There is a theorem due to Sato which says that any vector bundle over $\mathbb CP^{\infty}$ decomposes as a direct sum of line bundles (which is a generalization of Grothendieck's result over $\mathbb CP^1$). I am wondering if there is a similar theorem for vector bundles over $\mathbb RP^{\infty}$ or a criterion for splitting. In my case I have a rank $2n$ grassmann bundle over $\mathbb RP^{\infty}$. Does it split as a direct sum of line bundles ? Thanks.
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1$\begingroup$ How is it a generalization of Grothendieck's result? Sorry to be pedantic, but a generalization of a theorem should reduce to that theorem in a special case, right? $\endgroup$– Qiaochu YuanCommented Mar 20, 2017 at 17:01
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$\begingroup$ Sorry, do you mean Grassmann bundle or vector bundle? I don't see how a Grassmann bundle would split. $\endgroup$– Sebastian GoetteCommented Mar 20, 2017 at 20:35
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$\begingroup$ Yes, the question is for a rank $2n$ Grassmann bundle over $\mathbb RP^{\infty}$. $\endgroup$– jackCommented Mar 20, 2017 at 23:10
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You are essentially asking about the set $[B\mathbb{Z}/2,BSO(2n)]$. This bijects with the set of conjugacy classes of homomorphisms from $\mathbb{Z}/2$ to $SO(2n)$, or in other words real, oriented representations of $\mathbb{Z}/2$. Essentially the same thing works for $[BP,BG]$ whenever $P$ is a finite $p$-group and $G$ is a compact connected Lie group. In fact, one can describe the full homotopy type of the space $\text{Map}(BP,BG)$: it is the classifying space of the groupoid of homomorphisms $P\to G$, and conjugacies between them. In this strong form the theorem is due to Dwyer and Zabrodsky. There are probably earlier references for the special case that you need. There is an exposition with references at http://www.math.purdue.edu/~wilker/papers/bzpztobg.pdf.
I am not sure exactly what to say if you want $O(2n)$ instead of $SO(2n)$, but I expect that there is some kind of straightforward reduction.
answered Mar 20, 2017 at 17:00
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$\begingroup$ Thanks for the reference. Could you please explain me how is the question equivalent to describing the set $[B\mathbb Z/2, BSO(2n)]$ ? $\endgroup$– jackCommented Mar 20, 2017 at 17:24
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1$\begingroup$ For any paracompact space $X$, vector bundles of dimension $n$ over $X$ are classified by homotopy classes of maps from $X$ to the Grassmannian $G_n(\mathbb{R}^\infty)$. This is Theorem 1.16 of math.cornell.edu/~hatcher/VBKT/VB.pdf, for example. Also, the Stiefel manifold $V_n(\mathbb{R}^\infty)$ is contractible and has a free action of $O(n)$, so the quotient $G_n=V_n/O(n)$ is a model for the classifying space $BO(n)$. Also, $\mathbb{R}P^\infty$ is a model for $B\mathbb{Z}/2$. $\endgroup$ Commented Mar 20, 2017 at 17:32
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$\begingroup$ Thanks. Then what is the splitting of a bundle equivalent to in terms of a homotopy class ? $\endgroup$– jackCommented Mar 20, 2017 at 17:41
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1$\begingroup$ @jack It is equivalent to proving that the map $[X,BO(1)^n]\to [X,BO(n)]$ induced by the inclusion of diagonal matrices $O(1)^n\to O(n)$ is surjective. $\endgroup$ Commented Mar 20, 2017 at 18:27
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$\begingroup$ It is maybe worth mentioning that the Dwyer-Zabrodsky lemma in fact holds in case $P$ is a $p$-toral group, i.e. an extension of a $p$-group by a torus. This reference might be useful: hopf.math.purdue.edu/Jackowski-Oliver/bg-bu.pdf $\endgroup$ Commented Mar 21, 2017 at 4:42