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There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor product of two " difference bundles" representing it as a new " difference bundle".

However the proof ( at least the original proof of Atiyah) of the tensor product formula is quite indirect and non-intuitive. I wonder if there is a "straightforward bare-hands" proof of this seemingly not very complicated formula. It would be interesting to see such a proof even assuming that all bundles under consideration are trivial.

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    $\begingroup$ If you are interested in questions in this general area, you should be sure to read Thomason's paper "Beware the phony multiplication on Quillen's $\mathcal{A}^{-1}\mathcal{A}$". $\endgroup$
    – Neil Strickland
    Commented Dec 3, 2016 at 9:49
  • $\begingroup$ I looked at this interesting Thomason's paper, but I think the issue here is a bit different. In the paper it is explained that construction of some K-theoretic tensor product can not be too simplified. However in the Atiyah's tensor product of "difference bundles" the resulting formula IS very simple, and what is not that simple is its proof. $\endgroup$
    – Brennan
    Commented Dec 3, 2016 at 15:02

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