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Algebraic Topology-The Brouwer Degree

Hi there,

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 !