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Re: What is pi_0?
Posted:
Jan 19, 2013 6:37 PM


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 nsphere 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 0sphere S^0 is the unit sphere in 1dimensional 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 pathcomponents of X. In general, pi_0(X,*) has a distinguished point consisting of the pathcomponent of the point *.
In the case X is a topological group with * the identity, pi_0(X,*) is the set of pathcomponents 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



