Homologous fibres and total spaces (Q1802970)

The aim is to describe for a given space \(B\) various classes of fibrations \(F\to E \to B\) over \(B\) such that the acyclicity of \(B\) is equivalent to `\(H_ *(F) \to H_ *(E)\)'. is an isomorphism for all fibrations of this class. Homology here is with trivial integer coefficients. The space \(B\) and also the spaces \(F\), \(E\) in the fibrations \(F\to E\to B\) over \(B\) are assumed to be of the homotopy type of connected CW-complexes. 1) A ``fundamentally torsion-generated (ftg) space \(B\)'' is a space, such that there is a wedge-product \(\bigvee_ \pi K(\pi,1)\) where \(\pi\) runs through a set of finite groups, and a map \(f: \bigvee_ \pi K(\pi,1) \to B\), inducing an isomorphism in \(\pi_ 1\). It is proved: Theorem 2.1. A ftg space \(B\) is acyclic iff for any fibration \(F \to E\to B\) with \(F\) of finite type, \(H_ *(F)\to H_ *(E)\) is an isomorphism. 2) Let \({\mathcal E}_ 0(F)\) (resp. \({\mathcal E}(F)\)) denote the group of free (resp. based) homotopy classes of self-homotopy equivalences of \(F\), and let \(F^ +\) be the plus-construction on \(F\) (so that \(\pi_ 1(F^ +)\) has no non-trivial perfect subgroups). It is proved: Theorem 3.1. \(B\) is acyclic iff \(H_ *(F) \to H_ *(E)\) is an isomorphism, whenever \(\mathcal{E}_ 0(F)\) or \({\mathcal E}_ 0(F^ +)\) has no non-trivial perfect subgroup. 3) Combining this with results of Tsukiyama it follows: Corollary 3.3. The following are equivalent: (i) \(B\) is acyclic; (ii) \(H_ *(F) \to H_ *(E)\) is an isomorphism whenever \(F\) or \(F^ +\) is nilpotent and \(H_ j(F) \neq 0\) for only finitely many \(j\); (iii) \(H_ *(F)\to H_ *(E)\) is an isomorphism whenever \(F\) or \(F^ +\) is homotopy equivalent to a connected CW-complex whose Postnikov system is finite and whose homotopy groups all have solvable automorphism groups.