In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.



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> What is G'?
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I know there is concrete description in terms of pairs (M,f) with M in G and f a function (see Lemma 2.14 of Huybrechts <http://arxiv.org/pdf/1111.1745v2.pdf>). But I'd like to know if G' is diffeomorphic to some other manifold.


At some point I convinced myself that G was actually diffeomorphic to $\mathbb C^\* \times \mathbb C$ and therefore $G'$ was $\mathbb C^2$. But I've never seen this written down, so I'm getting suspicious.


EDIT: For those who have voted to close: I realize this question was very elementary. However, none of the resources on stability conditions I know of state this fact explicitly, which is quite frustrating. I think it would be very useful for people looking for a quick answer to have a place where this fact appears unambiguously.



what is the universal cover of GL(2,R)?
For the sake of completeness here is an explicit proof. Let $m\in GL\_{2}(\mathbb{R})^{+}$, then $m\rightarrow \frac{m}{\det(m)}$ maps it to an element in $SL\_{2}(\mathbb{R})$. And we know $SL\_{2}(\mathbb{R})\cong \mathbb{R}^2\times \mathbb{S}^1$ via appropriate [parametrization](https://math.stackexchange.com/questions/193993/are-there-any-good-ways-to-see-the-universal-cover-of-gl2-mathbbr/193995#193995). Thus $GL\_{2}(\mathbb{R})^{+}\cong \mathbb{R}^{+}\times \mathbb{R}^{2}\times \mathbb{S}^1\cong \mathbb{R}^{3}\times \mathbb{S}^1$ as a manifold. And its universal cover is $\mathbb{R}^4$. 


It might be overkill, but I think others were alluding to [Iwasawa decomposition](http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,R).pdf). 


