, i.e. after [[localisation of a space]], it<math>\mathbb{HP}^\infty</math> is an [[Eilenberg–Maclane space]] <math>K(\mathbb{Q},4)</math>. That is <math>\mathbb{HP}^{\infty}_{\mathbb{Q}} \simeq K(\mathbb{Z}, 4)_{\mathbb{Q}}</math>. (cf. the example [[K(Z,2)]]). See [[rational homotopy theory]].
