Let \(E,B\) be two spaces, \(\pi:E\to B\) a map, \(B^p\) the compact \(p\)-ball, \(S^p=\partial B^{p+1}\), and \(*\in S^p\) a base point. \(\pi\) is a homotopic submersion if for every map \(f:X\to E\) whose source \(X\) is a polytope, every germ-of-homotopy for \(\pi\circ f\) lifts to a germ-of-homotopy for \(f\). A vanishing \(p\)-cycle is a fibred map \(f:S^p\times [0,1]\to E\) such that for each \(t>0\), the map \(f_t\) is null-homotopic in its fibre. An emerging \(p\)-cycle is a fibred map \(f:S^p \times [0,1]\to E\) such that \(f(*,t)\) has a limit for \(t\to 0\). This paper investigates necessary and sufficient conditions for a submersion to have the homotopy lifting property. The author proves
Theorem 1. A surjective map is a fibration if and only if it satisfies the following three conditions: it is a homotopic submersion, all vanishing cycles of all dimensions are trivial, and all emerging cycles of all dimensions are trivial.
A vertical domain is a \(p\)-dimensional compact submanifold of a fibre with a smooth boundary. A family \(VD\) of vertical domains is exhaustive if every compact subset of every fibre is contained in some \(X\in VD\). The author also proves
Theorem 2. A surjective smooth submersion \(\pi:E\to B^q\) is a (locally trivial fibre) bundle if and only if it admits an exhaustive isotopy, \((q-1)\)-fibred family of vertical domains.
Let \(VE(X,E)\) be the space of vertical embeddings of \(X\) into \(E\), and \(VE^0(X,E)\) the connected component of \(VE(X,E)\) containing the original inclusion \(X\to E_b\), \(E_b\) is a fiber. \(VD\) is isotopy invariant if for every \(X\in VD\) and every \(\Phi\in VE^0(X,E)\), \(\Phi (X)\in VD\). The result is applied to spaces of embeddings of compact domains into the fibers. From this results some corollaries are obtained.