I am reading an article about overtwisted contact structure and I am stuck at some point. I will not add all the context because it is quite long but I can summarize my question as follows:
I would like to know how many connected components are there in $$\text{Diff}^+(S^3 \times \mathbb R),$$ the group of orientation preserving diffeomorphism of the $3$-sphere $S^3$ times $\mathbb R$. But I do not really see how to do that. I know that there is only one connected component in $$\text{Diff}^+(S^3) \quad \text{and} \quad \text{Diff}^+(\mathbb R).$$ Indeed, it is obvious for $\mathbb R$ and for $S^3$ we know that there is an homotopy equivalence between $$\text{Diff}(S^3) \to O(4)$$ where $O(4)$ is the orthogonal group of dimension 4, so that $$\pi_0(O(4)) \cong \pi_0(\text{Diff}(S^3))$$ and therefore it has only two connected components (and only one preserves the orientation). But I do not see if there is any way to connect these two results to $\text{Diff}^+(S^3 \times \mathbb R)$. Do any of you have an idea how to do that ?