Submanifold decompositions that induce approximate fibrations and approximation by bundle maps

Author

Young Ho Im

Date of Award

5-1991

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Robert J. Daverman

Committee Members

K. Johannson, M. Thistlethwaite, J. Dydak, K. Jeon

Abstract

Recently, R.J. Daverman showed that all closed surface N² with non-zero Euler characteristic have the property that all proper surjective mappings p from an (n+2)-manifold M to a 2-manifold B for which each p^-1(b) is homotopy equivalent to N² necessarily are approximate fibrations. In the first part of this dissertation, we extend so that any finite product of closed orientable surfaces with non-zero Euler characteristic possess the above property as well. Next we consider a question when approximate fibrations can be approximated by locally trivial bundles. We prove that if a proper map p from an (m+n) -manifold M (m+n ≥ 5) to a n-dimensional polyhedron B for which each fiber is homotopy equivalent to a closed manifold F^m is an approximate fibration so that S₀(TⁱxI^jxF) = 0 for i+j = n and the inclusion H → G induces monomorphism for O≤i≤n-2, then p can be approximated by a locally trivial bundle. Also we give several applications of this result.

Recommended Citation

Im, Young Ho, "Submanifold decompositions that induce approximate fibrations and approximation by bundle maps. " PhD diss., University of Tennessee, 1991.
https://trace.tennessee.edu/utk_graddiss/11138