what is meant by the space $BSU(2)$,

Question

I am aware of the group, $SU(2)$, but I came across a space, $BSU(2)$, but I have no idea of what this stands for?

I noticed that $BSU(2) \cong \mathbb{HP}^\infty$, can any one please explain how is this if this is true?

Answer 1

Recall $SU(2)\cong Sp(1)$, the group of unit quaternions. The (left) quaternionic projective $n$-space $\mathbb{HP}^n$ can be described as the quotient of the unit sphere $S^{4n+3}\subset\mathbb{H}^{n+1}$ by the (free continuous) scalar multiplication of $Sp(1)$ (on the left), hence $Sp(1)\to S^{4n+3}\to\mathbb{HP}^n$. The scalar multiplication respects the inclusion $\mathbb{H}^n\to\mathbb{H}^{n+1}$, so taking limit we have principal bundle $Sp(1)\to\varinjlim_n S^{4n+3}\to\varinjlim_n\mathbb{HP}^n$, i.e., $$ Sp(1)\to S^\infty\to\mathbb{HP}^\infty. $$ Since $S^\infty$ is contractible, we get $BSp(1)\simeq\mathbb{HP}^\infty$ (up to homotopy equivalence).