Geometric model for classifying spaces of alternating groups - MathOverflow

Geometric model for classifying spaces of alternating groups

2010-06-30T22:27:29Z

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...]

Answer by Kevin Walker for Geometric model for classifying spaces of alternating groups

2010-06-30T23:00:36Z

$n$ linearly independent points in $R^\infty$ together with an orientation of the $n$-plane which they span.

Answer by Greg Kuperberg for Geometric model for classifying spaces of alternating groups

2010-06-30T23:10:24Z

Probably the right thing to do is to express the classifying space of $A_n$ as the non-trivial double cover of the classifying space of $S_n$. A point in the classifying space is then a set of $n$ points in $\mathbb{R}^\infty$ with a "sign ordering". A sign ordering is an equivalence class of orderings of the points, i.e., ways to number them from 1 to $n$, up to even permutations. I coined the term "sign ordering" by analogy with a cyclic ordering. But that name aside, the idea comes up all the time in various guises. For instance an orientation of a simplex is by definition a sign ordering of its vertices.

This is in the same vein as your other examples and you can of course do something similar with any subgroup $G \subseteq S_n$. You can always choose an ordering of the points up to relabeling by an element of $G$.

A bit more whimsically, you could call the configuration space of $n$ sign-ordered points in a manifold "the configuration space of $n$ fermions". Although a stricter model of the $n$ fermions is the local system or flat line bundle on $n$ unordered points, in which the holonomy negates the fiber when it induces an odd permutation of the points. This local system is similar to the sign-ordered space in the sense that the sign-ordered space is the associated principal bundle with structure group $C_2$.