Necessary and Sufficient Conditions for S-hopfian Manifolds to Be Codimension-2 Fibrators

Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p : M → B defined on an (n + 2)-manifold M such that all fibre p−1(b), b ∈ B, are shape equivalent to N is an approximate fibration. The most natural objects N to study are s-Hopfian manifolds. In this note we give some necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators. Approximate fibrations, introduced and extensively studied by Coram and Duvall, form an important class of mappings, almost as efficacious as Hurewicz fibrations and cell-like maps. The advantage of this notion is that on one hand there exists an exact homotopy sequence but on the other hand there are more such approximate fibrations available. (See [3, 4] for the definition and usefulness of approximate fibrations.) Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p : M → B defined on an (n+2)-manifold M such that all fibres p−1(b), b ∈ B, are shape equivalent to N is an approximate fibration. The most natural objects N to study are s-Hopfian manifolds. All closed s-Hopfian manifolds with either trivial fundamental group or Hopfian fundamental group and nonzero Euler characteristic or hyperhopfian fundamental group are known to be codimension-2 fibrators [6, 7, 15, 16, 18]. Surprisingly few nonfibrators are known so far. The best well-known nonfibrators are closed manifolds that cyclically cover themselves nontrivially, such as S × S and RP#RP [6, Theorem 4.2 ]. Recently Daverman gave some other kind of nonfibrator, which is L(p, q) × S [8, Example 2.1]. None of them gives necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators in general cases. In this note we give some necessary and sufficient conditions in some special cases. Received by the editors October 19, 1999. 2000 Mathematics Subject Classification. Primary 57N15, 55M25; Secondary 57M10, 54B15.