### 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.