Characteristic classes and differentiable manifolds
These lectures might be more accurately titled “The application of characteristic classes to geometric problems on manifolds.” In particular we will be interested in studying vector fields on manifolds. This first lecture gives the basic definitions we will need throughout the course.
1. Smooth manifolds. Let R denote the real numbers and Rn, for n ≥ 1, the space of n-tples (x 1,…,x n ), x i∈R. Let U be an open set in Rn. A map f: U → R will be called smooth if its partial derivatives of all orders exist and are continuous. More generally, a map f: U → Rq will be called smooth if each coordinate function f j is smooth, where f j: U → R is given by the following composition:
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Books and Lecture Notes
- M.lJ. Milnor, Lectures on characteristic classes, Mimeographed notes, Princeton University, 1957.Google Scholar
- M.3——, Lectures on Morse Theory, Annals of Math. Studies (No. 51), Princeton, 1963.Google Scholar
- St.lN. Steenrod, The Topology of Fiber Bundles, Princeton University Press, 1951.Google Scholar
- St.2N. Steenrod and D. Epstein. Cohomology operations, Annals of Math. Studies (No. 50), 1962Google Scholar
- 4.J. Adem, The relations on Steenrod powers of cohomology classes, in Algebraic Geometry and Topology, Princeton, 1957, 191–230.Google Scholar
- 5.J. Adem and S. Gitler, Secondary characteristic classes and the immersion problem, Bol. Mat. Mex., 1963, 53–78.Google Scholar
- 10.——, Zur topologie der Komplexen mannigfaltigkeiten, in Studies and Essays presented to R. Courant, Interscience, 1941, 167–186.Google Scholar
- 17.R. Thorn, Espaces fibrés en spheres et carrés de Steenrod, Ann. Ecole Norm, Sup., 69 (1952), 109–182.Google Scholar
- 18.E. Thomas, Postnikov variants and higher order cohomology operations, Annals of Math., to appear.Google Scholar
- 19.——, The index of a tangent 2-field, Comment, Math. Helv., to appear.Google Scholar
- 20.J.H. Whitehead, On the groups πr (Vn, m) and sphere-bundles (with Corrigen-duum), Collected Works, Vol. II, Pergamon Press, 303–362.Google Scholar