Stable Homotopy Theory

# Stable homotopy theory

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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)<sup>th</sup> homotopy group of its i<sup>th</sup> iterated suspension, &pi;<sub>n+i</sub> (Σ<sup>i</sup>X), becomes stable (i.e., isomorphic after further iteration) for large but finite values of i. For instance,

ℤ<Id<sub>S<sup>1</sup></sub>> = &pi;<sub>1</sub>(S<sup>1</sup>) &cong; &pi;<sub>2</sub>(S<sup>2</sup>) &cong; &pi;<sub>3</sub>(S<sup>3</sup>) &cong; &pi;<sub>4</sub>(S<sup>4</sup>) &cong; ... and
ℤ<η> = &pi;<sub>3</sub>(S<sup>2</sup>) &rarr; &pi;<sub>4</sub>(S<sup>3</sup>) &cong; &pi;<sub>5</sub>(S<sup>4</sup>) &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;<sub>n</sub>(S<sup>n</sup>) &cong; ℤ<Id<sub>S<sup>n</sup></sub>...

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