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From: Bacle on 2 May 2010 19:36
Hi, everyone:

Let CP^2- be a copy of CP^2 with reversed orientation.

I am trying to show that there is no orientation-preserving diffeo. between the two.

Would someone suggest and/or please check my work.?

I cannot prove that diffeos. preserve orientation numbers
(do they.?) but I think this works:

We have that the intersection form for CP^2 is <1>
since H_2(CP^2) is generated by CP^1 , which
self-intersects once; e.g., take the embedded
copies [x:0:1] and [0:1:y] of CP^1 in CP^2
, meeting at 1 point only. (And, as complex
submanifolds, their self-intersection is positive).
while the intersection form for CP^2- is <-1>

Now, we pass to cohomology, and use deRham's theorem
to define the intersection number, using the
fundamental form w for CP^1:

1=Integral_..X w/\w (with /\=wedge)

**Now** this is where things are unclear:

if w is a fundamental form for CP^1 (embedded in
CP^2) , and, by deRham, a nowhere-zero volume form
for CP^1 , is it then fair to say that -w is
a nowhere-zero volume form for CP^1 embedded in
-CP^2.?.

I know this is wrong, since it implies that
we cannot have negative self-intersections, but
I cannot see why this is wrong.

Any Ideas.?

Thanks in Advance.
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