Obstructions to the existence of stable (and unstable?) complex structures?

Question

Let $V$ be a real vector bundle on a space $X$, perhaps the tangent bundle of a smooth compact manifold. I'm interested in understanding the obstructions to $V$ admitting a stable complex structure, and also understanding how much this differs from the unstable story (if we ask for $V$ to admit a complex structure). Here are some things I know.

On the one hand, there are necessary conditions coming from characteristic classes. Namely, the odd Stiefel-Whitney classes $w_{2k+1}(V) \in H^{2k+1}(X, \mathbb{F}_2)$ must vanish, and the even Stiefel-Whitney classes must be in the image of the reduction map $H^{2k}(X, \mathbb{Z}) \to H^{2k}(X, \mathbb{F}_2)$, or equivalently the odd integral Stiefel-Whitney classes $\beta w_{2k}(V) = W_{2k+1}(V) \in H^{2k+1}(X, \mathbb{Z})$ must vanish, where $\beta$ is the Bockstein map $H^{2k}(X, \mathbb{F}_2) \to H^{2k+1}(X, \mathbb{Z})$. I don't know if this is sufficient in general. It certainly doesn't suffice for complex, rather than stable complex, structures, since even spheres have no odd cohomology but already $S^4$ doesn't have an almost complex structure.

On the other hand, by obstruction theory we need to look at the fibration

$$O/U \to BU \to BO$$

since lifting the stable classifying map $X \to BO$ of $V$ to a classifying map $X \to BU$ is equivalent to finding sections of an associated bundle with fibers $O/U$. The obstructions to doing this are cohomology classes in $H^{i+1}(X, \pi_i(O/U))$. Now, Bott periodicity implies that $O/U \cong \Omega O$, so its homotopy groups are known: they are periodic with period $8$ and the nontrivial ones are

$$\pi_{8k}(O/U) \cong \pi_{8k+7}(O/U) \cong \mathbb{Z}_2$$

and

$$\pi_{8k+2}(O/U) \cong \pi_{8k+6}(O/U) \cong \mathbb{Z}.$$

So there are obstructions living in $H^{8k+1}(X, \mathbb{Z}_2), H^{8k+8}(X, \mathbb{Z}_2), H^{8k+3}(X, \mathbb{Z})$, and $H^{8k+7}(X, \mathbb{Z})$. Three of these live in the same groups as the characteristic classes above so one might hope that they are in fact the same obstructions, but the obstructions living in $H^{8k+8}(X, \mathbb{Z}_2)$ don't match up. What's up with those?

For the unstable picture, when $\dim V = 2n$ we need to look at the fibration

$$O(2n)/U(n) \to BU(n) \to BO(2n).$$

There are induced maps $O(2n)/U(n) \to O(2n+2)/U(n+1)$ which induce isomorphisms on $\pi_k$ for (if I've calculated this correctly) $k \le 2n - 2$, so for example if we only cared about tangent bundles the stable and unstable stories almost match up except for the possibility of a mismatch involving classes in $H^{2n}(X, \pi_{2n-1}(O(2n)/U(n))$. For example, when $n = 2$ we have $O(4)/U(2) \cong S^2 \sqcup S^2$ and $\pi_0, \pi_1, \pi_2$ match up with the stable values above but $\pi_3$ is $\mathbb{Z}$ instead of being trivial.

Here are some questions I have.

How does the Stiefel-Whitney class story match up to the obstruction theory story? To what extent can we identify the obstructions involved in the two stories with each other? And how different are the stable and unstable obstructions?

This question is closely related but I don't think it completely answers my questions.

Answer 1

I accidentally ran into this older question and noticed that the unstable situation wasn't discussed yet. So I thought I point out a couple of references discussing the obstruction classes for the existence of almost complex structures.

• W.S. Massey. Obstructions to the existence of almost complex structures. Bull. Amer. Math. Soc. 67 1961 559–564. (Link to paper on Project Euclid)

Theorem I of Massey's paper describes a relation between the integral Stiefel-Whitney classes and the obstruction classes in integer cohomology in the stable range. The relation is that the integral Stiefel-Whitney classes are certain multiples of the obstruction classes for the existence of a complex structure. In the presence of torsion, the integral Stiefel-Whitney classes can vanish while the obstruction are non-trivial. (This would describe the relation between the Stiefel-Whitney class story and the obstruction class story in the cases $8k+3$ and $8k+7$.)

Massey also describes the obstruction classes related to the first unstable homotopy group of $SO(2n)/U(n)$, living in the "mismatch group" $H^{2n}(X,\pi_{2n-1}(SO(2n)/U(n)))$ mentioned in the question. Assume that we have an almost complex structure over the $2n-1$-skeleton of the space $X$. Theorem II describes exactly the obstruction class to extend this to the $2n$-skeleton, for $n=2k$ even (which is the case where the obstruction groups have integer coefficients): the obstruction class encodes the failure of a natural relation between the Pontryagin class $p_k$ and the Chern classes of the almost complex structure on the $2n-1$-skeleton.

Finally, concerning the mismatch between Stiefel-Whitney and obstruction story in the case $8k+8$. There are partial results in Theorem III of Massey's paper. Much more precise statements concerning the obstruction class in this case can be found in the following two papers.

• E. Thomas. Complex structures on real vector bundles. Amer. J. Math. 89 1967 887–908. (Link to paper on JSTOR)

• H. Yang. A note on stable complex structures on real vector bundles over manifolds. Topology Appl. 189 (2015), 1–9. (Link to paper on ScienceDirect)

It seems the obstruction class for the case $8k+8$ involves discussion of quite a couple of cohomology operations. (Also check out references in these papers for further results.)

Answer 2

I would recommend doing this stably with spectra, at least to start with. A key ingredient is this theorem of Reg Wood: there is an equivalence $KO/\eta\simeq KU$, where $\eta\in\pi_1KO=\widetilde{KO}^0(\mathbb{R}P^1)$ corresponds to the tautological bundle minus one. (In the comments Qiaochu Yuan refers to a "mysterious map" $BO\to O$; this is essentially $V\mapsto\eta\otimes V$.) This gives a cofibre sequence $$ \Sigma KO \xrightarrow{\eta} KO \xrightarrow{f} KU \xrightarrow{g} \Sigma^2KO. $$ Here $f$ is the obvious complexification map. To describe $g$, let $\nu\in\pi_2KU$ be the usual generator (which is invertible), and let $h\colon KU\to KO$ denote the forgetful map; then it can be shown that $g=h\circ\nu^{-1}$. This can now be twisted around to give a cofibre sequence $$ \Sigma^{-2}KO \xrightarrow{\nu^{-1}\circ f} KU \xrightarrow{h} KO \xrightarrow{\eta} \Sigma^{-1}KO, $$ which gives information about the image of $h$, as required. The zeroth spaces in the above sequence are $$ O/U \to \mathbb{Z}\times BU \to \mathbb{Z}\times BO \to O. $$ The first map is trivial in mod 2 (co)homology. If we just look at the base components, the other three spaces give a diagram of Hopf algebras $$ \mathbb{F}_2[c_k|k>0] \xleftarrow{} \mathbb{F}_2[w_k|k>0] \xleftarrow{} \mathbb{F}_2[w_{2k-1}|k>0] $$ The first map sends $w_{2k}$ to $c_k$ and $w_{2k-1}$ to $0$; the second map is the obvious inclusion.

All of these things are covered in an integrated way in my thesis using Hopf rings, although of course all individual pieces of the story are much older.