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Tools of Unstable Homotopy Theory
2/24/2022
Contents
The functor \(\Omega \) gives an equivalence between pointed connected spaces and group-like \(\EE _1\)-algebras in spaces (\(\infty \)-groups), so we can approach our study of spaces by studying groups. A nilpotent discrete group is effectively studied via its Lie algebra, and the higher homotopy groups in \(\Omega X\) should be thought of as a nilpotent thickening of the group \(\pi _1X\). Thus if \(X\) is a nilpotent space, we can expect the data of \(X\) to be largely captured by Lie algebra data associated to \(\Omega X\).