Codimension two submanifold decompositions that induce approximate fibrations

An extensive variety of closed n-manifolds N, called codimension-2 (orientable) fibrators, automatically induce approximate fibrations, in the sense that all proper maps ƒ: M → B from any (respectively, orientable) (n + 2)-manifold M to a metric space B (equivalently, to a 2-manifold B) such that each f^-1(b) has the same homotopy type (or, more generally, the same shape) as N are approximate fibrations. R.J. Daverman showed that closed hopfian manifolds with hopfian fundamental group and nonzero Euler characteristic, as well as closed hopfian manifolds with hyperhopfian fundamental group are codimension-2 orientable fibrators. Using the concept of s-hopfianness, we generalize Daverman's results about codimension-2 orientable fibra tors to the orientation-free version as follows:

Closed s-hopfian manifolds with either hyperhopfian fundamental group or hopfian fundamental group and nonzero Euler characteristic are codimension-2 fibrators.

In the second part of this dissertation, we study the behavior of codimension-2 fibrators under the connected sum operation. As a consequence, we will get the neat result on closed 4—manifolds as follows;

Given a closed 4—manifold N which is a nontrivial connected sum of two 4—manifolds N₁ and N₂, where π₁(N₁ and π₁(N₂) are hopfian, then N is a codimension-2 fibrator if and only if N is not homotopy equivalent to RP⁴#RP⁴. In the next section we determine which closed n-manifolds (n ≤ 4) with geometric structure are codimension-2 fibrators. In closing, we mention some unsettled topics.