Could you help me with some hint or reference for the following questions?

I'm reviewing the Milnor-Stasheff for references.

  1. Is there some $(2k)$-manifold with stably quasi-complex structure, such that the structure is not complex?

  2. Is there some $(2k+1)$-manifold that is stably quasi-complex?

  3. Is there any embedding of $\mathbb{CP}^4$ in $\mathbb{R}^{11}$?

  4. For which values of $n$ does it have that $\mathbb{CP}^n$ has structure Spin?


a) Let $ E \rightarrow M $ real vector bundle of range $ 2n $, a quasi-complex structure in the bundle is a function $ J: E \rightarrow E $ which is $ \mathbb{R} $ - linear in each fiber and such that $ J \circ J = -Id $

b) Stably quasi-complex structure for $M$ means that there exists a trivial vector bundle $ \eta: \mathbb{R}^k \rightarrow M $ such that $ \eta \oplus \tau M $ (the Whitney sum of bundles) has a complex structure for $ M $ (with $\tau M$ the tangent bundle of $M$).

First of all, Thanks

  • 1
    $\begingroup$ For $4.$ you can use Theorem 4.5 to calculate the Chern classes of $\mathbb{C}P^n$. The theorem is stated in terms of the SW classes of $\mathbb{R}P^n$, but the same proof works. Then you should know the relationship between $c_1$ and $w_2$, and what this implies for spin structure. $\endgroup$
    – Tyrone
    Apr 25, 2019 at 9:21
  • 3
    $\begingroup$ By 'quasi-complex' do you mean 'almost complex'? $\endgroup$
    – Tyrone
    Apr 25, 2019 at 9:22
  • $\begingroup$ I do not know the definition of almost-complex, but I leave you the definition of quasi-complex in the publication, thanks $\endgroup$ Apr 26, 2019 at 21:42
  • 1
    $\begingroup$ This is what is usually called an almost complex structure, en.wikipedia.org/wiki/Almost_complex_manifold . Milnor and Stasheff also use this term in the 1974 printing. $\endgroup$
    – Tyrone
    Apr 27, 2019 at 9:00
  • 1
    $\begingroup$ For 1., spheres are stably parallelizable, so an even dimension sphere is stably almost complex. However the only spheres which are almost complex are $S^2$ and $S^6$. Of these $S^2$ is complex, and as far as I understand it is still undecided as to whether $S^6$ is complex or not. $\endgroup$
    – Tyrone
    Apr 27, 2019 at 12:40


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