I've been trying to understand principal bundles, and to that end have been looking at the bundle $$ \pi: SU(2) \to \mathbb{CP}^1,~~~ (a_{ij}) \mapsto [a_{11},a_{21}], $$ with fibre $U(1)$. I assumed that the bundle would be trivial over the standard nbds $U_1,U_2 \subset \mathbb{C}$, but can't seem to identify the local trivializations. Now $$ \pi^{-1}(U_1) = \{\left( \array{a & - \overline{b}\\\ b & \overline{a}} \right)|~ a \neq 0\}, ~~~ \pi^{-1}(U_2) = \{\left( \array{a & - \overline{b}\\\ b & \overline{a}} \right)|~ b \neq 0\}, $$ and any trivialization $\alpha_1:\pi^{-1}(U_i) \to U_i \times U(1)$, will map $$ \alpha_1:\left( \array{a & - \overline{b}\\\ b & \overline{a}} \right) \mapsto ([a,b],h_{a,b}^1), $$ for some $h_{a,b}^1$. Defining $h^1_{a,b} = arg(a) = \frac{a}{|a|}$, and similarly $h^2$, works, but then the transition functions are not in $U(1)$.
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$\begingroup$ For your "similarly" is $h^2 = arg(b)$? $\endgroup$– Ryan BudneyCommented Feb 3, 2010 at 18:23
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$\begingroup$ "I assumed that the bundle would be trivial over the standard nbds....". More generally, every fiber bundle trivializes over any contractible open set. Of course, this won't neccesarily help you CHOOSE you transition functions or show that everything works (which seems to be your issue), but it can certainly be helpful for later problems where you suspect something is a fiber bundle. $\endgroup$– Jason DeVito - on hiatusCommented Feb 3, 2010 at 18:28
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$\begingroup$ Yes, I mean $h^2_{a,b} = arg(b)$. $\endgroup$– Dyke AclandCommented Feb 3, 2010 at 18:54
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$\begingroup$ Note that this is the Hopf fibration. $\endgroup$– SebastianCommented Mar 18, 2010 at 7:15
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You have two charts $U_0=\Bbb C$ and $U_\infty=(\Bbb C\setminus 0)\cup \infty$. The transition function on the intersection $\Bbb C\setminus 0$ is $g(z)=z/|z|$.
answered Feb 3, 2010 at 18:04
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$\begingroup$ What about the general bundle $SU(n) \to \mathbb{CP}^{n-1}$ with fibre $U(n-1)$? What are the trivialisations $\alpha_k:\pi^{-1}(U_k) \to U_k \times U(n-1)$. My guess is that one embeds $U(n)$ into $SU(n) \times U(1)$, and then maps $(a_{ij})$ to $(h^k_{a_{ij}},\frac{a_{k1}}{|a_{k1}|})$. I 'feel' that $h^k_{a_{ij}}$ should somehow be related to the minor of $a_{i1}$ but can't see how to define it. $\endgroup$ Commented Feb 3, 2010 at 20:13
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$\begingroup$ Well, there are many different ways to define these trivializations. Basically, for each of the standard charts $V_1,...,V_n$ on the projective space, choose sections $s_i: V_i\to U(n)$, and then compute the transition functions as ratios of these sections on intersections of the charts. $\endgroup$ Commented Feb 3, 2010 at 20:30
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$\begingroup$ But surely I need the trivializations to define the sections. $\endgroup$ Commented Feb 3, 2010 at 20:42
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$\begingroup$ yes. So let $V_1$ be the first chart, where the 1-st coordinate is not zero. Then to a point $z=(z_1,...,z_n)$ of this chart one should attach a unitary matrix $u(z)$ with the first row proportional to $z$, and do so continuously in $z$ when $z$ is in $V_1$. This is a linear algebra problem: basically you need to construct a frame continuously depending on $z$ whose first vector is a multiple of $z$. This is not hard to do, but I don't really know the best way of doing this. And then you do similarly for $V_i$, $i>1$. $\endgroup$ Commented Feb 3, 2010 at 20:59
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$\begingroup$ Yes, this is this problem I was talking about in my earlier comment. As I said before I think the best approach is to map $V_i$ to $U(n)$ viewed as contained in $SU(n) \times U(1)$, but I'm not too sure how to do so exactly. As you said this is really a linear algebra problem (I can't see continuity being a problem) and I might re-post it as such $\endgroup$ Commented Feb 3, 2010 at 21:21