Is it true that every (smooth) homomorphism of lie groups can be written a submersion followed by an immersion?

Question

This isn't clear to me!

Any help would be much appreciated!

Hint: given $\varphi:G \to H$, consider $G \twoheadrightarrow \operatorname{Im}\varphi \hookrightarrow H$> — Sammy Black, Nov 18 '13 at 18:56

score 2 · Answer 1 · answered Nov 18 '13 at 18:51

2

Let $f:G\to H$ be a smooth map of Lie groups. Its kernel $K$ is a closed normal subgroup of $G$, so $G/K$ is a Lie group and $f$ factors as the composition of the quotient map $G\to G/K$, which is a submersion, and an injection $G/K\to H$, which is an immersion.

answered Nov 18 '13 at 18:51

Mariano Suárez-Álvarez

135,076

One has to do this slightly contorted thing because $G/K$ need not be a closed subgroup of $H$; its topology can be very different to the topology of its image in $H$. – Mariano Suárez-Álvarez Nov 18 '13 at 18:53
Hi Mariano. Why is it clear that the quotient map is a lie group morphism? And moreover, why are these maps submersions and immersions? Could you elaborate please, it would be much appreciated! – will hart Nov 18 '13 at 23:09
You should check a book like Lee's book on smooth manifolds. – Mariano Suárez-Álvarez Nov 19 '13 at 02:33

Is it true that every (smooth) homomorphism of lie groups can be written a submersion followed by an immersion?

1 Answers1