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

DOI: http://dx.doi.org/10.12775/TMNA.2016.020

#### 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 \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.

#### Keywords

Morse theory; homotopy type; flow selector

#### References

J. Milnor, Morse theory, based on lecture Notes by M. Spivak and R. Wells, Ann. Math. Stud. No. 51. Princeton University Press, Princeton, N.J., 1963.

J. Weber, Stable foliations and semi-flow Morse homology, Ann. Scuola Norm. Sup. Pisa Cl. Sci., arXiv 1408.3842 (to appear).

J. Weber, Contraction method and Lambda-Lemma, São Paulo J. Math. Sci. 9 (2) (2015), 263–298.

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