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Topic: What is pi_0?
Replies: 2   Last Post: Jan 20, 2013 3:31 AM

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 W. Dale Hall Posts: 71 Registered: 2/11/05
Re: What is pi_0?
Posted: Jan 19, 2013 6:37 PM

Kaba wrote:
> Hi,
>
>
> 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

Date Subject Author
1/19/13 Kaba
1/19/13 W. Dale Hall
1/20/13 Kaba