Classical Morse theory revisited - I backward $\lambda$-lemma and homotopy type

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in \cite{weber:2014c}, as a \emph{dynamical thickening of the stable manifold}. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem \cite{milnor:1963a}. Dynamical thickening leads to a conceptually simple and short proof