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, π<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,

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 π<sub>*n*</sub>(*S*<sup>*n*</sup>) ≅ ℤ<Id<sub>*S*<sup>*n*</sup></sub>...

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- ℤ<Id<sub>
*S*<sup>1</sup></sub>> = π<sub>1</sub>(*S*<sup>1</sup>) ≅ π<sub>2</sub>(*S*<sup>2</sup>) ≅ π<sub>3</sub>(*S*<sup>3</sup>) ≅ π<sub>4</sub>(*S*<sup>4</sup>) ≅ ... and - ℤ<η> = π<sub>3</sub>(
*S*<sup>2</sup>) → π<sub>4</sub>(*S*<sup>3</sup>) ≅ π<sub>5</sub>(*S*<sup>4</sup>) ≅ ...

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 π<sub>

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