Criteria of a trivial G bundle of a 3-manifold

It is said that

If G is a simply connected compact Lie group, then a G bundle on a 3-manifold is necessarily trivial, see e.g. here.

I think, this means that a bundle or fiber bundle is trivial if it is isomorphic to the cross product of the base space and a fiber.

• How do we show this?

• Is this below true that

"If G is a simply connected compact Lie group, then a G bundle on any $d$-manifold is necessarily trivial?" For $d=1,2,3,4,...$? What is the limitation of $d$?

• What is the criteria of trivial G bundle on a $d$-manifold? (Other than saying the definition that the bundle is the cross product of the base space and a fiber.)

Solution 1:

A principal $G$-bundle $P \to B$ is trivial if and only if there exists a section $\sigma: B \to P$. (Then $B \times G \cong P$, given by sending $(b,g) \mapsto \sigma(b)g$.) Whenever you want to know if a bundle has a section, the right place to look is obstruction theory.

Given a fiber bundle $F \to E \to B$, there is a sequence of classes $o_i(E) \in H^{i+1}(E; \pi_i F)$, where $o_i$ is defined provided that $o_{i-1} = 0$.

(Two comments deserve to be made to make this precise: 1) this sequence of classes only exists assuming that $F$ is simple, meaning that $\pi_1 F$ acts trivially on $\pi_n F$ for all $n$; 2) these cohomology groups use local coefficients, depending on the action of $\pi_1 B$ on $\pi_i F$ determined by the fiber bundle. The first is true when $F$ is a Lie group, and the second is true for any principal $G$-bundle.)

If $o_i(E) = 0$ for all $i \leq j$, then there exists a section of $E$ defined over the $(j+1)$-skeleton of $B$. If all classes $o_i(E) = 0$, then there is a section of $E$ defined over all of $B$.

When $G$ is a simply connected Lie group, then $\pi_0 G = \pi_1 G = 0$ is the trivial group by assumption. It is furthermore a theorem (I believe of Bott) that $\pi_2 G = 0$ is trivial as well; see here for more details. Then $\pi_3 G \cong \Bbb Z^t$ is nontrivial unless $G$ is the trivial group (also due to Bott).

Now, on a prinicipal $G$-bundle $P \to B$ over a 3-manifold $B$, you may run the obstruction theory machine: because $\pi_0 G = \pi_1 G = \pi_2 G = 0$, the elements $o_i(P) \in H^{i+1}(B; \pi_i G)$ are zero for $0 \leq i \leq 2$. So by the above there exists a section over the 3-skeleton; you are working over a 3-manifold, and thus you have defined a section everywhere. (The higher classes $o_i(P)$ tautologically vanish because $H^k(B) = 0$ when $k > 3$.) In fact, we never used the manifold structure here (except implicitly in assuiming that $B$ has a CW-complex structure of dimension 3, but this is harmless).

It is worth mentioning here that if $G$ is a nontrivial simply connected compact Lie group, there will always be a nontrivial $G$-bundle over a 4-manifold (because $\pi_3 G$ is nonzero).

Here is a by-hands proof encoding the above obstruction theory. Pick a cell decomposition of your 3-manifold $B$ (no need to assume orientability). Pick a section above each point in the $0$-skeleton; this is just picking a point in the fiber, so is clearly possible.

Now we induct. Assume we have constructed a section over $B^{(i)}$, the $i$-skeleton, for $i \leq 2$. (If $i = 3$, we are finished.)

For every $(i+1)$-simplex $e: \Delta^{i+1} \to B$ in the $(i+1)$-skeleton, the bundle $e^*P \cong \Delta^{i+1} \times G$ is trivial over $\Delta^{i+1}$, because the simplex is contractible; and we have already constructed a section over the $i$-skeleton, which pulls back to a map $\partial \Delta^{i+1} \to G$ which we want to extend to a map defined over all of $\Delta^{i+1}$. Note that $\partial \Delta^{i+1} \cong S^i$, and constructing an extension over all of $\Delta^{i+1}$ is constructing a null-homotopy of the map from $S^i$. Now $i \leq 2$, and $\pi_i G = 0$ in this range: every map from $S^i$ is null-homotopic. So we may choose our null-homotopy arbitrarily, and this is how we extend over this simplex $e(\Delta^{i+1}) \subset B$. Run this same argument over the rest of the simplices and you have a section defined over $B^{(i+1)}$, as desired.