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A survey of homotopy nilpotency and co-nilpotency. (English) Zbl 1469.55005

Recall the concept of the homotopy nilpotency and co-nilpotency of spaces as follows. First, let \(X\) be an \(H\)-space, \(\varphi_{X,1}=\mbox{id}_X:X\to X\) and let \(\varphi_{X,2}:X^2\to X\) the commutator map. Then define the family of maps \(\{\varphi_{X,n}:X^n\to X\}_{n\geq 1}\) inductively by the equality \(\varphi_{X,n+1}=\varphi_{X,2}\circ (\mbox{id}_X\times \varphi_{X,n}).\) An \(H\)-space \(X\) is called homotopy nilpotent of class \(n\) if \(\varphi_{X,n+1}\) is null-homotopic but \(\varphi_{X,n}\) is not. In this case, we write \(\mbox{nil }X=n\). Dually we can define the concept of co-nilpotency of a co-\(H\)-space. Next, define the concept of the homotopy nilpotency (resp. homotopy co-nilpotency) of a pointed space \(X\) by means of its loop space \(\Omega X\) (resp. its suspension space \(\Sigma X\)).
This paper is concerned with reviewing known results and stating new results on the homotopy nilpotency and co-nilpotency of spaces. Especially, the author considers the systematic study of the homotopy nilpotency of a homogenous space \(G/K\) for a Lie group \(G\) with its closed subgroup \(K<G\). In particular, he studies the homotopy nilpotency of the loop spaces \(\Omega G_{n,m}(\mathbb{K})\), \(\Omega F_{n_1,n_2,\dots ,n_k}(\mathbb{K})\) and \(\Omega V_{n,m}(\mathbb{K})\) for \(\mathbb{K}=\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{H}\), where \(G_{n,m}(\mathbb{K})\), \(F_{n_1,n_2,\dots ,n_k}(\mathbb{K})\) and \(V_{n,m}(\mathbb{K})\)) denote the Grassmann, flag and the Stiefel manifold of the field of real and complex numbers or the skew \(\mathbb{R}\)-algebra of quaternions.

MSC:

55P15 Classification of homotopy type
20F18 Nilpotent groups
55P45 \(H\)-spaces and duals
55P60 Localization and completion in homotopy theory
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