Stable homotopy theory

In mathematics, stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that for a given CW-complex X the (n+i)^th homotopy group of its i^th iterated suspension, π_n+i (ΣⁱX), becomes stable (i.e., isomorphic after further iteration) for large but finite values of i. For instance,

ℤ<Id_S¹> = &pi;₁(S¹) &cong; &pi;₂(S²) &cong; &pi;₃(S³) &cong; &pi;₄(S⁴) &cong; ... and

ℤ<η> = &pi;₃(S²) &rarr; &pi;₄(S³) &cong; &pi;₅(S⁴) &cong; ...

In the two examples above all the maps between homotopy groups are applications of the suspension functor. Thus the first example is a restatement of the Hurewicz theorem, that &pi;_n(Sⁿ) &cong; ℤ<Id_Sⁿ...
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