Invitation to Morse Theory

As the the title suggests, the goal of this little book is to give the reader a taste of the 'unreasonable effectiveness' of Morse theory. The main idea behind this technique can be easily visualized. Suppose M is a smooth, compact manifold which for simplicity we assume is embedded in an Euclidean space E. We would like to understand basic topological invariants of M such as its homology and we attempt a 'slicing' technique. We fix a unit vector u in E and we start slicing M with the family of hyperplanes perpendicular to u. Such a hyperplane will in general intersect M along a submanifold (slice). The manifold can be recovered by continuously stacking the slices on top of each other in the same order as they were cut out of M. Think of the collection of slices as a deck of cards of various shapes. If we let these slices continuously pile up in the order they were produced we notice and increasing stack of slices. As this stack grows we notice that there are moments of time when its shape suffers a qualitative change. Morse theory is about extracting quantifiable information by studying the evolution of the shape of this growing stack of slices.Nicolaescu, Liviu I. is the author of 'Invitation to Morse Theory', published 2007 under ISBN 9780387495101 and ISBN 038749510X.