A Glance at S. Novikov’s Theory of Multivalued Morse Functions

François Laudenbach ()
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François Laudenbach: Université de Nantes, Laboratoire de Mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences et Techniques

Chapter Chapter 3 in Essays on Geometry, 2025, pp 21-45 from Springer

Abstract: Abstract This chapter provides a brief presentation of the so-called Morse-Novikov complex associated with a closed manifold, equipped with a closed non-exact one-form α $$\alpha $$ and a descending gradient. We are interested in the homoclinic bifurcation of this gradient when crossing the stratum of gradients which have a simple homoclinic orbit, based at a non-degenerate zero of α $$\alpha $$ . The effect of such a crossing on the Morse-Novikov complex is analyzed. Moreover, an original construction, described in this chapter, shows that the phenomenon of homoclinic bifurcation is very easy to make up. Finally, a surprising doubling phenomenon is associated to every such a bifurcation.

Date: 2025
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DOI: 10.1007/978-3-031-76257-4_3

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