Topology of Closed 1-Forms on Manifolds with Boundary

Abstract

The topological structure of a manifold can be eectively revealed by studying the
critical points of a nice function assigned on it. This is the essential motivation of
Morse theory and many of its generalisations from a modern viewpoint. One fruitful
direction of the generalisation of the theory is to look at the zeros of a closed 1-form
which can be viewed locally as a real function up to an additive constant, initiated
by S.P. Novikov, see [32] and [33]. Extensive literatures have been devoted to the
study of so-called Novikov theory on closed manifolds, which consists of interesting
objects such as Novikov complex, Morse-Novikov inequalities and Novikov ring.
On the other hand, the topology of a space, e.g. a manifold, provides vital
information on the number of the critical points of a function. Along this line, a
whole dierent approach was suggested in the 1930s by Lusternik and Schnirelman
[25] and [26]. M. Farber in [9], [10], [11] and [12] generalised this concept with
respect to a closed 1-form, and used it to study the critical points and existence of
homoclinic cycles on a closed manifold in much more degenerate settings.
This thesis combines the two aspects in the context of closed 1-forms and attempts
a systematic treatment on smooth compact manifolds with boundary in the
sense that the transversality assumptions on the boundary is consistent thoroughly.
Overall, the thesis employs a geometric approach to the generalisation of the
existing results.