# Algebraic Topology-The Brouwer Degree

• Jan 14th 2012, 12:00 AM
WannaBe
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 $\displaystyle \sum_{i=0}^{6} [a_i, a_{i+1} ] +[a_7,a_0]$ which generates $\displaystyle sd K$ is being mapped to the generator $\displaystyle 2 ( [a_0,a_2 ] +[a_2,a_4] + [a_4,a_6]+[a_6,a_0] )$ which generates K. As far as I know, it implies that $\displaystyle deg f =2$, as we need.

My problem is with part (b) :
If we take one of the $\displaystyle a_i$ ' s and denote by $\displaystyle \pi$ the radial projection, we get that:
$\displaystyle h( \pi ( a_i) ) = f(a_i)$ . But why do we need this radial projection in the first place? isn't $\displaystyle |K|$ 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 $\displaystyle \frac{f(x)}{||f(x)||}$?" ) .

Please help me understand where am I wrong and solve this (b) part .

Thanks in advance !