Homotopy classes of self-maps and induced homomorphisms of homotopy groups. (English) Zbl 1112.55007

For a based space \((X,x_0)\), let \([X,X]\) denote the homotopy classes of based self-maps. With composition \([X,X]\) has the structure of a monoid. The units in \([X,X]\) form a group \({\mathcal E}(X)\), the group of homotopy classes of homotopy equivalences \(X\to X\). The focus of the paper is the subgroups of \({\mathcal E}(X)\) defined as follows: \[ \begin{aligned} {\mathcal E}_\Omega(X) &= \{\alpha \in {\mathcal E}(X) \mid \Omega\alpha = \text{ id}\}\cr {\mathcal E}_{\# n}(X) &= \{\alpha \in {\mathcal E}(X) \mid \alpha_{\#} = \text{ id} : \pi_i(X) \to \pi_i(X), \text{\;for all\;}i \leq n\}\cr {\mathcal Z}_\Omega(X) &= \{\alpha \in [X,X] \mid \Omega\alpha = 0\}\cr {\mathcal Z}_{\# n}(X) &= \{\alpha\in [X,X] \mid \alpha_{\#} = 0 : \pi_i(X) \to \pi_i(X) \text{\;for all\;}i\leq n\}\end{aligned} \] From the definitions, there are inclusions \[ {\mathcal E}_\Omega(X) \subset {\mathcal E}_{\#\infty}(X) \subset {\mathcal E}_{\#}(X), \quad {\mathcal Z}_\Omega(X) \subset {\mathcal Z}_{\#\infty}(X) \subset {\mathcal Z}_{\#}(X) \] where \({\mathcal E}_{\#}(X)\) and \({\mathcal Z}_{\#}(X)\) denote \({\mathcal E}_{\# n}(X)\) and \({\mathcal Z}_{\# n}(X)\) when \(n = \dim X\) as a CW-complex. These groups and semigroups have been widely studied by Arkowitz, Maruyama, Oshima, and Pavešić. An interesting question explored in the paper and in work of Pavešić is whether there is a finite-dimensional CW-complex \(X\) for which \({\mathcal E}_\Omega(X) \neq {\mathcal E}_{\# \infty}(X)\). In the case of the semigroups \({\mathcal Z}(X)\) there is a finite complex with \({\mathcal Z}_\Omega(X) \neq {\mathcal Z}_{\#\infty}(X)\) and there is an example of an infinite-dimensional complex \(X\) with \({\mathcal E}_\Omega(X) \neq {\mathcal E}_{\# \infty}(X)\). The paper contains all sorts of subtle results that help describe the groups studied in other ways. For example,
\[ {\mathcal E}_\Omega(X) = \{\alpha \in {\mathcal E}(X) \mid \alpha_* = \text{ id}: [\Sigma A,X] \to [\Sigma A, X]\text{\;for every space\;}A\} \]
and there is analogous characterization for \({\mathcal Z}_\Omega\). It follows that if \(X\) is a co-H-space, then \({\mathcal E}_\Omega(X) = \{\text{id}\}\) and \({\mathcal Z}_\Omega(X) = \{0\}\). When the homotopy groups of \(X\) are a quotient of the homotopy groups of a wedge of spheres, there are identifications \({\mathcal E}_{\#\infty}(X) = {\mathcal E}_{\# N}(X)\) when \(N\) bounds the dimension of the wedge. When \(X\) is group-like (satisfies the axioms for a group up to homotopy), the function \(\alpha \mapsto \alpha + \text{ id}\) determines a bijection between the \({\mathcal E}\)-groups and the \({\mathcal Z}\)-groups. For such spaces, the order of nilpotence of \([X,X]\) provides lower bounds on the number of elements in localizations of the \({\mathcal E}\)-groups and \({\mathcal Z}\) groups. In the case that \(X\) is a Lie group, certain localizations yield homotopy types that are products of odd spheres (regular primes), to which earlier results apply. The authors consider products of low rank compact Lie groups and completely determine the \({\mathcal E}\)-groups and the \({\mathcal Z}\)-groups.


55P10 Homotopy equivalences in algebraic topology
55P45 \(H\)-spaces and duals
55P60 Localization and completion in homotopy theory
55Q05 Homotopy groups, general; sets of homotopy classes
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