Morse theory: a microlocal perspective

Abstract

[en] The main goal of this work is to introduce the notion of micro-support and explore its connections to Morse theory. First, the main results of classical Morse theory are introduced, with a special focus to Morse inequalities. Some applications of classical Morse theory are outlined without proof, Freudenthal suspension theorem and Bott periodicity theorem being the most well-known. A schematic, proof-free, overview of sheaf theory and derived categories is presented in the second chapter. Its aim is to introduce some notions that will be essential in the last chapter and to fix notation. Extensive bibliography is given for the interested reader. In the third chapter the notion of micro-support of a sheaf on a manifold is motivated through the idea of propagating a sheaf in a given direction. Some basic properties of the micro-support are derived, together with some examples. Finally, the connection between micro-support and classical Morse theory is investigated and complemented with three illustrative examples.