Non-singular closed forms and finiteness properties of groups

Let M-n a closed, connected manifold, n greater than or equal to 6. Denote by Nov(M) and Nov(st)(M) the minimal number of zeros of a generic closed on e-form defined on M, resp. the minimal number of zeros of a generic closed one-form on M x R-k which is almost quadratic at infinity. Under an assumption on the Bieri-Neumann-Strebel and Bieri-Renz invariants of pi(n), we show the implication Nov(st) = 0 => Nov = 0. If we drop this a ssumption, the implication above turns out to be false in general. In fact, our assumption is closely related to the implication FP2 => F-2, therefore , it's in some sense optimal. Applications in Novikov homology theory and in symplectic topology are give n. (C) 2000 Editions scientifiques et medicales Elsevier SAS.