Must a continuous variation through compact simply connected Lie groups preserve topology

Question

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 ?

Answer 1

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, 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