```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, thehomotopy group pi_n(X,*) is the set of homotopy classes of maps fromthe 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 abelianfor 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 ofpoints {-1, +1}, and holding one of these (say, -1) to be the thedistinguished point, we find the homotopy set pi_0(X,*) as the set ofhomotopy classes of maps from S^0 into X, sending the distinguishedpoint to * in X.Note that for two maps f,g : (S^0, *) --> (X,*) to b homotopic, theremust 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 justthe set of path-components of X. In general, pi_0(X,*) has adistinguished 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 componentX_id of X is normal, pi_0(X,*) is then X/X_id, which can be seen to be a group.Dale
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