Morse theory and the euler characteristic daniel mitsutani abstract. Morse theory is a beautiful subject that sits between differential geometry, topol. Brouwers definition, in 1912, of the degree of a mapping. Rutgers university, new brunswick, new jersey 08903 and felix browder rutgers university, new brunswick, new jersey 08903 received september 11, 1997 contents 1. The first was to give an introduction to morse theory from a topological point of view. He is the author of topology from the differential viewpoint, singular points of complex hypersurfaces, morse theory, introduction to algebraic k theory, characteristic classes with james stasheff, and lectures on the hcobordism theorem princeton. M is a critical point of f if the differential dfp. They present some topics from the beginnings of topology, centering about l. A manifold is a topological space which locally looks like cartesian nspace.
S2d3 hauptseminar differentialtopologie morsetheorie. In this post we will see a course of differential geometry and topology a. In differential topology, morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. Introduce an important and oftenused technique in differential topology. Algebras and bott periodicity, topology, 4 196566, pp. S4d2 graduate seminar on topology morse theory s2d3. Wallace, and others, including a proof of the generalized poincare hypothesis in high dimensions. These problems do not belong so much to the realm of pure homotopy theory as to a special kind of homotopy theory connected with vector space bundles and the like, as exemplified by work around the bott periodicity theorems. We then use the basics of morse theory and the poincar ehopf theorem to prove that the euler characteristic equals the sum of the alternating betti. The methods used, however, are those of differential topology, rather.
Morse theory has provided the inspiration for exciting developments in differential topology by. An invitation to morse theory university of notre dame. A related slicing technique was employed in the study of the topology of algebraic manifolds called the picardlefschetz theory. Morse a nd in the b o ok lectures on t he h cob ordism theorem b y j. Soon after winning the fields medal in 1962, a young john milnor gave these nowfamous lectures and wrote his timeless topology from the differentiable viewp. Preface these lectures were delivered at the university of virginia in december 1963 under the sponsorship of the pagebarbour lecture foundation.
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