Self-maps on twisted Eilenberg-MacLane spaces
Let L be a space having at most two homotopy groups: thus \(L=K(\pi,1)x_{\phi}K(A,n)\) say where \(\phi: \pi\to \text{aut }A\) gives the action of the fundamental group. The author gives the monoids \((L,L)\) and \((L,L)\) ofbased and unbased homotopy classes of self-maps in terms of group-theoretic invariants. Given \(\alpha\in \text{end }\pi\), define \[ \text{end}(A)_{\alpha}=\{\phi \in \text{end }A:\;\forall \zeta \in \pi,\;a\in A\text{ we have }\phi(\zeta a) = \alpha(\zeta)\phi(a)\}. \] A monoid structure is defined on the set \[ \cup_{\alpha \in \text{end }\pi}H^ n(B\pi:\;\alpha^*A)\times end(A)_{\alpha} \] and it is shown to be isomorphic to \((L,L)^*\). In particular there is a short exact sequence \[ 1\to \text{ext}^ n_{\pi}({\mathbb{Z}},A)\rtimes \text{end }\pi (A)\to (L,L)^*\to \text{end }\pi \to 1. \] Similar results are given for \((L,L)\).
scientific article; zbMATH DE number 4138778