Abstract
The main reason for introducing smooth structures was to enable us to define smooth functions on manifolds and smooth maps between manifolds. In this chapter we carry out that project. We begin by defining smooth real-valued and vector-valued functions, and then generalize this to smooth maps between manifolds. We then focus our attention for a while on the special case of diffeomorphisms, which are bijective smooth maps with smooth inverses. If there is a diffeomorphism between two smooth manifolds, we say that they are diffeomorphic. The main objects of study in smooth manifold theory are properties that are invariant under diffeomorphisms. At the end of the chapter, we introduce a powerful tool for blending together locally defined smooth objects, called partitions of unity. They are used throughout smooth manifold theory for building global smooth objects out of local ones.
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References
Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Clarendon Press, New York (1990)
Freedman, Michael, Quinn, Frank: Topology of 4-Manifolds. Princeton University Press, Princeton (1990)
Kervaire, Michel A., Milnor, John W.: Groups of homotopy spheres: I. Ann. Math. 77, 504–537 (1963)
Milnor, John W.: On manifolds homeomorphic to the 7-sphere. Ann. Math. 64, 399–405 (1956)
Moise, Edwin E.: Geometric Topology in Dimensions 2 and 3. Springer, New York (1977)
Munkres, James R.: Obstructions to the smoothing of piecewise differentiable homeomorphisms. Ann. Math. 72, 521–554 (1960)
Nestruev, Jet: Smooth Manifolds and Observables. Springer, New York (2003)
Smale, S.: On the structure of manifolds. Am. J. Math. 84, 387–399 (1962)
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Lee, J.M. (2013). Smooth Maps. In: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol 218. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9982-5_2
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DOI: https://doi.org/10.1007/978-1-4419-9982-5_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9981-8
Online ISBN: 978-1-4419-9982-5
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