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

Abstract

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 [J. Weber, Stable foliations and semi- ow Morse homology, arXiv 1408.3842], as a 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 [J. Milnor, Morse theory, Ann. Math. Stud. No. 51. Princeton University Press, Princeton, 1963]. Dynamical thickening leads to a conceptually simple and short proof.