Incidence coefficients in the Novikov Complex for Morse forms: rationality and exponential growth properties - Andrei Pajitnov (Pracì Mìžnarodnogo Geometričnogo Centru, 2021)
Let f : M → S 1 be a Morse map, v a transverse f -gradient. The construction of the Novikov complex associates to these data a free chain complex C ∗ (f, v) over the ring Z[t]][t −1 ], generated by the critical points of f and computing the completed homology module of the corresponding infinite cyclic covering of M . Novikov’s Exponential Growth Conjecture says that the boundary operators in this complex are power series of non-zero convergence raduis. In [12] the author announced the proof of the Novikov conjecture for the case of C 0 -generic gradients together with several generalizations. The proofs of the first part of this work were published in [13]. The present article contains the proofs of the second part. There is a refined version of the Novikov complex, defined over a suitable com- pletion of the group ring of the fundamental group. We prove that for a C 0 -generic f -gradient the corresponding incidence coefficients belong to the image in the Novikov ring of a (non commutative) localization of the fundamental group ring. The Novikov construction generalizes also to the case of Morse 1-forms. In this case the corresponding incidence coefiicients belong to a certain completion of the ring of integral Laurent polynomials of several variables. We prove that for a given Morse form ω and a C 0 -generic ω-gradient these incidence coefficients are rational functions. The incidence coefficients in the Novikov complex are obtained by counting the algebraic number of the trajectories of the gradient, joining the zeros of the Morse form. There is V.I.Arnold’s version of the exponential growth conjecture, which concerns the total number of trajectories. We confirm this stronger form of the conjecture for any given Morse form and a C 0 -dense set of its gradients. We give an example of explicit computation of the Novikov complex.