Classifying space of the symmetric group

Groups first appear when we study undergraduate algebra. The group we will talk about today is one of the first that undergraduates meet, it is called the symmetric group  and elements of the nth symmetric group permute the numbers {1,2,…,n}.

For instance an element of the 7th symmetric group permutes {1,…,7} and might send the elements to each other like this:

1 >>> 3

2 >>> 5

3 >>> 4

4 >>> 1

5 >>> 6

6 >>> 7

7 >>> 2

Sometimes we want to use geometry and topology to talk about algebraic things such as groups. An example of this is homology of a group, which is a nice invariant. Homology is only defined for spaces though, and so to even be able to talk about the homology of a group, we need to associate the group with a space. This space is called the classifying space  of the group. We will define the classifying space for the symmetric group, using some pictures I made for a talk last week.

For the nth symmetric group, the classifying space is constructed by first considering configurations (conf) of n points in infinite real space. What does is mean to describe a point in ‘infinite real space’? It means is that there are an infinite number of coordinates describing the position of the point, but only finitely many are non-zero. Below is a picture of a configuration of 7 points, {1,…,7}.

Configuration space 1

The nth symmetric group permutes the labels of these n points: below is the same configuration of 7 points, with the labels permuted by the element of the 7th symmetric group we described earlier. This permutation of the labels is called an action of the symmetric group on the configuration space.

Configuration space 2

When we have an action, we can ‘divide’ by that action. In the case of our action, this ‘dividing’ means that we think of all the different ways of labeling this configuration with {1,…,7} as the same: we no longer care about how the points are numbered, just which 7 points we chose. It leaves us with the space of  unlabeled configurations: this is the classifying space of the symmetric group.
Classifying space


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