# Algebraic Topology-The Brouwer Degree

• January 14th 2012, 01:00 AM
WannaBe
Algebraic Topology-The Brouwer Degree
Hi there,

As for part (a): the chain $\sum_{i=0}^{6} [a_i, a_{i+1} ] +[a_7,a_0]$ which generates $sd K$ is being mapped to the generator $2 ( [a_0,a_2 ] +[a_2,a_4] + [a_4,a_6]+[a_6,a_0] )$ which generates K. As far as I know, it implies that $deg f =2$, as we need.
If we take one of the $a_i$ ' s and denote by $\pi$ the radial projection, we get that:
$h( \pi ( a_i) ) = f(a_i)$ . But why do we need this radial projection in the first place? isn't $|K|$ 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 $\frac{f(x)}{||f(x)||}$?" ) .