Q: The Thom space of a Whitney sum of vector bundles Let $\xi$ and $\eta$ be vector bundles over the same base space $X$. Their Whitney sum is a bundle $\xi\oplus\eta$ over $X$. I read somewhere (without proof) that its Thom space is given by
$$
T(\xi\oplus\eta) = T(\eta|_{D(\xi)})/T(\eta|_{S(\xi)}),
$$
where the restrictions are really pullbacks of $\eta$ under the projection maps $D(\xi),S(\xi)\to X$.

Can anyone provide a proof, or a reference to a proof? 


A: The Thom space is the homotopy cofiber of the Unit Sphere Projection:
$$ \mathbb{S}_\xi X \to X \to T(\xi) $$
while the unit sphere in a sumspace $\mathbb{S}[U\oplus V]$ is a join
$$ \begin{array}{c} \mathbb{S}[U] \times \mathbb{S}[V] & \rightarrow & \mathbb{S}[U] \\ \downarrow & & \downarrow \\
\mathbb{S}[V] & \rightarrow & \mathbb{S}[U\oplus V] 
\end{array} $$ (homotopy-pushout).  By one of "Mather's" Cube Theorems (it's a kind of distrutivity) this fiberwise pushout description integrates to a homotopy pushout of bundles:
$$ \begin{array}{c} \mathbb{S}_\xi \times_X \mathbb{S}_\eta &\rightarrow & \mathbb{S}_\xi \\ \downarrow & & \downarrow \\
\mathbb{S}_\eta & \rightarrow & \mathbb{S}_{\xi\oplus \eta}
\end{array} $$ The top and the left maps are both, as it happens, sphere fibrations, the ones relevant to your question.  Consider the composable arrows
$$ \mathbb{S}_\xi \times_X \mathbb{S}_\eta \to \mathbb{S}_\xi \to X $$ and form the following small squares as homotopy pushouts:
$$ \begin{array}{c} \mathbb{S}_\xi \times_X \mathbb{S}_\eta &\rightarrow & \mathbb{S}_\xi & \rightarrow & \mathbb{S}_{\xi\oplus\eta} & \to & X \rlap{\simeq D\eta} \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ * & \to & T(\eta|_{S\xi}) & \to 
& T(\eta|_{S\xi}) \vee T(\xi|_{S\eta}) & \to & W \\ & & \downarrow & & \downarrow & & \downarrow \\
& & * & \to & T(\xi|_{S\eta}) & \to & T(\xi) \rlap{\simeq T(\xi|_{D\eta})}\\ 
&& && \downarrow & & \downarrow \\ & & & & * & \to & T(\xi\oplus\eta) \end{array} $$ where $W$ doesn't matter for the rest of the argument.
Now, I've sneakily applied the homotopy pushout pasting lemma, several times: to identify the $-\vee-$ in the second row (which gives the Thom space below it) and to identify the other Thom spaces we want.  That says the homotopy type we want is the homotopy cofiber of a map of Thom spaces; and the particular fine result you want is: the inclusion $T(\xi|_{S\eta}) \to T(\xi|_{D\eta})$ is a cofibration and represents the homotopy class of the pushout structure map indicated by the diagram.

A: I think Statement (3.6) in the following paper of Becker and Schultz
Becker, J.C.; Schultz, R.E.
Equivariant function spaces and stable homotopy theory. I. 
Comment. Math. Helv. 49, 1-34 (1974).
is in the line of what you are after.
