  We generalize the Novikov inequalities for 1-forms in two different
directions: first, we allow non-isolated critical points (assuming that they
are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the
inequalities by means of twisting by an arbitrary flat bundle. We also obtain
an $L^2$ version of these inequalities with finite von Neumann algebras.
  The proof of the main theorem uses Bismut's modification of the Witten
deformation of the de Rham complex; it is based on an explicit estimate on the
lower part of the spectrum of the corresponding Laplacian.
