1. Basics on Morse Homology

This brief chapter is intended to provide the reader with an overview of the construction of Morse (co)homology for finite-dimensional manifolds. We present the main notions and results in a concise way and give references to detailed presentations and proofs whenever appropriate. Moreover, we establish some notation that will be employed throughout this book. Except for notational conventions, a reader familiar with Morse homology might skip this chapter without disadvantages. There are several detailed and recommendable references on Morse homology, see e.g. the textbooks Schwarz, Morse homology, Birkhäuser, Basel, 1993, [Sch93], Banyaga et al. Lectures on Morse homology, Kluwer Academic Publishers Group, Dordrecht, 2004, [BH04], Jost, Riemannian geometry and geometric analysis, Springer, Berlin, 2008, [Jos08, Chap. 7], Nicolaescu, An invitation to Morse theory, Springer, New York, 2011, [Nic11] or Audin and Damian, Morse theory and Floer homology, Springer, London, [AD14] as well as the set of lecture notes Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), UC Berkeley, 2002, [Hut02] and the article Weber, Expo Math, 24(2), 127–159, 2006, [Web06].