TITLE: Closed formulas for topological K-theory?
QUESTION [8 upvotes]: 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 $E = \bigoplus_{\alpha}{L_\alpha}$.

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 $[L_\alpha]\in K(X)$?
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 $[L_\alpha]\in K(X)$?
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 $[L_\alpha]\in K(X)$?
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 $K(X \times \mathbb{P}^n)=K(X)[T]/(T)^n$?

REPLY [10 votes]: 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 $\prod_\alpha u_\alpha$.