Ebook: An introduction to Morse theory

Chapter 1. Morse Theory on Surfaces 1 -- 1.1. Critical points of functions 1 -- 1.2. Hessian 3 -- 1.3. The Morse lemma 8 -- 1.4. Morse functions on surfaces 14 -- 1.5. Handle decomposition 22 -- a. The case when the index of po is zero 26 -- b. The case when the index of po is one 26 -- c. The case when the index of po is two 29 -- d. Handle decompositions 30 -- Chapter 2. Extension to General Dimensions 33 -- 2.1. Manifolds of dimension m 33 -- a. Functions on a manifold and maps between manifolds 33 -- b. Manifolds with boundary 34 -- c. Functions and maps on manifolds with boundary 38 -- 2.2. Morse functions 41 -- a. Morse functions on m-manifolds 41 -- b. The Morse lemma for dimension m 44 -- c. Existence of Morse functions 47 -- 2.3. Gradient-like vector fields 56 -- a. Tangent vectors 56 -- b. Vector fields 61 -- c. Gradient-like vector fields 63 -- 2.4. Raising and lowering critical points 69 -- Chapter 3. Handlebodies 73 -- 3.1 Handle decompositions of manifolds 73 -- 3.3. Sliding handles 105 -- 3.4. Canceling handles 120 -- Chapter 4. Homology of Manifolds 133 -- 4.1. Homology groups 133 -- 4.2. Morse inequality 141 -- a. Handlebodies and cell complexes 141 -- b. Proof of the Morse inequality 147 -- c. Homology groups of complex projective space CP[superscript m] 147 -- 4.3. Poincare duality 148 -- a. Cohomology groups 148 -- b. Proof of Poincare duality 150 -- 4.4. Intersection forms 158 -- a. Intersection numbers of submanifolds 159 -- b. Intersection forms 159 -- c. Intersection numbers of submanifolds and intersection forms 163 -- Chapter 5. Low-dimensional Manifolds 167 -- 5.1. Fundamental groups 167 -- 5.2. Closed surfaces and 3-dimensional manifolds 173 -- a. Closed surfaces 173 -- b. 3-dimensional manifolds 181 -- 5.3. 4-dimensional manifolds 186 -- a. Heegaard diagrams for 4-dimensional manifolds 186 -- b. The case N = D[superscript 4] 190 -- c. Kirby calculus 194 -- A View from Current Mathematics 199