How can we detect the existence of almost-complex structures? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:37:16Z http://mathoverflow.net/feeds/question/63439 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63439/how-can-we-detect-the-existence-of-almost-complex-structures How can we detect the existence of almost-complex structures? Aaron Mazel-Gee 2011-04-29T16:46:39Z 2011-05-03T20:45:39Z <p>Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto $O_k$, then $BGL_{k}(\mathbb{R})\simeq BO_k$, which is a cute way (though it's certainly overkill) of proving that every smooth manifold admits a Riemannian metric. An almost-complex structure, on the other hand, is equivalent to a reduction of the structure group from $GL_{2n}(\mathbb{R})$ to $GL_n(\mathbb{C})$, which is the same as asking for a lift of the classifying map through $BU_n\simeq BGL_n(\mathbb{C})\rightarrow BGL_{2n}(\mathbb{R})$.</p> <blockquote> <p>Can we detect the nonexistence of a lift entirely using characteristic classes? If not, what else goes into the classification?</p> </blockquote> <p>I'd imagine these don't suffice themselves. I know that $w_{2n}(TM) \equiv_2 c_n(TM)$, so this holds in the universal case $H^\ast(BO_{2n};\mathbb{Z}/2) \rightarrow H^\ast(BU_n;\mathbb{Z}/2)$. And certainly there are necessary conditions like $w_1(TM)=0$ (which of course just means that $TM$ is an orientable bundle, which is the same as asking that $M$ be an orientable manifold). But I have no idea of what sufficient conditions would look like. I've heard that this problem is indeed solved. Maybe it takes some characteristic class &amp; cohomology operation gymnastics, or maybe it even needs extraordinary characteristic classes. Or maybe there's yet another ingredient in the classification?</p> <p><strong>Edit</strong>: Apparently I misquoted my source, and this is only known stably (which makes sense, in light of Joel's answer and Tom's comments on it).</p> http://mathoverflow.net/questions/63439/how-can-we-detect-the-existence-of-almost-complex-structures/63443#63443 Answer by Sean Tilson for How can we detect the existence of almost-complex structures? Sean Tilson 2011-04-29T17:06:18Z 2011-04-29T17:06:18Z <p>This is more of a comment:</p> <p>Maybe you have already thought of this, or maybe there is something I am missing. Or perhaps this isn't the answer you want.</p> <p>Suppose $M$ is a smooth (finite-dimensional) compact manifold (without boundary). Let $f: M \to BO$ be the map that classifies it's stable normal bundle (or "equivalently" it's tangent bundle). Then an almost complex structure is a lift of $f$ over $i: BU \to BO$ to $\widetilde{f} : M \to BU$. So to see if we can get such a lift can't we just compose with $f$ with $j: BO \to Cof(i)$? Perhaps this is easier said than done. But we can apply homtopy and see what we get.</p> http://mathoverflow.net/questions/63439/how-can-we-detect-the-existence-of-almost-complex-structures/63472#63472 Answer by Francesco Polizzi for How can we detect the existence of almost-complex structures? Francesco Polizzi 2011-04-29T20:51:05Z 2011-04-29T20:51:05Z <p>If $M$ is a $4$-manifold, the existence of an almost-complex structure can be often detected by using the following result due to Wu:</p> <p><strong>Theorem.</strong> A $4$-manifold $M$ admits an almost-complex structure $J$ if and only if there exists $h \in H^2(M, \mathbb{Z})$ such that </p> <p>$h^2=3 \sigma(X)+2 \chi(X) \quad \textrm{and} \quad h \equiv w_2(X) (\ \textrm{mod} \ 2).$</p> <p>In this case $h=c_1(M, J).$</p> <p>See Gompf-Stipsicz, "4-manifolds and Kirby calculus", p. 30 for more details.</p> http://mathoverflow.net/questions/63439/how-can-we-detect-the-existence-of-almost-complex-structures/63476#63476 Answer by Joel Fine for How can we detect the existence of almost-complex structures? Joel Fine 2011-04-29T21:46:16Z 2011-05-03T20:45:39Z <p><strong>Edit:</strong> Now updated to include reference and slightly more general result. <strong>Edit 2:</strong> Includes remark about integrability.</p> <p>Similar to Francesco Polizzi's answer, there is the following Theorem concerning 6-manifolds.</p> <p>A closed oriented 6-dimensional manifold $X$ without 2-torsion in $H^3(X,\mathbb{Z})$ admits an almost complex structure. There is a 1-1 correspondence between almost complex structures on $X$ and the integral lifts $W \in H^2(X, \mathbb{Z})$ of $w_2(X)$. The Chern classes of the almost complex structure corresponding to $W$ are given by $c_1 = W$ and $c_2 = (W^2 - p_1(X))/2$.</p> <p>In fact, a necessary and sufficient condition for the existence of an almost complex structure is that $w_2(X)$ maps to zero under the Bockstein map $H^2(X,\mathbb{Z}_2) \to H^3(X,\mathbb{Z})$. </p> <p>I think the reason for results such as this and the one mentioned by Francesco is the following. To find an almost complex structure amounts to finding a section of a bundle over $X$ with fibre $F_n=SO(2n)/U(n)$. The obstructions to such a section existing lie in the homology groups $H^{k+1}(X, \pi_k(F_n))$. When $n$ is small I would guess we can compute these homotopy groups and so have a good understanding of the obstructions. For example, in the case mentioned above, n=3, $F_n = \mathbb{CP}^3$ and so the only non-trivial homotopy group which concerns us is $\pi_2 \cong \mathbb{Z}$. This is what leads to the above necessary and sufficient condition concerning 2-torsion. On the other hand when $n$ is large I don't know what $F_n$ looks like, let alone its homotopy groups...</p> <p>For the proof of the above mentioned result see the article "Cubic forms and complex 3-folds" by Okonek and Van de Ven. (I highly recommend this article, it's full of interesting facts about almost complex and complex 3-folds.)</p> <p>It is worth pointing out that in real dimension 6 or higher there is <em>no known obstruction to the existence of an integrable complex structure</em>. In other words, there is no known example of a manifold of dimension 6 or higher which has an almost complex structure, but not a genuine complex structure. By the classification of compact complex surfaces, those 4-manifolds admitting integrable complex structures are well understood.</p>