I'm supposed to show that there is no neighborhood of
on which my copy of
is globally cut out by independent functions, and I think this should just be a byproduct of
, but I cannot see how to show this, as far as I can see I should be able to separate
from itself since
with a copy of
glued in, so why not just move the sphere away from itself into this copy or Euclidean space. Any help appreciated.
To be able to move the sphere away from itself into the copy of
means you have a global section of the principal
and hence you would get
, which is nonsense.
Another way to see why
is using the cohomology ring of
has degree 2 and represented by the dual of any hyperplane