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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.

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What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

Combinatorial descriptions of homotopy groups of spheres have been studied in the unstable world rather than the stable world. Jie Wu found a description of homotopy groups of $S^2$ in his thesis and …
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