
#1
Dec1710, 01:01 AM

P: 19

why is S^n/S^m homotopic to S^nm1. the book just made this remark how do you see this geometrically.
how do you compute fundamental groups of matrices like O(3) and SO(3) or SL(2) and whatnot. 



#2
Dec1910, 01:15 PM

P: 662

If I understand you well, when, you do a quotient S^n/S^m ; n>= m, you
identify S^m (seen as a subspace of S^n ) to a point. This collapses a subset/subspace of S^n to a single point, which (meaning selfintersection ) does not happen in S^k. Re O(n) , etc., AFAIK, you identify them as a subset of points in R^n, or , if you know any of these is the covering space of some top space X, you may, e.g., use a SES in homotopy given by fibration, or properties of covering spaces. Maybe someone else can expand on this. HTH. 


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