Rationality and exponential growth properties of the boundary operators in the Novikov complex (Q1815698)

The paper is concerned with the generalization to maps \(f:M\to S^1\) of the Morse-Thom-Smale theory associated with real valued mappings from a closed and connected manifold \(M\). The analog corresponding to the Morse complex is a free chain complex over the ring \(\mathbb{Z}((t))\) of formal power series with integer coefficients and finite negative part. To be more explicit, let \(P:\overline M\to M\) be the connected infinite cyclic covering for which \(f \circ P\) is homotopic to zero, \(F: \overline M\to \mathbb{R}\) a lifting of \(f\circ P\) and \(t\) the generator of the structure group of \(P\) such that for every \(x\in \overline M\) we have \(F(xt) < F(x)\). For every critical point \(x\) of \(f\) choose a lifting \(\overline x\) of \(x\) to \(\overline M\). Choose orientations of stable manifolds of the critical points. The boundary operator \(\partial: C_m(F) \to C_{m-1} (f)\) of the complex depends on the choice of a gradient-like vector field \(v\) for \(f\). It is defined in terms of incidence coefficients \(n_k(x,y;v)\), the algebraic number of \((-v)\)-trajectories joining \(\overline x\) to \(\overline yt^k\), each trajectory being counted with the sign arising from the orientations. One defines \(\partial x= \sum_y yn(x,y;v)\) with \(n(x,y;v)= \sum_{k\in Z} n_k (x,y;v)t^k\), where \(x\) is a critical point of \(f\) of index \(m\) and the sum ranges over the critical points of \(f\) of index \(m-1\). It is proved that generically, the coefficients \(n(x,y;v)\) are rational functions in \(t\) thus confirming the conjecture of S. P. Novikov from the 60's on this issue.