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Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see here for instance) shows that there is a homotopy equivalence $h\colon\mathbb{Z}\times BU \rightarrow \Omega^2 (\mathbb{Z}\times BU)$, hence there is a natural isomorphism of set-valued functors $KU(\_ )\rightarrow KU^{-2}(\_ )$. Karoubi calls this "weak Bott periodicity" in his book.

Now, is it possible to deduce "strong Bott periodicity" from that? I.e. that external multiplication by the Bott element is an isomorphism?

It would suffice to show that the maps $KU(X)\rightarrow KU^{-2}(X)$ are $KU(X)$-module homomorphisms, i.e. I think one has to show that a diagram like $$\begin{matrix} \mathbb{Z}\times BU\wedge\mathbb{Z}\times BU &\stackrel{\otimes}{\rightarrow}&\mathbb{Z}\times BU\\ \downarrow{h\wedge id}&&\downarrow{h}\\ \Omega^2(\mathbb{Z}\times BU)\wedge \mathbb{Z}\times BU&\stackrel{\otimes}{\rightarrow}&\Omega^2(\mathbb{Z}\times BU) \end{matrix} $$ commutes up to homotopy (the lower $\otimes$ should be something like pointwise multiplication of maps $S^2\rightarrow \mathbb{Z}\times BU$ by elements of $\mathbb{Z}\times BU$). Of course, the problem is that the map $h$ is not very explicit.

Is it at all possible to do that? Many thanks.

Edit: I also asked this at mathoverflow.net here

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1  
+1 for typesetting that diagram :-) –  joriki Mar 25 '12 at 20:29
2  
i think there is a comment about this by peter may on mathoverflow. you might want to check that. –  mland May 1 '12 at 15:30

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