I'll be glad to receive some guidance to the attached question.
This is what I tried:
As for part (a): the chain which generates is being mapped to the generator which generates K. As far as I know, it implies that , as we need.
My problem is with part (b) :
If we take one of the ' s and denote by the radial projection, we get that:
. But why do we need this radial projection in the first place? isn't the unit sphere anyway? How can I define this radial projection explicitly for all the sphere and show that it's an homotopy? ( I thought of making it a linear homotopy, but it will only imply that it's an homotopy for every quadrant separately... Is this radial projection just ?" ) .
Please help me understand where am I wrong and solve this (b) part .
Thanks in advance !