On the spaces of self homotopy equivalences for fibre spaces. II (Q1083713)

The author extends, to the non-simply connected case, an earlier result from part I [Proc. Japan Acad., Ser. A 61, 15-18 (1985; Zbl 0575.55009)] on the weak homotopy type of the space \(G_ 0(E)\) of self-homotopy equivalences of a space, E, with precisely two non-vanishing homotopy groups: remarkably \(G_ 0(E)\) has the weak homotopy type of the product \[ R\times H^ n(B;G)\times \prod^{n-1}_{i=1}K(H^{n-i}(B,b_ 0;G),i) \] where \(K(G,n)\to E\to K(\pi,m)=B\) is classified by \((B,*)\to (L(G,n+1),*)\), where the cohomology is taken with local coefficients \(B\to K(aut G,1)\) and R is a subgroup of aut \(\pi\times aut G.\) There is the fibration \(K(G,n+1)\to L(G,n+1)\to K(aut G,1)=W\) where \(L(G,n+1)\) is the classifying space for fibrations with fibre K(G,n). The space of maps over (W,*) is shown to have the weak homotopy type of the product \[ map_ 0(X,L(G,n+1))_ W\simeq H^{n+1}(X,x_ 0;G)\times \prod^{n}_{i=1}K(H^{n+1-i}(X,x_ 0;G),i) \] again with local coefficients for the cohomology. Furthermore, for a suitable fibration \(F=K(G,n)\to E\to B\), with \((B,*)\to (L(G,n+1),*)\), the space \(map_ 0(B,L(G,n))_ W\) is shown to have the weak homotopy type of the space \({\mathcal G}(E mod F)\) of self-fibre-homotopy equivalences of E leaving F fixed.