Doctoral Dissertations
Date of Award
8-1998
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Major
Mathematics
Major Professor
Robert J. Daverman
Committee Members
Klaus Johannson, Morwen B. Thistlethwaite, Jerzy Dydak, Kenneth C. Gilbert
Abstract
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.
Recommended Citation
Kim, Yongkuk, "Codimension two submanifold decompositions that induce approximate fibrations. " PhD diss., University of Tennessee, 1998.
https://trace.tennessee.edu/utk_graddiss/9292