Cartan form plus exact is base
Suppose that
represents a cohomology class in the Cartan model:
When
acts freely on
, we can find a
-exact form
such that:
This
is the corresponding base form.
This is easier to see using the Weyl model. They Weyl complex is
.
The existence of the homotopy operator
(
cp. Eq. (
4)) implies that this complex has the same
cohomology as
.
More explicitly:
Introduce new variables:
Eq. (
11) implies that we can remove
and
from the cocycle, by adding a
-exact expression,
effectively replacing
with
and
with
.
Because of the -invariance, replacement of with corresponds to the horizonthal projection.
On the next page we will give the explicit formula for passing to the base form.
, in order to build the horizonthal combination
.
