Group structure on the classifying space of abelian groups

Question

Let $G$ be a topological abelian group. Its classifying space $BG$ is (at least sometimes) also a topological group. For example if $G$ is a finite abelian group, higher Eilenberg-MacLane spaces are $K(G,n)=B^nG=B(B^{n-1}G)$, implying that $B^{n-1}G$ must be a group.

1. When is it true that $BG$ has a group structure?

2. How can we describe the multiplication in $BG$ in terms of the multiplication in $G$?

I would like to understand this directly from the realization of $BG$ as the quotient $EG/G$, where $EG$ is a contractible space with a free a action of $G$. Or at least I would like to see the explicit connection of that model with a one more suitable for providing a product structure.

Answer 1

Basically this depends on how you set up the classifying space. A good choice is to define $BG$ to be the geometric realization of the nerve of $G$. Then $B$ is a monoidal functor from topological groups to topological spaces, and so it sends group objects (topological abelian groups) to group objects (topological groups).

Answer 2

The space $BG$ has a model as a labeled configuration space. The points of $BG$ consist of configurations of points in $\mathbb{R}$ labeled with elements of $G$ topologized such that if two points collide we multiply their values according to the order of the collision. Further, elements labeled by the identity are allowed to disappear or reappear, and points in a configuration may disappear to infinity. It is not difficult to argue this is homeomorphic to the nerve of the one object category associated to G.

If $G$ is abelian, then the overlay map $BG \times BG \rightarrow BG$ is well defined since if points in our two configurations coincide, we may add them in whatever order we see fit and get a single output. This operation makes $BG$ into an abelian monoid.