The classifying space of the nth symmetric group $S\_n$ is well-known to be modeled by the space of subsets of $R^\infty$ of cardinality $n$. Various subgroups of $S\_n$ have related models. For example, $B(S\_i \times S\_j)$ is modeled by subsets of $R^\infty$ of cardinality $i + j$ with $i$ points colored red and $j$ points colored blue. More fun: the wreath product $S\_i \int S\_j \subset S\_{ij}$ has classifying space modeled by $ij$ points partitioned into $i$ sets of cardinality $j$ (but these sets are not "colored").


My question: is there a geometric model, preferably related to these, for classifying spaces of alternating groups? [Note: since any finite group is a subgroup of a symmetric group one wouldn't expect to find geometric models of arbitrary subgroups, but alternating groups seem special enough...]



Geometric model for classifying spaces of alternating groups
$n$ linearly independent points in $R^\infty$ together with an orientation of the $n$-plane which they span.


