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My question, or probably my confusion, is about the structure group of $\mathbb{Z}_2$-graded vector bundle and its classifying space.

Let say we have a $\mathbb{Z}_2$-graded complex vector bundle $E\to M$, where $M$ is an ordinary real smooth manifold. That is, $E=E^+\oplus E^-$ and there is an involution $\rho$ on $E$, where $\rho|_{E_\pm}=\pm 1$, defining the $\mathbb{Z}_2$-grading. To me, the structure group of $E\to M$ should be $GL(n, m)$, where $n$ and $m$ are the ranks of the $E^+\to M$ and $E^-\to M$ respectively, and $GL(n, m)$ should be the general linear supergroup. But somehow I don't know if the correct structure group of $E\to M$ should just be $GL(n+m)$, possibly with some ($\mathbb{Z}_2$) grading. In the latter case its classifying space and its cohomology are well known, but if the former case is true then, as I have asked a similar question in mathoverflow, I don't know what should be a suitable notion of classifying space is.

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    $\begingroup$ The structure group is $GL(n) \times GL(m)$. $\endgroup$ – Qiaochu Yuan Sep 12 '15 at 18:11

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