# differential topology – Quotients of a set manifold by a set Lie group Answer

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differential topology – Quotients of a set manifold by a set Lie group

Let $$M$$ a linked paracompact differentiable manifold. Let $$G$$ a linked Lie group. I’m within the workable “regular” (e.g. flush) quotients of $$M$$ by actions of $$G$$. What topological properties can I anticipate from them (beside the fibration lengthy require sequences)? Or possibly they will breathe organized in some kindly of house?

The particular illustration which pursuits me is that of flush quotients by free actions of a non-compact Lie group (principal actions in a liberal sense). That being stated, outcomes for compact teams (or capable actions) are welcome.

A associated query is that of classifying the Lie group actions of $$G$$ on $$M$$ as much as diffeomorphism: a diffeomorphism conjugating two actions induces a diffeomorphism between the quotient areas (and identifies the orbit house classifying maps). In the transitive illustration, the corresponding Lie algebra motion is represented by a Maurer-Cartan figure, so possibly Lie algebra actions are simpler to categorise.

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