Date: Jan 19, 2013 6:37 PM
Author: W. Dale Hall
Subject: Re: What is pi_0?
Kaba wrote:

> Hi,

>

> In this page

>

> http://en.wikipedia.org/wiki/Indefinite_orthogonal_group

>

> there is the notation pi_0 in the topology section. What does it refer

> to? I don't see how the homotopy groups could cover n = 0...

>

Recall that for a space X with a distinguished point * in X, the

homotopy group pi_n(X,*) is the set of homotopy classes of maps from

the n-sphere S^n into X, sending a distinguished point to * in X.

Actually, this is generally a group only for n > 0, which is abelian

for n > 1.

The 0-sphere S^0 is the unit sphere in 1-dimensional Euclidean space

(otherwise known as the real line R). We find that S^0 is the pair of

points {-1, +1}, and holding one of these (say, -1) to be the the

distinguished point, we find the homotopy set pi_0(X,*) as the set of

homotopy classes of maps from S^0 into X, sending the distinguished

point to * in X.

Note that for two maps f,g : (S^0, *) --> (X,*) to b homotopic, there

must be a path connecting the images f(+1) and g(+1) of the non-

distinguished points in X. In short, the homotopy set of (X,*) is just

the set of path-components of X. In general, pi_0(X,*) has a

distinguished point consisting of the path-component of the point *.

In the case X is a topological group with * the identity, pi_0(X,*)

is the set of path-components of X. Noting that the identity component

X_id of X is normal, pi_0(X,*) is then X/X_id, which can be seen to be a

group.

Dale