# Elementary proof of the third Lie theorem

I would like to understand a (not well known) proof due to G.M; Tuynman, of the third Lie theorem, which asserts that for any given finite dimensional Lie algebra $\mathcal{G}$ there exists a (simply connected) Lie group $G$ whose Lie algebra is $\mathcal{G}$.

The proof is similar to using the theorem of Ado, but requires some tools:

1) The notion of universal cover.

2) The fact that for a simply connected Lie group $G$ not only the first de Rham cohomology space $H^1(G)=\{0\}$ but also $H^2(G)=\{0\}$.

My problem is:

1) To acquire quickly the concepts of algebraic topology: simple connectedness, universal covering, fundamental group ...

2) Review the fact that $H^2(G)=\{0\}$ (unfortunately, I do not have the book of Godement). 3) Understanding the example discussed in the Tuynman's paper

http://ifile.it/hy0q139

Thanks for your help!

I would like also to add, I get a copy , I find it excellent to deepen his knowledge of algebraic topology. I ran on page 202 of the Godbillon book. This is the theorem Künneth that identifies the cohomology of product manifolds with the tensor product of their cohomologies: $$K : H(M)\otimes H(N)\rightarrow H(M\times N)$$ If $M$ is compact, $K$ is an isomorphism. There are two things I can not understand, among others:

1) Godbillon consider $d=K^{-1}\circ\mu^\star$, where $\mu$ is the multiplication on the Lie group $G$, and $K$ is invertible, while the compactness is essential for $K$ to be an isomorphism?!

2) Godbillon theorem shows in his page 202, with $d$, if $G$ is connected then the smallest integer $q>0$ such that $H^q(G)\neq 0$ is odd. First, $G$ must be compact? Why $H^1(G)=0$? This is not a consequence of the above theorem where compactness is essential!

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Is mathoverflow.net/questions/8784/… of any interest? –  Gerry Myerson Aug 11 '11 at 22:41
Ah, I see that this question was first posted as an answer to the MO question. It would have been good of amine to have included the MO reference here, and it would have been good of amine to edit a reference to m.se into the answer at MO. –  Gerry Myerson Aug 12 '11 at 1:59
Please take a look at the related question on mathoverflow: mathoverflow.net/questions/8784/… as mentioned by Gerry! –  amine Aug 12 '11 at 4:48