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The left exactness of the smooth left Puppe sequence

  • Paul Cherenack
Part of the Mathematics and Its Applications book series (MAIA, volume 350)

Abstract

In a previous paper using ideas due to Frohlicher and Kriegl [2] and Lawvere, Schanuel and Zame [5], we showed how to extend the category of differential manifolds to the category of smooth spaces which is topological over sets and Cartesian closed. We also showed that the absolute smooth homotopy groups exist in a natural way in smooth homotopy. Let f : A → B be a smooth map between smooth finite dimensional differentiable manifolds. Using techniques from differential topology, we demonstrated that on applying the smooth II0 to the smooth left Puppe sequence:

...→ ΩM f→ ΩA→ΩBM fAB (1)

one obtains the exact sequence of pointed sets:

... →II1 M f→IIA→II1 B→II0 M f→II0 A→II0 B.

Here we show how one can argue directly, using methods internal to the category of smooth spaces for the more general left exactness of (1) in the sense of Whitehead [8] for a more general map between smooth spaces. We also show that the smooth suspension functor is left adjoint to the smooth loop functor, determine a representation of the n-th suspension Σn S 0 of the 0-th sphere as a quotient of Rn and obtain the long exact sequence of a smooth pointed pair.

Keywords

Exact Sequence Homotopy Group Smooth Structure Smooth Space Leave Adjoint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. Cherenack: Smooth Homotopy, (to appear)Google Scholar
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    A. Frohlicher and A. Kriegl: Linear Spaces and. Differentiation Theory. New York, John Wiley and Sons, 1988Google Scholar
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    M. Grandis: Homotopical algebra: a two dimensional categorical setting info Preprint Dipartimento di Matematica, Universita di Genova 191 (1991), 1–50.Google Scholar
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    M.W. Hirsch: Differential Topology. Springer-Verlag. 1976, BerlinzbMATHGoogle Scholar
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    L. Lawvere, S. Schanuel and W. R. Zame: On C°°-function Spaces, preprintGoogle Scholar
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    J. Rotman: An Introduction to Algebraic Topology. Springer-Verlag, 1988, BerlinzbMATHCrossRefGoogle Scholar
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    G.W. Whitehead: Homotopy Theory. Berlin, Springer-Verlag, 1978zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Paul Cherenack
    • 1
  1. 1.University of Cape TownRondeboschSouth Africa

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