I'm trying to prove proposition $4.8$ of Husemöller (pay 148)
Here it is the text:
Where the only hints are:
$[L^{n+1}(\xi),L^{n+1}(zp)] \approx [L^{n}(\xi),L^{n}(p)] \oplus [\xi,z]$
$[L^{n+1}(\xi),L^{n+1}(p)] \approx [L^{n}(\xi),L^n(p)] \oplus [\xi , Id]$
Notations: $\xi$ is a vector bundle over $X$. $L^n(\xi)=(n+1)\xi$. With $[ \xi , f]$ i'll denote the usual clutch construction, here I found a short explanation of it. With $(L^{n+1}(\xi,p))_{\pm}$ I'll denote the usual $\pm$ splitting with the property that $L^n(\xi) = L^n(\xi,p)_+ \oplus L^n(\xi,p)_-$ and for a linear clutching function $L^n(q)$, the following formula holds $[L^n(\xi),L^n(p)] \approx [(L^n(\xi,p))_+,Id]\oplus [(L^n(\xi,p))_-,z]$
I tried writing down the hints and using the splitting, and I obtain at the left side of the $\approx$ the term $[(L^{n+1}(\xi,p))_-,z]$ and at the right side $[(L^{n}(\xi,p))_-,z]$. I don't know if I can do something to this pair of bundles. They are quite similar and I'm tempted to "simplify" the $n+1$ but I don't think it makes sense. So I don't know any other way to solve it. Maybe the cause is that I can't handle very well the linearization of a polynomial clutching function, but again, I don't know where to study the properties I don't know.
Thanks for every advices/hints/solutions
I'll give you some hints about the prop.
You have to prove two facts
fact 1 The clutch construction preserve direct sum
fact 2 The $\pm$ splitting is unique and preserve the direct sum
Using these facts you can easily prove your statement
here you can find one of my old answer with some calculation done:)