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→ΩB→ M f→ A→B (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.
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References
P. Cherenack: Smooth Homotopy, (to appear)
A. Frohlicher and A. Kriegl: Linear Spaces and. Differentiation Theory. New York, John Wiley and Sons, 1988
M. Grandis: Homotopical algebra: a two dimensional categorical setting info Preprint Dipartimento di Matematica, Universita di Genova 191 (1991), 1–50.
M.W. Hirsch: Differential Topology. Springer-Verlag. 1976, Berlin
L. Lawvere, S. Schanuel and W. R. Zame: On C°°-function Spaces, preprint
Mac Lane: Categories for the Working Mathematician. Springer-Verlag, 1971, Berlin
J. Rotman: An Introduction to Algebraic Topology. Springer-Verlag, 1988, Berlin
G.W. Whitehead: Homotopy Theory. Berlin, Springer-Verlag, 1978
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© 1996 Kluwer Academic Publishers
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Cherenack, P. (1996). The left exactness of the smooth left Puppe sequence. In: Tamássy, L., Szenthe, J. (eds) New Developments in Differential Geometry. Mathematics and Its Applications, vol 350. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0149-0_5
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DOI: https://doi.org/10.1007/978-94-009-0149-0_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6553-5
Online ISBN: 978-94-009-0149-0
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