Incidence coefficients in the Novikov Complex for Morse forms: rationality and exponential growth properties

Let f : M → S 1 be a Morse map, v a transverse f -gradient. Theconstruction of the Novikov complex associates to these data a free chain complexC ∗ (f, v) over the ring Z[t]][t −1 ], generated by the critical points of f and computingthe completed homology module of the corresponding infinite cyclic covering of M .Novikov’s Exponential Growth Conjecture says that the boundary operators in thiscomplex are power series of non-zero convergence raduis.In [12] the author announced the proof of the Novikov conjecture for the case ofC 0 -generic gradients together with several generalizations. The proofs of the firstpart of this work were published in [13]. The present article contains the proofs ofthe 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 -genericf -gradient the corresponding incidence coefficients belong to the image in the Novikovring of a (non commutative) localization of the fundamental group ring.The Novikov construction generalizes also to the case of Morse 1-forms. In thiscase the corresponding incidence coefiicients belong to a certain completion of thering of integral Laurent polynomials of several variables. We prove that for a givenMorse form ω and a C 0 -generic ω-gradient these incidence coefficients are rationalfunctions.The incidence coefficients in the Novikov complex are obtained by counting thealgebraic number of the trajectories of the gradient, joining the zeros of the Morseform. There is V.I.Arnold’s version of the exponential growth conjecture, whichconcerns the total number of trajectories. We confirm this stronger form of theconjecture 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.

The number of split points of a Morse form and the structure of its foliation

Sharp bounds are given that connect split points — conic singularities of a special type — of a Morse form with the global structure of its foliation.

Number of minimal components and homologically independent compact leaves for a morse form foliation

The numbers m ( ω ) of minimal components and c ( ω ) of homologically independent compact leaves of the foliation of a Morse form ω on a connected smooth closed oriented manifold M are studied in terms of the first non-commutative Betti number b ′ 1 ( M ). A sharp estimate 0 ≦ m ( ω ) + c ( ω ) ≦ b ′ 1 ( M ) is given. It is shown that all values of m ( ω ) + c ( ω ), and in some cases all combinations of m ( ω ) and c ( ω ) with this condition, are reached on a given M . The corresponding issues are also studied in the classes of generic forms and compactifiable foliations.

Analytic embedded atom method potentials for face-centered cubic metals

The universal form of embedding function suggested by Banerjea and Smith together with a pair-potential of the Morse form are used to obtain embedded atom method (EAM) potentials for fcc metals: Cu, Ag, Au, Ni, Pd, and Pt. The potential parameters are determined by fitting to the Cauchy pressure (C12 − C44)/2, shear constant GV = (C11 − C12 + 3C44)/5, and C44, the cohesive energy and the vacancy formation energy. The obtained parameters are utilized to calculate the unrelaxed divacancy binding energy and the unrelaxed surface energies of three low-index planes. The calculated quantities are in reasonable agreement with the experimental values except perhaps the divacancy energy in a few cases. In a further application, lattice dynamics of these metals are discussed using the present EAM potentials. On comparison with experimental phonons, the agreement is good for Cu, Ag, and Ni, while in the other three metals, Au, Pd, and Pt, the agreement is not so good. The phonon spectra are in reasonable agreement with the earlier calculations. The frequency spectrum and the mean square displacement of an atom in Cu are in agreement with the experiment and other calculated results.