Existence of Complex Structures and Characteristic Classes

[WWGD]

Hi,
Just curious if someone knows of any Characteristic class used to determine if a manifold allows
a Complex structure? It seems strange that Complex Space C^n is topologically Identical to R^{2n}
yet I believe not all R^{2n}s ( if any) allow Complex structures. Thanks for any comments, refs.

EDIT: I believe we would be working with classifying spaces associated to , respectively,
almost-complex structures and GL(2n) , and then using the inclusion of almost-complex
into GL(2n) and then working with the associated inclusion i of the classifying spaces
and lifts for the associated classifying maps.
But we would then need to know about the cohomology associated to each, in order to
figure out the obstructions to the existence . Am I on the right track, and, if not, please correct me,
if so, p-lease help me take the next step on the details.
Thanks.

 

[Ben Niehoff]

Not sure, but I have a feeling that if the obstruction to a complex structure were a characteristic class that we could easily describe, then the existence of a complex structure on S^6 would not be an open question.

 

[fresh_42]

Not sure, but I have a feeling that if the obstruction to a complex structure were a characteristic class that we could easily describe, then the existence of a complex structure on S^6 would not be an open question.

That's a funny example. A quick search offered me several papers for each position. And not of the kind "I've proven Fermat."
Could be an interesting topic on its own.

 

[lavinia]

This isn't much of an answer but maybe it points to areas of research.

This will be for obstructions to almost complex structures.

A 2n-dimensional orientable manifold has an almost complex structure if there is a section of the quotient bundle of the principal SO(2n) bundle with fiber SO(2n)/U(n).

For a CW-complex or simplicial complex obstructions to sections of the bundle over the skeleta of the complex lie in the cohomology groups with twisted coefficients equal to the homotopy groups of SO(2n)/U(n). One needs to know the homotopy groups of SO(2n)/U(n). I am sure there is a lot research on this.

If there is an almost complex structure then the bundle has Chern classes and as is well known the mod 2 reduction of Chern classes are the even dimensional Stiefel-Whitney classes. By the Bockstein cohomology sequence for $0 →Z→Z→Z_{2}→0$ it follows that the integer Stiefel-Whitney classes (the images of the Stiefel-Whitney classes under the Bockstein connecting homomorphism) must be zero. For instance, for a 6 manifold this requires that both the second and fourth Stiefel Whitney classes of the tangent bundle map to zero under the Bockstein connecting homomorphism.

One wonders how almost complex structures on a 6 manifold correspond to integer lifts of the second Stiefel Whitney class.

 

[zinq]

As is often mentioned, the only spheres having an almost-complex structure are the 2-sphere (which clearly has an actual complex structure as ℂℙ¹) and the 6-sphere (where the existence of an actual complex structure is a long-unsolved open problem).

To get an almost-complex structure on S⁶, identify this sphere with the set of pure octonions of unit length. Then a bundle map

J: T(S⁶) → T(S⁶)​

of the tangent bundle to itself satisfying J² = -I can be defined via a linear isomorphism

J_x: T_x(S⁶) → T_x(S⁶)​

given on each tangent space T_x(S⁶) by

J_x(y) = x y​

for any pure unit octonions x and y, where xy denotes the octonion product of x and y. Then we have

J_x²(y) = x (x y) = (x x) y = -y​

because the product x y z of octonions x, y, z is well-defined when any two of x, y, z are equal (even though the octonions are not associative in general), and because any pure unit octonion is a square root of -1. QED

________________________________________________
Note: The real division ring of all octonions is identified with R⁸, and may be given an orthonormal basis

B = {1, e₀, ..., e₆}.​

To define multiplication of gracefully: Multiples of 1 form the real numbers within the octonions, and the rest are multiplied using the rules

a) e_j² = -1

b) e_j e_j+1 = e_j+3

(index addition is understood modulo 7), and

c) e_j e_k = -e_k e_j,

for all j, k in the range 0 ≤ j, k ≤ 6.

and these rules are extended to make multiplication on all of bilinear.

Then these rules imply that

(e₀ e₁) e₂ = e₃ e₂ = -e₂ e₃ = -e₅,​

but

e₀ (e₁ e₂) = e₀ e₄ = -e₁,​

showing that octonion multiplication is not, in general, associative.