Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line bundle decomposition $E = \bigoplus\_{\alpha}{L\_\alpha}$.


1. Let $S(E)$ the pointed sphere bundle obtained by fiberwise compactification. $K(S(E))$ is a $K(X)$ algebra. Is there a formual in terms of $[E]\in K(X)$? In terms of all the $[L\_\alpha]\in K(X)$?
2. Let $\mathbb{P}(E)$ the associated projective bundle obtained fiberwise projectivization. $K(\mathbb{P}(E))$ is a $K(X)$ algebra. Is there a formual in terms of $[E]\in K(X)$? In terms of all the $[L\_\alpha]\in K(X)$?
3. Let $Fl(E)$ the flag bundle of $E$ obtained by associated bundle construction from the frame bundle via the representation on the flag $GL\_n/B\_n$. $K(Fl(E))$ is a $K(X)$ algebra. Is there a formula in terms of $[E] \in K(X)$? In terms of all the $[L\_\alpha]\in K(X)$?
4. Let $Fr(E)$ be the frame bundle of $E$. $K(Fr(E))$ is a $K(X)$ algebra. Is there a formula in terms of $[E] \in K(X)$?


Even if all none of the above admit closed formulas references for computations of these will help me a lot too.


A last tiny question:


Is this correct $K(X \times \mathbb{P}^n)=K(X)[T]/(T)^n$?



Closed formulas for topological K-theory?
Put $u\_\alpha=[\mathbb{C}]-[L\_\alpha]$.
It is a standard fact, known as the projective bundle theorem, that
$$K(\mathbb{P}(E))=K(X)[t]/\prod\_{\alpha}(t-u\_\alpha)$$
One can express $Fl(E)$ as the top of a tower in which each level is the projective bundle associated to a vector bundle over the level below. Using this one can obtain
$$ K(Fl(E)) = K(X)[t\_1,\dotsc,t\_d]/(r\_1,\dotsc,r\_d), $$
where $r\_i$ is the difference between the $i$'th elementary function in the variables $t\_j$, and the $i$'th elementary function in the variables $u\_\alpha$. Next, there is a natural cofibration sequence $S(E)\xrightarrow{}X\xrightarrow{}X^E$, where $X^E$ is the Thom space. In combination with the Thom isomorphism, this gives a long exact sequence 
$$ K^\*(X) \xrightarrow{f} K^\*(X) \xrightarrow{} K^\*(S(E)) \xrightarrow{d} K^{\*+1}(X). $$
Here $f$ is just multiplication by the $K$-theory Euler class, which is $\prod\_\alpha u\_\alpha$.


