How can you geometrically see the homotopy between S^n/S^m and S^n-m-1?

[mich0144]

why is S^n/S^m homotopic to S^n-m-1. 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.

 

[Bacle]

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
self-intersection ) 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.