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In page 76 of Atiyah's K-theory, he discussed the possibilities of extending Bott periodicity to the case when $\mathbb{S}^{2}\times X$ is replaced by a fibration over $X$ with structure group $U_{1}$. He quote an example as:

"..In particular let $$E=L\oplus 1$$ be the sum of a line-bundle $L$ and the trivial-bundle $L$. Then $\mathbb{P}(E)$ has fibre the complex projective line on $\mathbb{S}^{2}$. Moreover on $\mathbb{P}(E)$ we have the natural tautologous line-bundle $H^{*}$. Consider of determinants then shows that, on $\mathbb{P}(E)$, we have an exact sequence $$0\rightarrow H^{*}\rightarrow \mathbb{P}^{*}(E)\rightarrow H\otimes p^{*}L\rightarrow 0$$ where $p: \mathbb{P}(E)\rightarrow X$ denotes the projection. This exact sequence shows that, in the $K(X)$ module $K(\mathbb{P}(E))$, the element $[H]$ satisfies the relation $$[E]=[H][L]+[H]^{*}"$$

This "particular case" was later expanded by Atiyah into a theorem claiming $K(\mathbb{P}(L\oplus 1))$ is generated by $[H]$ and is subject to the single relation $([H]-[1])([H][L]-[1])=0$.

My questions are:

Suppose $H$ is the canonical line bundle over $\mathbb{S}^{2}\cong \mathbb{C}\mathbb{P}_{1}$, how do we get this exact sequence? I know the question is kind of dumb but I do not see how $P(E)$ (I understand as the projectification of the fibre in E) has fibre the complex projective line on $\mathbb{S}^{2}$, since $\mathbb{P}(E)$ is a vector bundle over $X$ and not over $\mathbb{S}^{2}$.

Also I do not understand what Atiyah meant that "consideration of determinants". Someone suggested me that this may mean by applying $\wedge $ operation on the sequence to see they are actually the same; unfortunately the fact I stuck in the former question made me unable to proceed to exterior powers.

I am suspecting that Atiyah actually meant $H$ to be something different, since he used $H^{*}$ for "the tautologous line bundle", it maybe the pull back of some $H$ defined on $X$. But this looks quite unlikely so I decided to ask. Originally I thought this is simply technical but later Atiyah used similar constructions to get incredible results like: $$K(X)[t]/\prod^{n}_{i=1}(t-[L^{*}_{i}])\cong K(\mathbb{P}(L_{1}\otimes...\otimes L_{n})$$ with $L_{i}$s being line bundles over $X$ and $H$ be "the standard bundle over $\mathbb{P}(L_{1}\otimes...\otimes L_{n})$.

And $$K(\mathbb{P}(\mathbb{C}^{n}))\cong \mathbb{Z}[t]/(t-1)^{n}$$ under $t\rightarrow H$.

A proof in detail (as I am an undergraduate) would be mostly welcome.

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    «Then $P(E)$ has fibre the complex projective line on $S^2$» is a typo. Since $E$ is a vector bundle with fibers of dimension $2$, its projectivization $P(E)$ is a bundle whose fibers are copies of $P(\mathbb C^2)$, so homeomorphic to $S^2$.2011-03-07
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    That is what I think. But so far I have never found any typo in Atiyah's book, so I feel it is very unlikely to be appeared in here. Another reason is $H$ is used in here: the standard usage of $H$ is the canonical projective line bundle over $\mathbb{C}\mathbb{P}^{1}$. I suspect Atiyah meant something more subtle in here - maybe using the canonical bundle as $H^{*}$ and the dual bundle as $H$? I just feel very confused.2011-03-07
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    still waiting for some response.2011-03-09
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    what edition of the book do you have...it's not on page 76 in my version and I can't find that sequence anywhere.2011-03-11
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    http://www.amazon.com/K-theory-Advanced-Classics-Michael-Atiyah/dp/0201407922/ref=sr_1_1?ie=UTF8&qid=1299893233&sr=8-1 This version I think..I had been reading the original 1960s old one.2011-03-12
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    BTW for your $K(\mathbb{R}\mathbb{P}_{n})$ problem, I understand what you said, I think you can do it via covering map $S^{n}\rightarrow \mathbb{R}\mathbb{P}_{n}$ in the six term sequence.2011-03-12
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    That's the version of the book I have but I couldn't find what you quoted. What section is it in?2011-03-12
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    Right after the proof of Bott periodicity.2011-03-12
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    @user7887: it is not clear to me why you flagged this (your own) post for moderator attention. Please also note that for other commenters to be notified of your response (in this case @Eric), you need to put the @ symbol before their username, as I have done here. (You, as the owner of this post, will get notified of any comments regardless of the @ symbol.) Please also see [this](http://meta.tex.stackexchange.com/questions/598/when-writing-a-comment-how-do-i-ensure-that-the-person-it-is-addressed-to-gets-n) for more detailed description.2011-03-17
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    @Willie Wong♦: I think there is something wrong that caused you to find I flagged my own post. I see.2011-03-19
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    @Eric did you got any progress?2011-03-19
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    This proof is also given in "on the periodicity theorem for complex vector bundles", by Atiyah and Bott.2015-04-23

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