suppose $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is surjective. Prove that $\operatorname{coker}\phi$ is discrete.

My attempts:

  • Prove that $\phi(G)$ is open which will lead to $H/\phi(G)$ is discrete: we have no topological information to use here.

  • prove that $Lie(H/\phi(G))=0$.

I don't know how to proceed and I don't see how to use the surjectivity of $d\phi$.

Thank you for your help.

  • 1
    $\begingroup$ Why do you say you have no topological information to use here? You know that $\phi$ is a submersion. Submersions are locally projections, and are thus open maps. $\endgroup$ – jgon Apr 19 at 14:44
  • $\begingroup$ Thank you! this is useful! $\endgroup$ – PerelMan Apr 19 at 14:52

Let $N$ be the kernel of $\phi$, $G/N$ is a Lie group and there exists a Lie hmomorphism $f:G/N\rightarrow H$, the differential of $f$ at any point of $G/N$ is an isomorphism, the local inverse mapping theorem implies that $f$ is open.

  • $\begingroup$ Thank you @Aristide. Why is $d\phi$ a local isomorphism? it is surjective but is it injective? $\endgroup$ – PerelMan Apr 19 at 14:58
  • $\begingroup$ $f$ the quotient map $G/N\rightarrow H$ is a local isomorphism. $\endgroup$ – Tsemo Aristide Apr 19 at 15:00

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