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. The proof 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.
In particular, we obtain a new analytic proof of the degenerate
Morse inequalities of Bott.