A stable homotopy group of a based space 
is the direct limit of the
homotopy groups of its iterated reduced suspensions:
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The transition maps are suspension homomorphisms. These are the homotopy groups of the suspension
spectrum of 
. For 
they give the stable homotopy groups 
of spheres, equivalently those of the sphere
spectrum. For example, 
. Adams spectral
sequences and Morava K-theory localizations
are tools for investigating these groups.
