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Prof. Alexander F. Ritter Associate Professor, University of Oxford |
In addition to the notes below, there are also notes from a 2024 course on Morse Homology at Oxford, which are less formal, and possibly more helpful to graduate students: Morse Homology (TCC Graduate course) : Lecture notes here. ⋄ CAMBRIDGE PART III COURSE ON MORSE HOMOLOGY (LENT TERM 2011) LECTURE NOTES I would be very grateful for your feedback in the form of typos, suggestions or mistakes you found, because these notes will be published as a book. My contact details: find the contacts here Combined PDF of all 24 Lectures and Homeworks: all Lectures and homeworks in one package Combined PDF of all 24 Lectures (no homeworks) Combined PDF of all Homeworks Lecture 1: (PDF)(PS) Morse functions, topology of sublevel sets, Morse homology, number of critical points of a generic function, geometry vs functional analysis, Poincare duality via Morse homology Lecture 2: (PDF)(PS) connections, parallel transport, Levi-Civita, geodesics, exp map, diffeomorphisms, regular maps Lecture 3: (PDF)(PS) Sard's thm, transversality, intersection numbers, stability, embedding thm, tubular nbhd thm Lecture 4: (PDF)(PS) Parametric transversality, perturbing a map to make it transverse to a submfd, zeros of sections, Morse functions, properties of Morse functions. Lecture 5: (PDF)(PS) Banach spaces, C^{infinity} is not Banach, Banach manifolds, Inverse/implicit fn thm, why Fredholm maps Lecture 6: (PDF)(PS) Fredholm operators, Sard-Smale theorem, zeros of Fredholm sections Lecture 7: (PDF)(PS) Flowlines, properties of negative gradient flows, energy estimates, cobordisms, h-cobordism thm Lecture 8: (PDF)(PS) handle attachments, topology of sublevel sets, stable and unstable mfds, Lecture 9: (PDF)(PS) Morse-Smale metrics, Morse homology = cellular homology for self-indexing Morse functions, Lecture 10: (PDF)(PS) Moduli spaces, overview: Floer homology, symplectic homology, Lagrangian Floer homology, Fukaya category and mirror symmetry, Heegaard-Floer homology, Instanton homology, Seiberg-Witten homology Lecture 11: (PDF)(PS) Sobolev spaces, Sobolev embedding thm, Rellich compactness thm, Sobolev on manifolds Lecture 12: (PDF)(PS) Banach space setup for the transversality thm: checking the spaces are Banach mfds Lecture 13: (PDF)(PS) showing the section is C^k, proof of the transversality thm Lecture 14: (PDF)(PS) reducing to the local setup d/ds + A_s, proof it is Fredholm Lecture 15: (PDF)(PS) formal adjoints, index calculation for d/ds + A_s Lecture 16: (PDF)(PS) spectral flow for d/ds + A_s with A_s converging to hyperbolic matrices, homotopy invariance of the index Lecture 17: (PDF)(PS) compactness of the moduli spaces up to breaking Lecture 18: (PDF)(PS) gluing theorem Lecture 19: (PDF)(PS) definition of Morse homology, invariance theorem Lecture 20: (PDF)(PS) invariance of Morse homology under changing Morse function and Riemannian metric Lecture 21: (PDF)(PS) Poincare' duality, Kunneth Theorem, Morse inequalities, Intersection/cup product via Morse-graph flows Lecture 22 and 23: (PDF)(PS) (Non-examinable) Spectral sequences, Leray-Serre spectral sequence in Morse homology Lecture 24: (PDF)(PS) (Non-examinable) Morse-Bott homology, spectral sequence for Morse-Bott theory (Lecture 24 was the last lecture) Course Description: description of the Part III course Syllabus: Initial Syllabus of the course I handed out Exam Guide: What the key topics are, what is examinable, and what isn't |