Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:17:48Z http://mathoverflow.net/feeds/question/52396 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52396/are-there-topological-restrictions-to-the-existence-of-almost-quaternionic-struct Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds? Andrei Moroianu 2011-01-18T10:20:59Z 2012-06-05T17:21:31Z <p>I start with some background, but people familiar with the subject may jump directly to the question. </p> <p>Let $M^{4n}$ be a compact oriented smooth manifold. Recall that an <em>almost hypercomplex structure</em> on $M$ is a 3-dimensional sub-bundle $Q\subset End(TM)$ spanned by three endomorphisms $I$, $J$ and $K$ satisfying the quaternionic identities: $I^2=J^2=-Id$, $IJ=-JI=K$. </p> <p>An <em>almost quaternionic structure</em> on $M$ is a 3-dimensional sub-bundle $Q\subset End(TM)$ which is <em>locally</em> spanned by three endomorphisms with the above property. </p> <p>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 $\mathrm{Sp}(n)$ or $\mathrm{Sp}(1)\mathrm{Sp}(n)$ respectively, but this is not relevant for the question below.</p> <p>Notice that in dimension $4$ every manifold has an almost quaternionic structure (since $\mathrm{Sp}(1)\mathrm{Sp}(1)=\mathrm{SO}(4)$), 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:</p> <blockquote> <p>Are there any known topological obstructions to the existence of almost quaternionic structures on compact manifolds of dimension $4n$ for $n\ge 2$?</p> </blockquote> <hr> <p><strong>EDIT:</strong> Thomas Kragh has shown in his answer that there is no almost quaternionic structure on the sphere $S^{4n}$ for $n\ge 2$. I have found further obstructions in the litterature and summarized them in my answer below.</p> http://mathoverflow.net/questions/52396/are-there-topological-restrictions-to-the-existence-of-almost-quaternionic-struct/52436#52436 Answer by Thomas Kragh for Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds? Thomas Kragh 2011-01-18T21:33:28Z 2011-01-20T18:18:36Z <p>It seems to me that if I understood the comments to my comment correctly that the map</p> <p>$$\mathrm{Sp}(1) \times \mathrm{Sp}(n) \to \mathrm{SO}(4n)$$</p> <p>induced by right unit quarternionic multiplaction on $\mathbb{H}^n$ of the left factor and right matrix multiplication on $\mathbb{H}^n$ of the left factor has kernel $\{ \pm 1\}$. Since the source is simply connected it must lift to the spin group. So we have a map</p> <p>$$\mathrm{Sp}(1) \times \mathrm{Sp}(n) \to \mathrm{Spin}(4n)$$.</p> <p>Covering the map</p> <p>$$\mathrm{Sp}(1)\mathrm{Sp}(n) \to \mathrm{SO}(4n)$$</p> <p>Since the covering fiber is $\mathbb{Z}/2\mathbb{Z}$ and we can check that after taking the functor $B$ both fibers are $K(\mathbb{Z}/2\mathbb{Z},1)$-spaces we see that</p> <p>$$\begin{matrix} B(\mathrm{Sp}(1)\times \mathrm{Sp}(n)) &amp; \longrightarrow &amp; B(\mathrm{Spin}(4n)) \\ \downarrow &amp;&amp; \downarrow \\ B(\mathrm{Sp}(1)\mathrm{Sp}(n)) &amp; \longrightarrow &amp; B(\mathrm{SO}(4n)) \end{matrix}$$</p> <p>is homotopy cartesian.</p> <p>So if $M$ is spinable <em>and</em> has an almost Quarternionic structure it means that its classifying map lifts to $B(\mathrm{Sp}(1) \times \mathrm{Sp}(n))$</p> <p>Edit: The conclusion (which is now removed) was wrong, but at least it seems to simplify the picture when $M$ is spin.</p> <p><strong>Added</strong>: For spheres $S^{4n}$ we may use the above on the $4n$th homotopy group and deloop. This implies that if we had a quartenionic structure on $S^{4n}$ we would have that the image of the map</p> <p>$$\pi_{4n-1}(\mathrm{Sp}(1)\times \mathrm{Sp}(n) ) \to \pi_{4n-1} (\mathrm{SO}(4n))$$</p> <p>contains the image of the map $\mathbb{Z} \cong \pi_{4n-1}(\Omega S^{4n}) \to \pi_{4n-1}(\mathrm{SO}(4n)) \cong \mathbb{Z}\times \mathbb{Z}$ (*) induced by the delooping of the classifying map for the tangent bundle of $S^{4n}$.</p> <p>We know that not having <strong>an almost hypercomplex</strong> structure implies that the image of $\pi_{4n-1}(\mathrm{Sp}(n)) \to \pi_{4n-1} (\mathrm{SO}(4n))$ never contains this image, and since $\pi_{4n-1}(\mathrm{Sp}(1))$ is torsion for $n\geq 2$ the above map can not do so either for $n\geq 2$.</p> <p>(*) $\pi_{4n-1}(\mathrm{SO}(4n)) \cong \mathbb{Z}\times\mathbb{Z}$ follows WHEN $n\geq 4$ from the paper </p> <p>Barratt, M. G.; Mahowald, M. E. The metastable homotopy of O(n). Bull. Amer. Math. Soc. 70 1964 775-760.</p> <p>I think this is true in general. Indeed, it is true for $n=1$ where the above is not a contradiction because there $\pi_3(\mathrm{Sp}(1))\cong \mathbb{Z}$. Andrei pointed out in a comment that this is also true for $n=1,2$.</p> http://mathoverflow.net/questions/52396/are-there-topological-restrictions-to-the-existence-of-almost-quaternionic-struct/52510#52510 Answer by Andrei Moroianu for Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds? Andrei Moroianu 2011-01-19T12:52:11Z 2011-02-08T10:11:44Z <p>Thomas' proof 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 the <a href="http://jlms.oxfordjournals.org/content/s1-40/1/705.full.pdf+html" rel="nofollow">following article</a> by W. Sutherland shows that this can only happen for $n=1$. </p> <p>Of course, the general question is still open. I do not know, in particular, whether the complex projective spaces $\mathbb{C}\rm{P}^{2n}$ have almost quaternionic structures for $n\ge 2$, although I strongly suspect they don't.</p> <hr> <p><strong>EDIT:</strong> 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 $\mathbb{C}\rm{P}^{4n}$. </p> <p>Moreover, in dimension 8, Cadek and Vanzura not only have found further obstructions (e.g. $4p_2(M)=p_1^2(M)+8e(M)$), but they also gave <em>sufficient</em> topological conditions for the existence of a $\rm{Sp}(1)\rm{Sp}(2)$ structure on 8-dimensional manifolds. Their article is available <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ojm/1200787905" rel="nofollow">here</a>. </p> http://mathoverflow.net/questions/52396/are-there-topological-restrictions-to-the-existence-of-almost-quaternionic-struct/98889#98889 Answer by Oldřich Spáčil for Are there topological restrictions to the existence of almost quaternionic structures on compact manifolds? Oldřich Spáčil 2012-06-05T17:21:31Z 2012-06-05T17:21:31Z <p>I know this is a bit late, but as you mentioned Cadek's and Vanzura's paper, I'd like to point out (selfishly?) that there's also my <a href="http://www.sciencedirect.com/science/article/pii/S0926224510000227" rel="nofollow">paper</a> which uses a bit of their work and gives some integrality conditions on the existence of quaternionic structures on closed manifolds - an example is below. I should emphasize that I really mean honest quaternionic not just almost quaternionic here, although the referee believed that the same should hold for only almost quaternionic structures too. </p> <p>Theorem: Let M be an 8-dimensional compact quaternionic manifold with Pontryagin classes $p_1(TM)$ and $p_2(TM)$ and a fundamental class $[M]$. Then the following expressions are integers $\biggl(\frac{143}{960}p_{1}^{2}-\frac{89}{240}p_{2}\biggr)[M], \quad \biggl( -\frac{17}{480}p_{1}^{2}+\frac{71}{120}p_{2}\biggr)[M].$</p>