# Is the space of complex structures a loop space?

Define the space of (normalized) complex structures $$\mathcal{J}_{2k}$$ on $$\mathbb{R}^{2k}$$ as the orthogonal transformations in $$SO(2k)$$ that square to minus the identity.

My question is if there exists a space $$X$$ such that $$\Omega (X) \simeq \mathcal{J}_{2k}$$?

I suspect this is not the case - if so then how would one prove this?

As far as I am aware, a space can be de-looped if and only if it can be given an $$A_\infty$$-structure. Since $$\mathcal{J}_{2k}$$ can be identified with the quotient $$SO(2k)/U(k)$$ (and $$U(k)$$ is not a normal subgroup of $$SO(2k)$$ for all $$k\neq 1$$*) then there is no obvious well-defined product to put on the quotient, and hence $$\mathcal{J}_{2k}$$.

Another note is that the space of complex structures can be identified with the space of minimal geodesic paths between two fixed points in $$SO(2k)$$. So to de-loop $$\mathcal{J}_{2k}$$ is to de-loop $$\Omega^m_{p,q}(SO(2k))$$.

A final point of possible interest is that stably (in the colimit) $$\mathcal{J}:= SO/U \simeq \Omega SO$$** can be de-looped.

*Note: I am identifying $$U(k)$$ with the subgroup of $$SO(2k)$$ of those transformations that commute with a fixed complex structure $$J \in \mathcal{J}_{2k}$$

**Note: This is a part of the story of Bott periodicity - see eg. Milnor's Morse theory.

No, it is not even an H-space.

I prefer slightly different notation, I write $$J_k$$ to mean $$SO(2k)/U(k)$$. Topologically your space is the product of this with a two point set.

1. There is a fiber sequence $$J_k \to J_{k+1} \to S^{2k}.$$

I leave this as an exercise.

2. Euler characteristic is multiplicative under fibrations. It follows from $$J_1 = \{*\}$$ that $$\chi(J_k) = 2^{k-1}$$.

3. A finite-dimensional H-space has zero Euler characteristic. In fact, the alternating sum of dimensions in any finite dimensional Hopf algebra is zero. This follows from their basic structure theory; see Hatcher 3C.4.

QED.

• Thanks for the nice answer! Good point about $SO(2k)$ versus $O(2k)$ - I have updated the question to the correct space. Jul 12, 2021 at 1:15