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The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ gives two disjoint sets.

How do we show that $M_2=S^2 \sqcup S^2$? Moreover, I'm a bit confused by dimensions: how does $\dim_{\mathbb R}M_2=8$ relates with the latter decomposition?

This is exercise 1.2.1 from the book Complex geometry by Huybrechts.

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As you noted, $M$ is not diffeomorphic to $S^2\coprod S^2$ for dimension reasons.

On the other hand, what is true is $M$ is homotopy equivalent to $S^2 \coprod S^2$.

(The following argument is partly adapted from a paper of Montgomery)

To see this, it's enough to show that $Gl^+(4,\mathbb{R})/Gl(2,\mathbb{C})$ is homotopy equivalent to $S^2$, where $Gl^+$ denotes those matrices of positive determinant.

Now, consider the subgroups $U(2)\subseteq Gl(2,\mathbb{C})$ and $SO(4)\subseteq Gl^+(4,\mathbb{R})$.

It's relatively well known that $Gl(2,\mathbb{C})$ is diffeomorphic to $U(2)\times \mathbb{R}^4$ and that $Gl^+(4,\mathbb{R})$ is diffeomorphic to $SO(4)\times \mathbb{R}^{10}$. Further, in the usual inclusion $Gl(2,\mathbb{C})\rightarrow Gl^+(4,\mathbb{R})$, $U(2)$ becomes a subgroup of $SO(4)$.

Now, the chain of subgroups $U(2)\subseteq SO(4)\subseteq Gl^+(4,\mathbb{R})$ gives rise to a homogeneous fibration $$SO(4)/U(2)\rightarrow Gl^+(4,\mathbb{R})/U(2)\rightarrow Gl^+(4,\mathbb{R})/SO(4).$$ In light of the above diffeomorphisms, $Gl^+(4,\mathbb{R})/SO(4)$ is diffeomorphic to $\mathbb{R}^{10}$. Since Euclidean spaces are contractible, it follows that the fibration is trivial, so $Gl^+(4,\mathbb{R})/U(2)$ is diffeomorphic to $SO(4)/U(2)\times \mathbb{R}^{10}$. In particular, $SO(4)/U(2)$ is homotopy equivalent to $Gl^+(4,\mathbb{R})/U(2)$.

Now, consider the chain of subgroups $U(2)\subseteq Gl(2,\mathbb{C})\subseteq Gl^+(4,\mathbb{R})$. This gives rise to a homogeneous fibration $$Gl(2,\mathbb{C})/U(2)\rightarrow Gl^+(4,\mathbb{R})/U(2) \rightarrow Gl^+(4,\mathbb{R})/Gl(2,\mathbb{C}).$$ In this case, the fiber is diffeomoprhic to $\mathbb{R}^4$, which immediately implies that $Gl^+(4,\mathbb{R})/U(2)$ is homotopy equivalent to $Gl^+(4,\mathbb{R})/Gl(2,\mathbb{C})$.

Putting the last two paragraphs together, we now know that $SO(4)/U(2)$ is homotopy equivalent to $Gl^+(4,\mathbb{R})/Gl(2,\mathbb{C})$.

To finish off the argument, we need to show that $SO(4)/U(2)$ is diffeomorphic to $S^2$.
To see this, first note that $U(2)$ intersects the center $Z(SO(4)) = \{\pm I\}$ of $SO(4)$. It follows that $$SO(4)/U(2) \cong [SO(4)/Z(SO(4)]/[U(2)/(Z(SO(4))\cap U(2))].$$

But $SO(4)/Z(SO(4))\cong SO(3)\times SO(3)$ and $U(2)/(Z(SO(4))\cap U(2)) \cong SO(3)\times S^1$. So, $SO(4)/U(2)\cong (SO(3)\times SO(3))/(SO(3)\times S^1)\cong SO(3)/S^1$.

But the standard action of $SO(3)$ on $S^2$ is transitive with stabilizer $S^1$, so $SO(3)/S^1 \cong S^2$.

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    $\begingroup$ This argument works for any $n$ to show $Gl^+(2n,\mathbb{R})/Gl(n,\mathbb{C})$ is always homotopy equivalent to $SO(2n)/U(n)$. I don't think $SO(2n)/U(n)$ has a nicer description for most $n$. But, when $n=1$, it's trivial. When $n=2$, it's $S^2$, and when $n=3$, it's $SU(4)/U(3) = \mathbb{C}P^3$. $\endgroup$ – Jason DeVito Sep 8 '14 at 18:20
  • $\begingroup$ Thanks for the nice answer; I also found an alternative way of looking at this fact, though I'm not able to fully understand it. It goes as follows: a matrix in $GL^+(2n,\mathbb R)/GL(n,\mathbb C)$ has polar decomposition $J=SO=-O^{-1}S^{-1}=(O^tS^{-1}O)(-O^{-1})$. This is OK, and implies that $O=-O^{-1}=P^{-1}J_0P$ defines an element $U(n)P \in SO(2n)/U(n)$: what I don't understand of this last statement is how the properties of $O$ imply that $P \in GL(2n,\mathbb R)$ is also in $SO(2n)$. Then one could conclude observing that $SO(4)/U(2)$ is $S^2$. Could you help? $\endgroup$ – jj_p Sep 12 '14 at 16:45
  • $\begingroup$ Here $J_0$ is the standard symplectic form $\endgroup$ – jj_p Sep 12 '14 at 16:46
  • $\begingroup$ @jj_p: I'm not sure I understand all the notation. For example, why is $SO = -O^{-1} S$? Is $J$ a specific matrix? And are you claiming that any orthogonal matrix can be obtained from $J_0$ by conjugating? $\endgroup$ – Jason DeVito Sep 12 '14 at 18:29
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    $\begingroup$ @jj_p: Sorry for the late response. The farthest I've been able to get is that $(PP^t) J_0 (PP^t)^{-1} = J_0$. So, we conjugate $J_0$ by a symmetric matrix and get $J_0$ back. I've been unable to show that this implies $PP^t = I$, and I'd have no idea how to get to showing $\det P = 1$ from there. Sorry! $\endgroup$ – Jason DeVito Sep 17 '14 at 17:54

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