nLab: Morse Theory
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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.
More resources on Morse Theory
Wolfram MathWorld
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Morse Theory
John Milnor's 1963 Princeton lecture notes, the classic account of how nondegenerate critical points of a function build a manifold cell by cell. Continues into geodesics and the path-space calculus of variations, ending with a proof of Bott periodicity.
nLab
Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.