An Invitation to Morse Theory (2nd ed.)
by Liviu I. Nicolaescu · Springer Universitext (author-hosted PDF, University of Notre Dame)
Graduate text in three parts: foundations of real Morse theory (local structure, handle attachment, Morse inequalities, the Morse–Floer complex), applications such as moment maps and equivariant localization, and complex Picard–Lefschetz theory. Includes exercises with solutions.
More resources on Morse Theory
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Morse Theory
The nLab wiki entry summarizing how critical points of a smooth function determine a manifold's topology, including Morse functions, handle decompositions, Morse homology and the Morse inequalities, with a curated bibliography. Serves as a terse reference and reading map.
Morse Theory Indomitable
Gives the map of where Morse theory leads, from a central figure in the subject. Bott's 1988 survey of how Morse theory developed: Morse's calculus of variations, Morse–Bott functions, Bott periodicity, equivariant Morse theory, the Atiyah–Bott work on Yang–Mills, and Witten's supersymmetric approach. Published in Publications Mathématiques de l'IHÉS 68, pp. 99–114.
Lecture Notes on Morse Homology (with an Eye Towards Floer Theory and Pseudoholomorphic Curves)
Informal notes from a 2002 UC Berkeley topics course. They construct the Morse complex from gradient flow lines and prove d²=0 and invariance through compactness and gluing of moduli spaces. They also cover orientations and the link to Floer theory. Hutchings' 2005 seminar page notes that the notes contain some mistakes.
Morse Theory and Floer Homology
Graduate textbook translated from French. Chapters 1–4 build finite-dimensional Morse theory and Morse homology from pseudo-gradient fields and moduli spaces of trajectories. The rest constructs Hamiltonian Floer homology and proves the Arnold conjecture. Includes exercises.
Lectures on the h-Cobordism Theorem
Princeton seminar notes from 1963 that prove Smale's h-cobordism theorem with Morse functions. Covers rearranging and cancelling critical points, Whitney's trick, and the generalized Poincaré conjecture in dimensions five and up. About 116 pages.