CP^2 , and CP^2 -bar, are not diffeomorphic.

[WWGD]

Hi, Topologists/Geometers:

I am trying to show that CP^2 and CP^2-bar (meaning CP^2 with reverse choice of
orientation, are not diffeomorphic.

I am kind of rusty; all I can think of using is the intersection form, but I do not
remember the precise result that we can use to show this. I have checked Rohklin's
theorem and other results in intersection forms, without success.

I would appreciate your suggestions.

 

[zhentil]

Way #1: Remember H^2(CP^2,Z) = Z, call the positive generator a, and the volume form on CP^2 is a positive multiple of a ^ a. Then f*(a)=ca, which implies f*(a^a)=c^2 a^a. Then c^2 can't possibly equal negative one.

Way #2: CP^2 freely generates the oriented cobordism group in dimension 4. If CP^2 admits an orientation reversing diffeomorphism, it's 2-torsion in the cobordism group, contradiction.

 

[Bacle]

Zhentil:

Thanks Again. I wonder if this would also work: ( I will try to be more succint this time.)

We know that the intersection form for CP²2 is <1>, and that for CP²^ (defined as CP² with reversed orientation is <-1>.

This means that surfaces intersect positively , and at single points in CP², and
surfaces intersect negatively at single points in CP²^.

Now assume there is a diffeomorphism f: CP² -->CP²^

Then f is either i) orientation-preserving, or ii)orientation-reversing.

If we have:

i) , then the image under f of both homology 2-surfaces will still intersect

positively. If we have

ii), then the image under f of the 2-surfaces will also end up intersecting

positively at one point.

In neither case can the 2-homology surfaces intersect negatively at one point

the way they do in CP²^.

Does that Work.?

 

[zhentil]

You have to be careful (why does an orientation-reversing map reverse signs on homology? it doesn't in general), but in this case, you're translating my way #1 via Poincare duality.

 

[Bacle]

Way #1: Remember H^2(CP^2,Z) = Z, call the positive generator a, and the volume form on CP^2 is a positive multiple of a ^ a. Then f*(a)=ca, which implies f*(a^a)=c^2 a^a. Then c^2 can't possibly equal negative one.

Way #2: CP^2 freely generates the oriented cobordism group in dimension 4. If CP^2 admits an orientation reversing diffeomorphism, it's 2-torsion in the cobordism group, contradiction.

Zhentil: I am a little confused here: are you using some sort of deRham duality to construct a volume form by using the fundamental class.?. Otherwise, how do we go from a fundamental class in H_2(CP²,Z) , to a top form , i.e., a form in
H₄(CP²;Z).?. The fundamental class is , AFAIK, a purely
topological object, while a volume form assumes a differentiable structure.

I hope this is not too far off.

 

[zhentil]

The general answer is because CP^2 is Kahler. The more specific reason (and the reason this question is simple to answer) is because the cohomology ring of CP^2 is a truncated polynomial ring on one generator (i.e. a in H^2, a^3 = 0).

 

[Bacle]

Wow, you blew me away there, Zhentil. Are you using Chern classes or something.?
My pithy one semester of algebraic topology cannot deal with that.

If I am too far off, just ignore it.

 

[zhentil]

Based on one semester of algebraic topology, you can prove that the cohomology of CP^2 is Z in the even dimensions and 0 otherwise (by using the cell structure or Mayer-Vietoris). The only other bit of information comes from the fact that the generator of the top cohomology is actually the square of the generator of the middle cohomology. This is where the Kahler comment comes in: if you have a Kahler form, its top power actually gives you a volume form.

 

[bdolan]

I came across this thread the other day and thought you might be
interested in the fact that exactly this point is discussed in the review by Eguchi, Gilkey and Hanson, Physics Reports, vol 66 (1980) 213. On page 320 they discuss the Dolbeault complex and show that the index is only integral if one chooses the correct oriantation.

 

[Bacle]

Thanks, Dolan, I'll look it up.

 

[bdolan]

Sorry, I meant page 330, not 320!