$\DeclareMathOperator\End{End}\newcommand\Id{\mathrm{Id}}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$I start with some background, but people familiar with the subject may jump directly to the question.


Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an *almost hypercomplex structure* on $M$ is a 3-dimensional sub-bundle $Q\subset \End(TM)$ spanned by three endomorphisms $I$, $J$ and $K$ satisfying the quaternionic identities: $I^2=J^2=-\Id$, $IJ=-JI=K$.


An *almost quaternionic structure* on $M$ is a 3-dimensional sub-bundle $Q\subset \End(TM)$ which is *locally* spanned by three endomorphisms with the above property.


In both cases one may assume (by an averaging procedure) that $M$ is endowed with a Riemannian metric $g$ compatible with $Q$ in the sense that $Q\subset \End^-(TM)$, i.e. $I$, $J$ and $K$ are almost Hermitian. Using this one sees that an almost hypercomplex or quaternionic structure corresponds to a reduction of the structure group of $M$ to $\Sp(n)$ or $\Sp(1)\Sp(n)$ respectively, but this is not relevant for the question below.


Notice that in dimension $4$ every manifold has an almost quaternionic structure (since $\Sp(1)\Sp(1)=\SO(4)$), but there are well-known obstructions to the existence of almost hypercomplex structures. For example $S^4$ is not even almost complex. Finally, here comes the question:



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> Are there any known topological obstructions to the existence of almost quaternionic structures on compact manifolds of dimension $4n$ for $n\ge 2$?
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**EDIT:** Thomas Kragh has shown in his [answer](https://mathoverflow.net/a/52436/2383) that there is no almost quaternionic structure on the sphere $S^{4n}$ for $n\ge 2$. I have found further obstructions in the literature and summarized them in my [answer](https://mathoverflow.net/a/52510) below.



Are there topological obstructions to the existence of almost quaternionic structures on compact manifolds?
Thomas' [proof](https://mathoverflow.net/a/52436) for the fact that $S^{4n}$ has no almost quaternionic structure is correct, but I have found an alternative argument for this statement using the twistor space. Indeed, if $S^{4n}$ has an almost quaternionic structure $Q$, then the twistor space $S(Q)$ is an $S^2$-bundle over $S^{4n}$ whose total space has an almost complex structure. On the other hand, Theorem 1.4 in [A Note on Almost Complex and Weakly Complex Structures](https://doi.org/10.1112/jlms/s1-40.1.705) by W. Sutherland shows that this can only happen for $n=1$.


Of course, the general question is still open. I do not know, in particular, whether the complex projective spaces $\mathbb{C}{\operatorname{P}^{2n}}$ have almost quaternionic structures for $n\ge 2$, although I strongly suspect they don't.




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**EDIT:** In fact there are topological obstructions! The first one (which I should have been aware of) is that the second Stiefel–Whitney class of an almost quaternionic manifold of real dimension $8n$ vanishes. This was first noticed by Marchiafava and Romani in 1975, then by Salamon in 1982, and thus rules out the complex projective spaces $\mathbb{C}{\operatorname{P}^{4n}}$.


Moreover, in dimension 8, Čadek and Vanžura not only have found further obstructions (e.g. $4p\_2(M)=p\_1^2(M)+8e(M)$), but they also gave *sufficient* topological conditions for the existence of a $\operatorname{Sp}(1)\operatorname{Sp}(2)$ structure on 8-dimensional manifolds. Their article is [Almost quaternionic structures on eight-manifolds](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-35/issue-1/Almost-quaternionic-structures-on-eight-manifolds/ojm/1200787905.full).


