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

Joa Weber



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.


Morse theory; homotopy type; flow selector

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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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