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Durham e-Theses
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Topology of Closed 1-Forms on Manifolds with Boundary

LI, TIEQIANG,TAN (2009) Topology of Closed 1-Forms on Manifolds with Boundary. Doctoral thesis, Durham University.



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.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:Transversality, Morse nondegeneracy, Closed 1-Form.
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2009
Copyright:Copyright of this thesis is held by the author
Deposited On:21 Dec 2009 11:45

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