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Let $P_1,P_2$ be $G_1,G_2$ principal bundles over a smooth manifold. Let $\alpha: G_1 \to G_2$ be a group homomorphism. A morphism of principal bundles over $\alpha$ is an $\alpha$-equivariant smooth bundle map.

I want to show that for a principal $G$-bundle if there exists a morphism over $\{ \operatorname{id}_G \} \to G$ then we can construct a global section.

I am bit confused because in my head the condition states that given $f: P_1 \to P_2$ then $f(p \cdot g)=f(p)\cdot \alpha(g)$ but in this case the first group is trivial (unless I don't understand the notation) so the condition would be $f(p\cdot e)=f(p)$.

I want to construct the section $s_f$ by specifying it on a trivializing open cover and then check that it behaves well under transition functions. I was thinking to define something like,

$$ U_i \to U_i \times G, \enspace p \mapsto (p,f(p,e))$$

but I don't know if this makes sense.

Abellan
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1 Answers1

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An $\{Id_G\}$-bundle is trivial since the transition functions are the identity. This bundle is thus isomorphic to $M\times \{e\}$. Suppose that you have an equivariant map $f:M\times \{e\}\rightarrow P$. Define the section $s$ of $P$, by $s(p)=f(p,e)$. You can specify it on a trivialisation of $P$ defined by the covering $(U_i)_{i\in I}$, by defined $f_i:U_i\times \{e\}\rightarrow P_{\mid U_i}$ by $s_i(p)=f_{\mid U_i\times\{e\}}(p,e)$.

Tsemo Aristide
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  • In the full exercise I am doing we have to show the following: P is trivial iff has a global section iff is exists a bundle map over the inclusion of the identity. Is this were true by your answer wouldn't every principal g bundle $P$ with a global section be $M \times { e }$ ? – Abellan Nov 28 '16 at 21:58
  • It has an $M\times{e}$ reduction. – Tsemo Aristide Nov 28 '16 at 22:00