Doctoral Dissertations

8-2004

Dissertation

Degree Name

Doctor of Philosophy

Mathematics

Major Professor

Robert J. Daverman

Committee Members

Jerzy Dydak, Morwen Thistlethwaite, Conrad Plaut, George Siopsis

Abstract

In the early 90s, R.Daverman defined the concept of the PL fibrator (). PL fibrators, by definition, provide detection of PL approximate fibrations. Daver- man defines a closed, connected, orientable PL n-manifold to be a codimension-k PL orientable fibrator if for all closed, connected, orientable PL (n + k)-manifolds M and PL maps p : MB, where B is a polyhedron, such that each fiber collapses to an n-complex homotopy equivalent to Nn, p is always an approximate fibration.

If N is a codimension-k PL orientable fibrator for all k > 0, N is called a PL orientable fibrator.

Until now only a few classes of manifolds are known not to be PL fibrators.

Following this concept of Daverman, in this dissertation we attempt to find to what extent such results can be obtained for PL maps p : Mn+k → B between manifolds, such that each fiber has the homotopy type (or more generally the shape) of N, but does not necessarily collapse to an n-complex, which is a severe restriction.

Here we use the following slightly changed PL setting: M is a closed, connected, orientable PL (n + k)-manifold, B is a simplicial triangulated manifold (not necessarily PL), p : Mn+kB a PL proper, surjective map, and N a fixed closed, connected, orientable PL n-manifold.

We call N a codimension-k shape msimplo-fibrator if for all orientable, PL (n + k)-manifolds Mn+k and PL maps p : Mn+kB, such that each fiber is homotopy equivalent to Nn, p is always an approximate fibration. If N is a codimension-k shape msimplo-fibrator for all k > 0, N is a shape msimplo-fibrator.

We are interested in PL manifolds N with π1(N ) /= 1, that force every map f : NN , with 1 /= f# (π1 (N )) π1 (N ), to be a homotopy equivalence. We call PL manifolds N with this property special manifolds. There is a similar group theoretic term: a group G is super Hopfian if every homomorphism φ : G → Gwith 1 /= φ(G)

In the first part of the dissertation we study which groups posses this property of being super Hopfian. We find that every non-abelian group of the order pq where p, q are distinct primes is super Hopfian. Also, a free product of non-trivial, finitely generated, residually finite groups at least one of which is not Z2 is super Hopfian.

Then we give an example of special manifolds to which we apply our main results in the second part of this dissertation.

First we prove that all orientable, special manifolds N with non-cyclic fun- damental groups are codimension-2 shape msimpl o-fibrators. Then we find which 3- manifolds have this property.

Next we prove which manifolds are codimension-4 shape msimpl o-fibrators.

Our main result gives that an orientable, special PL n-manifold N with a non-trivial first homology group is a shape msimplo-fibrators if N is a codimension-2 shape msimplo-fibrator. The condition of N being a codimension-2 PL shape orientable fibrator can be replaced with N having a non-cyclic fundamental group.

In the last section we list some open questions.

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