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Sep, 2019
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Stable Homotopy Theory

The word “stable” in stable homotopy theory means the stability under taking suspensions. By the Freudenthal suspension theorem, the map between homotopy groups

E : π   (ΣkX ) −→ π     (Σk+1X )
     n+k          n+k+1

induced by the suspension is an isomorphism for k large enough. This theorem suggested to define a category whose objects are based CW-complexes and the set of morphsims from X to Y is

{X,Y } = colim [ΣkX, ΣkY ]∗.
          k

The resulting category is the most elementary form of the stable homotopy category.

References