Let $V$ be a finite dimensional vector space over $\mathbb{R}$. Let $S$ be the vector space of multilinear maps from $V\times V$ to $V$. Let $L:\mathbb{I}\rightarrow S$ be a continuous map such that for every $t$ we have that $L(t)$ is a Lie bracket on $V$. For every $t$, let $G\_t$ be the simply connected Lie group whose Lie algebra is $(V,L(t))$. It is given that $G\_t$ is compact for every $t$.


**Question:** Must $G\_0$ and $G\_1$ be diffeomorphic ?



Must a continuous variation through compact simply connected Lie groups preserve topology
Yes, in fact in this situation you have more, all the Lie algebras $(V, L(t))$ (which I denote by $\mathfrak{g}\_t$ to abbreviate) are isomorphic:


Let $G$ be the simply connected Lie group integrating a finite dimensional real Lie algebra $\mathfrak{g}$. Let $W$ be a finite dimensional representation of $G$, and write $W$ also for the induced Lie algebra representation of $\mathfrak{g}$.


Then the differentiable Lie group cohomology of $G$ with values in $W$ vanishes in positive degrees (this can be proven by averaging cocycles with respect to a bi-invariant Haar measure to obtain primitives for them).


The group $G$ is 2-connected (connected and simply connected by assumption, and $\pi\_2(G)=0$ holds for any Lie group ~~by a result of Hopf~~ - see edit comment at the bottom), and so there is an isomorphism induced by the so-called Van est map in degrees up to 2, between differentiable Lie group and Lie algebra cohomology with values in $W$:


$$VE: H^k\_\mathrm{diff}(G,W)\stackrel{\cong}{\to} H^k(\mathfrak{g},W),\ \ \ k=0,1,2$$


In particular, considering $W$ to be the adjoint representation of a Lie algebra on itself, one conclusion is that, in the setting of the question, $H^2(\mathfrak{g}\_t, \mathfrak{g}\_t)=0$.


A result of Nijenhuis-Richardson says that any Lie algebra $\mathfrak{g}$ such that $H^2(\mathfrak{g}, \mathfrak{g})=0$ is rigid, i.e., nearby Lie algebras to $\mathfrak{g}$ are isomorphic to it, where nearby is with respect to the topology you have on $S$. One possible place to read about this is the following survey:
*M. Crainic, F. Schätz and I. Struchiner, [A survey on stability and rigidity results for Lie algebras](https://doi.org/10.1016/j.indag.2014.07.015), Indagationes Mathematicae, 25(2014), 957-976.* Theorem 5.3 there is the rigidity result.


Since every $\mathfrak{g\_t}$ is rigid and the path is continuous, they will end up being isomorphic, and so $G\_t$ are also isomorphic Lie groups.


Edit: the result of Hopf I had in mind only guarantees finite $\pi\_2(G)$. But the result that $\pi\_2(G)=0$ is true, actual proofs and references for the result are in this Mathoverflow question: [Homotopy groups of Lie groups](https://mathoverflow.net/q/8957/104042)


