Can the classifying space of a Lie group be also a Lie group?

Question

How to show a classifying space of a Lie group is also a Lie group or not a Lie group?

For example, $U(1)$ is a Lie group, let us consider the following classifying spaces:

• so $BU(1)=\mathbf{CP}^\infty$ can it be also be a Lie group, but how?

• so $B^2 U(1)=B\mathbf{CP}^\infty$ can it be also be a Lie group, but how?

• so $BSU(2)=\mathbf{HP}^\infty$ can it be also be a Lie group, but how?

Answer 1

Let $G$ be a topological group with classifying space $BG$. Suppose that $BG$ supports a group structure. Then there are homotopy equivalences $G\simeq \Omega BG\simeq\Omega^2B^2G$, so $G$ is a double loop space, and in particular supports a homotopy abelian H-space multiplication. However:

Theorem: Let $X$ be a non contractible, connected, finite complex which is a homotopy commutative H-space, then $X$ has the homotopy type of a torus.

This is Theorem 1.1 in John Hubbuck's paper On Homotopy Commutative H-spaces Top. 8 (1969), 119-126.

Thus the only compact connected Lie groups whose classifying spaces could potentially support a group structure are the tori. Since $BT^k\simeq (BS^1)^k$, to consider these we should study $BS^1$.

Now, it's well-known that $BS^1\simeq K(\mathbb{Z},2)$, and that there are topological groups within this homotopy type. In the comments below I argue that there is in fact an infinite-dimensional Banach Lie group in the homotopy type of $BS^1$. I personally do not consider this a postive answer to your question, but you may be happy with it.

We can also non-compact, non-connected Lie groups. In the connected non-compact world there are examples like $\mathbb{R}^n$ and $T^k\times\mathbb{R}^n$ which meet our critera. However these are essentially the only examples, because in general a noncompact Lie group will deformation retract onto its maximal compact subgroup, and Hubbuck's theorem then gives the same no-go result as before.

In the compact non-connected case there are torsion abelian groups. These do have classifying spaces which are non-compact, non-Lie topological groups (i.e. $\mathbb{R}P^\infty$ and other infinite lens spaces).

Finally in the non-compact non-connected case we introduce the torsion free abelian groups $\mathbb{Z}^k$. These groups do provide a positive answer to your question, since, as I mentioned in the comments, you have $B\mathbb{Z}^k\simeq T^k$.