Homoclinic bifurcation in Morse-Novikov theory a doubling phenomenon

François Laudenbach (1) , Carlos Moraga Ferrándiz (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Sans affiliation

We consider a compact manifold of dimension greater than 2 and a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued function has a gradient vector field with respect to some auxiliary Riemannian metric. According to S. Novikov's work and a complement by J.-C. Sikorav, under some genericity assumptions these data yield a complex, called today the Morse-Novikov complex. Due to the non-exactness of the form, its gradient has non-trivial dynamics in contrary to gradients of functions. In particular, it is possible that the gradient has a homoclinic orbit. The one-form being fixed, we investigate the codimension-one stratum in the space of (pseudo-) gradients formed by vector fields having one homoclinic orbit in a given homotopy class of loops. This stratum S breaks up into a left and a right part separated by a substratum. The algebraic effect on the Morse-Novikov complex of crossing S depends on the part, left or right, which is crossed. This difference makes necessary the doubling phenomenon mentioned in our title.

Mots clés: Closed one-form; Morse-Novikov theory; gradient; Morse-Smale; homoclinic bifurcation

Domaines: Topologie géométrique [math.GT]

François Laudenbach, Carlos Moraga Ferrándiz. Homoclinic bifurcation in Morse-Novikov theory a doubling phenomenon. 2016. ⟨hal-01304121v3⟩