A doubling phenomenon in Morse-Novikov homology.
François Laudenbach (Université de Nantes).
The first geometry seminar of the term takes place at 3pm on 15th September, in the Salle Solvay, on the 5th floor of building NO. Please note this is both a different room and time than normal! The speaker is François Laudenbach from Nantes and his abstract is as follows.
Morse-Novikov homology deals with closed 1-forms on a closed manifold. Such a form l is locally the differential of a function; so, globally, l can be thought of as a multi-valued function (up to an additive constant). In what follows, the cohomology class of l is fixed and non-zero. Generically, the zeroes of l are of Morse type. If X is a it descending gradient (that is, l(X)<0 apart from the zeroes), under some transversality condition it is possible to construct a complex associated with X, analogous to the Morse complex of a Morse function; this is due to S. Novikov first, the general definition being due to J.-C. Sikorav. This complex is based on the finite set of zeroes of l and the ring is the so-called Novikov completion of the group ring. The completion translates the fact that there may have connecting orbits of arbitrarily large length.
After an initiation, I will explain how the Morse-Novikov complex changes by change of X, more precisely the analogue of the so-called handle slides of the usual Morse theory. But here, the dynamics of X has recurrence. This generates strange phenomena, among which there are first the self-slides which themselves generate a doubling phenomenon, similar to the period doubling in the Andronov-Hopf bifurcation.