We discuss controlled connectivity properties of closed 1–forms and their
cohomology classes and relate them to the simple homotopy type of the
Novikov complex. The degree of controlled connectivity of a closed 1–form
depends only on positive multiples of its cohomology class and is related to the
Bieri–Neumann–Strebel–Renz invariant. It is also related to the Morse theory of
closed 1–forms. Given a controlled 0–connected cohomology class on a manifold
with
we
can realize it by a closed 1–form which is Morse without critical points of index 0, 1,
and
. If
and the
cohomology class is controlled 1–connected we can approximately realize any chain
complex
with the simple homotopy type of the Novikov complex and with
for
and
as the
Novikov complex of a closed 1–form. This reduces the problem of finding a
closed 1–form with a minimal number of critical points to a purely algebraic
problem.
