I'm studying an example provided by Hatcher in his K-theory and Vector Bundle book. I'm referring to example 1.13 pag. 24
The first part is clear, $z^2$ is the Kronecker Product of the two clutching function (or transition maps) for the taut-line-bundle, but then I can't follow is reasoning about the generalization.
1) Is $H \otimes H$ a special case of pullback of $H$?
2) Why is the matrix-representation (we fixed a point of $S^{k-1}$) for the pointwise product of clutching functions the Schur Product (or Hadamard or Pointwise product) of the two matrices?
3) Does $\left(f\oplus \text{Id} \right)\alpha_t\left(\text{Id} \oplus g \right)\alpha_t$ should be $\left(f\oplus \text{Id} \right)\left(\text{Id} \oplus g \right)\alpha_t$ ? I'm not sure, because if $t=1$ (or $t=0$) we have $\left(f\oplus \text{Id} \right)\left(\text{Id} \oplus g \right)$ which is $\left( f \oplus g \right)$, but for $t=0$ (or $t=1$) we have $\left(f\oplus \text{Id} \right)\left( g \oplus \text{Id} \right)$ but this is NOT equal to $\left( fg \oplus \text{Id} \right)$ because the matrix of $fg$ is the Schur product, not the usual matrix product). The problem is I don't understand the meaning of the $\alpha_t$ in the middle.
1) it is $H=E_z$ and so $H\otimes H=E_{z^2}$.
2) I think it is the usual Matrix product. The pointwise refers to points on $S^{k-1}$. For every point $x\in S^{k-1}$ you obtain two matrices by applying f and g. The product clutching function evaluated at any point should be the product of these two.
3) works fine as well if you take the usual matrix product