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Given a map f: M -B defined on an orientable (n + 2)-manifold with all point inverses having the homotopy type of a specified closed n-manifold N, we seek to catalog the manifolds N for which f is always an approximate fibration. Assuming H1 (N) finite, we deduce that the cohomology sheaf of f is locally constant provided N admits no self-map of degree d > 1 when H1 (N) has a cyclic subgroup of order d . For manifolds N possessing additional features, we achieve the approximate fibration conclusion.

Manifolds with finite first homology as codimension 2 fibrators