Cohomology fibrations

Čech cohomology fibrations in the sense of Ferry–McDuff–Segal are redefined for arbitrary proper maps. The main result of this paper is that R-cohomology fibrations resemble locally trivial bundles if the cohomology of the fibers is finitely generated. m e f :X→Y is a proper map and {Mk}k⩾0 is a sequence of finitely generated R-modules such that the following conditions are satisfied: (a) a submodule of Hk(X;R), nclusion induced homomorphism Hk(X;R)→Hk(f−1(y);R), y∈Y, sends Mk isomorphically onto Hk(f−1(y);R). (i) a point y in Y and m⩾0, there is a neighborhood U of y in Y such that for all compact subsets A of U, y∈A, there is an isomorphism φA :Hm(f−1(A);R)→Hm(A×f−1(y);R). Moreover, if y∈B⊂A⊂U, then the diagram (1) is commutative. , y∈A, is pathwise connected and compact, then Hm(f−1(A);R)∼Hm(A×f−1(z);R) for any z∈A.