  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 M
with n = dim M > 4 we can realize it by a closed 1-form which is Morse without
critical points of index 0, 1, n-1 and n. If n = dim M > 5 and the cohomology
class is controlled 1-connected we can approximately realize any chain complex
D_* with the simple homotopy type of the Novikov complex and with D_i=0 for i <
2 and i > n-2 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.
