# Characteristic classes and differentiable manifolds

• E. Thomas
Part of the C.I.M.E. Summer Schools book series (CIME, volume 41)

## Abstract

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:

## Books and Lecture Notes

1. A-M.
L. Auslander and R. MacKenzie, Introduction to Differentiable Manifolds, McGraw-Hill, 1963.
2. H.
P. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd Edition, Springer-Verlag, 1966.
3. Ha.
P. Halmos, Finite-dimensional vector spaces, 2nd Edition, Van Nostrand, 1958
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S. Hu, Elements of General Topology, Holden-Day, 1964.
5. M.l
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E. Spanier, Algebraic Topology, McGraw-Hill, 1966.
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N. Steenrod, The Topology of Fiber Bundles, Princeton University Press, 1951.Google Scholar
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E. Thomas, Seminar on Fiber Spaces, Lecture notes in Math., No. 13, Springer-Verlag, 1966.

## Papers

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