Notes on Topological Stability

By J. Mather. 2012

282 citations recorded by the source corpus.

These notes are part of the first chapter of a series of lectures given by the author in the spring of 1970. The ultimate aim of these notes will be to prove the theorem that the set of topologically stable mappings form a dense subset of C∞(N,P ) for any finite dimensional manifolds N and P where N is compact. The first chapter is a study of Thom–Whitney theory of stratified sets and stratified mappings. The connection of the material in these notes with the theorem on the density of topologically stable mappings appears in Section 11, where we give Thom’s second isotopy lemma. This result gives sufficient conditions for two mappings to be topologically equivalent.

Subfields: Control and Systems Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ab00db596fdb1709e897eb9357da04b2b4b0cfe0

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Citations in the corpus

• Regularity at infinity of real mappings and a Morse–Sard theorem (2011, background): “29] and later in [NZ], where it is called M-tameness. The name “Milnor set” 12 occurs in [NZ] too. Distance functions like ρare also central ingredients in defining regular stratifications, e.g. Mather [Ma], Kuo [Ku], Bekka [Be]. Example 5.4. Let ρ: Rn → R ≥0, ρ(x) = ( Pn i=1|xi| 2p i)1/2p, where (w 1”
• Singular Intersection Homology (2020, methodology): “In particular, we will not prove the existence and uniqueness of normalizations for topological stratified pseudomanifolds, instead referring to Padilla [180] or Matthews [159].”
• Localizing virtual cycles by cosections (2010, background): “Because M is algebraic and M(σ) is compact, such a neighborhood V always exists ([26]).”
• A Superficial Working Guide to Deformations and Moduli (2011, background): “By the use of stratified spaces (see [100]), and control data, one finds then a differentiable trivialization for the quotient analytic space S/G, hence a trivialization of the action.”