Compact Oriented Manifolds without Boundary have Nontrivial Homology
If we take to be a manifold equipped with an orientation given by an orientation form
. Then
is nowhere zero, and
for any positively oriented basis
of
at any point
.
Next we take to be an orientation-preserving embedding — a singular cube of top dimension. Then the pullback
for some strictly-positive function
. We conclude that
The integral of over all of
must surely be even greater than the integral over the image of
, since we can cover
by orientation-preserving singular
-cubes, and none of them can ever contribute a negative to the integral.
If we further suppose that is compact, we can cover
by finitely many such singular cubes, and the integral on each is well-defined. Using a partition of unity as usual this shows us that the integral over all of
exists and, further, must be strictly positive. In particular it’s not zero.
But now suppose that also has an empty boundary. Since
is a top form, we know that
— it’s closed in the de Rham cohomology. But we know that it cannot also be exact, for if
for some
-form
then Stokes’ theorem would tell us that
since is empty.
And so if is a compact, oriented
-manifold without boundary, then there must be some
-forms which do not arise from taking the exterior derivatives of
-forms. If
is pseudo-Riemannian, so we have a Hodge star to work with, this tells us that we always have some functions on
which are not the divergence of any vector field on
.
[…] — we see that any compact, oriented manifold without boundary cannot be contractible, since we know that they have some nontrivial homology! LD_AddCustomAttr("AdOpt", "1"); […]
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