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I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma 2.27.

i) How does this lemma show that a real vector bundle can be given by a pullback of direct sums of plane bundles?

ii) Assuming i) is clear, is this equation $\Pi_f(h^*E) = \prod_j g(p_1(P_j))$ then true for suitable real 2-plane bundles $P_j$ in which $h^*$ splits as direct sum?

iii) What happens in odd dimensions? Has it something to do with choosing $g(0)=1$?

I appreciate some literature or concise explaination. :)

EDIT: I've found a good reference [1]. Namely Proposition 11.2 together with Observation 11.8 gives a satisfying answer for me.

[1] H. Blaine Lawson and Marie-Louise Michelsohn. Spin Geometry.

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  • $\begingroup$ I don't understand your question (i). Obviously the pullback of a sum of plane bundles is again a sum of plane bundles, and not every even-dimensional bundle is of this form (for instance, pick a vector bundle such as $TS^8$ which doesn't have a complex structure). In your source, the lemma is proved by formal manipulation of symmetric functions. $\endgroup$
    – Bertram Arnold
    Commented Dec 3, 2019 at 12:22
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    $\begingroup$ As discussed there, you can also use the following real splitting principle: To any metrized real vector bundle $E\to X$ associate the flag bundle $p:F_E\to X$ whose fiber over $x$ are maximal collections (so of size $\lfloor \frac{\operatorname{dim}(E)}{2}\rfloor$) of mutually orthogonal $2$-dimensional subspaces of $E_x$. The map $p^*:H^*(X)\to H^*(F_E)$ is injective, and $p^*E$ splits as a sum of plane bundles and possibly one plane bundle, so that it suffices to check an identity between characteristic classes for bundles of this form. $\endgroup$
    – Bertram Arnold
    Commented Dec 3, 2019 at 12:25
  • $\begingroup$ The very last comment did answer my first question partially. Do you have a source on that? $\endgroup$
    – mjungmath
    Commented Dec 3, 2019 at 12:33
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    $\begingroup$ As explained in ncatlab.org/nlab/show/splitting+principle it boils down to the fact that for a compact connected Lie group $G$ with maximal torus $T$, the map $H^*(BG;\mathbb Q)\to H^*(BT;\mathbb Q)$ is injective (in fact, it is the inclusion of invariants under the Weyl group). See the link for integral statements. $\endgroup$
    – Bertram Arnold
    Commented Dec 3, 2019 at 12:39
  • $\begingroup$ @YoungMath Starts at page 66 of this reference. There the proof is given for K-theory, but it works verbatim for ordinary cohomology (and indeed for any complex orientable cohomology theory). $\endgroup$
    – Denis Nardin
    Commented Dec 3, 2019 at 16:09

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