Topological Stability of Smooth Mappings

Gibson, C. G.; Wirthmüller, Klaus; Plessis, A. A. Du; Looijenga, Eduard

doi:10.1007/bfb0095244

Cited by 369 publications

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“…The same is true for all y on the line between y and y . By simple counting (or by applying Thom's isotopy lemma, [18], [25]), we obtain that the Euler characteristics are equal, which means that…”

Section: Lipschitz Continuity Of Support Functionsmentioning

confidence: 99%

The normal cycle of a compact definable set

Bernig

2007

Isr. J. Math.

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Section: Lipschitz Continuity Of Support Functionsmentioning

confidence: 99%

The normal cycle of a compact definable set

Bernig

2007

Isr. J. Math.

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“…Unfortunately, Thom's treatment does not provide a complete proof of the density of topologically stable mappings. 4 Moreover, his proof that Whitney stratifications are topologically locally trivial is extremely difficult to understand and lacks essential details. Thom's outline is geometric and full of wonderful ideas, but it is written in a way that gives the reader very little guidance on how to start filling in the gaps.…”

Section: Mark Goreskymentioning

confidence: 99%

“…They became the standard source for the foundations of stratification theory, and they have often been summarized and paraphrased (for example, in [4,Chapter 2], [37,Chapter 1], [29,Chapter 3]). They are being formally published here for the first time (with minor corrections by John Mather).…”

Section: Mark Goreskymentioning

confidence: 99%

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Introduction to the papers of R. Thom and J. Mather

Goresky

2012

Bull. Amer. Math. Soc.

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Immediately following the commentary below, the previously published article by R. Thom is printed in its entirety: R. Thom, Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society 75 (1969), no. 2, 240-284 (French). This is followed by the first publication of the 1970 lecture notes of J. Mather, Notes on topological stability.. This includes the Cantor set, the Sierpiński sponge, the Snowflake, and other sets of fractional Hausdorff dimension. How does one prove that this sort of behavior cannot happen when f is an analytic function or an algebraic function? These questions were approached about 80 years ago when it was shown ([38, 15, 13, 16]) that algebraic sets could be triangulated. 1 For many years the 1932 paper [13] was cited as the only known proof that complex algebraic sets were locally contractible. But these papers are difficult to follow, and for decades the triangulability of algebraic and analytic sets was treated with some suspicion. Later articles, such as [30,3,18] and especially [10,11,8], finally put these questions to rest. However, the interesting structure of the singularities of an algebraic set is not easily2 described using simplices, and people began to look for a more intelligent way to decompose an algebraic set into (fewer, larger) pieces, starting with the nonsingular part.Hassler Whitney struggled with these questions for decades. In 1946 he wrote Complexes of manifolds [40], in which he considered spaces that were glued together out of smooth manifolds much in the way that a cell complex is glued together from cells. In 1957 Whitney showed in [41] that it is possible, in any algebraic set, to choose an open dense nonsingular part such that its complement is an algebraic set of smaller dimension. Therefore this procedure can be repeated so as to give a finite filtration by closed subsets X 0 ⊂ X 1 ⊂ · · · ⊂ X n such that X r+1 is obtained from X r by attaching a (possibly empty) smooth manifold S r+1 := X r+1 − X r of dimension r + 1.One might hope that the resulting decomposition into pieces X = r S r is locally trivial-that any two sufficiently nearby points x, y ∈ S r should have neighborhoods that are isomorphic (in a sense to be made precise below) by an isomorphism (a homeomorphism, or perhaps, a diffeomorphism) that preserves the induced filtrations. In 1962 René Thom made a first attempt in [34] to make precise such a notion of a locally trivial stratification. He proposed that each stratum S should have a 1 In other words, every compact algebraic set is homeomorphic to a finite simplicial complex.

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“…To put the results of this paper in their proper perspective, we start by recalling the local topological triviality of Whitney regular stratifications, see for details [9], [12].…”

Section: Introductionmentioning

confidence: 99%