The left exactness of the smooth left Puppe sequence
In a previous paper using ideas due to Frohlicher and Kriegl  and Lawvere, Schanuel and Zame , 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  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.
KeywordsExact Sequence Homotopy Group Smooth Structure Smooth Space Leave Adjoint
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