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For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy $$ f(\xi): B\longrightarrow BG, $$ $f(\xi)\in [B;BG]$, and the composition up to homotopy $$ g(\xi): B\overset{f}{\longrightarrow} BG\overset{i}{\longrightarrow}BO(n)\overset{j}{\longrightarrow}BO$$ where $BO=\lim _{n\to\infty} BO(n)$ and $g(\xi)\in [B;BO]$.

Suppose we have two such $n$-dimensional vector bundles $\xi_1$, $\xi_2$.

Question: Are there any formulas $$ g(\xi_1\oplus\xi_2)=? \text{ in terms of } g(\xi_1), g(\xi_2)? $$ And $$ f(\xi_1\oplus\xi_2)\in [B,BG\times BG]=?\text{ in terms of } f(\xi_1), f(\xi_2)? $$

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$BO(n)$ is the infinite-dimensional Grassmannian $Gr(n,\infty)$ of $n$-planes in ${\mathbf R}^\infty$. There is a natural direct sum operation $$\oplus\colon Gr(n,\infty)\times Gr(m,\infty)\to Gr(n+m,\infty)$$ (just taking the direct sum of linear subspaces) and it gives you the desired map $$BO(n)\times BO(m)\to BO(n+m).$$

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