Submanifold decompositions that induce approximate fibrations and approximation by bundle maps

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.