Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
«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 $S^2$. –  Mariano Suárez-Alvarez Mar 7 '11 at 8:24
    
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. –  Kerry Mar 7 '11 at 17:39
    
still waiting for some response. –  Kerry Mar 9 '11 at 5:50
    
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. –  Eric O. Korman Mar 11 '11 at 21:00
    
amazon.com/K-theory-Advanced-Classics-Michael-Atiyah/dp/… This version I think..I had been reading the original 1960s old one. –  Kerry Mar 12 '11 at 1:28
show 6 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.