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Finite Determinacy of Smooth Map‐Germs

Finite Determinacy of Smooth Map‐Germs C. T. C. WALL The analysis of the conditions for a map-germ to be finitely determined, and of the degree of determinacy, involves the most important of the local aspects of singularity theory. The foundations of the study were laid in an importan t series of papers [Mather I-VI]: we shall review these results as well as others which have appeared in the last decade. An equivalent condition to finite determinacy is possession of a finite dimensional unfolding. Unfoldings are the key to nearly all applications of the theory (in pure and in applied mathematics), and we shall give them some prominence. The present article concentrates on the analysis of smooth map-germs (W, 0) -> (U , 0), or complex holomorphic maps. We emphasise estimates and invariants likely to be important in the classification of such germs. In my earlier exposition [Wall, 1971b] it seemed important to point to examples. Now there are several useful lists [Mather VI], [Damon, 1975], [Arnold, 1976], [du Plessis, 1980a] and further ones can easily be produced, but systematisation is not yet in sight. In Mather's articles the emphasis was on C^-stability, and the question of density of stable maps. For the local case http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Finite Determinacy of Smooth Map‐Germs

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/13.6.481
Publisher site
See Article on Publisher Site

Abstract

C. T. C. WALL The analysis of the conditions for a map-germ to be finitely determined, and of the degree of determinacy, involves the most important of the local aspects of singularity theory. The foundations of the study were laid in an importan t series of papers [Mather I-VI]: we shall review these results as well as others which have appeared in the last decade. An equivalent condition to finite determinacy is possession of a finite dimensional unfolding. Unfoldings are the key to nearly all applications of the theory (in pure and in applied mathematics), and we shall give them some prominence. The present article concentrates on the analysis of smooth map-germs (W, 0) -> (U , 0), or complex holomorphic maps. We emphasise estimates and invariants likely to be important in the classification of such germs. In my earlier exposition [Wall, 1971b] it seemed important to point to examples. Now there are several useful lists [Mather VI], [Damon, 1975], [Arnold, 1976], [du Plessis, 1980a] and further ones can easily be produced, but systematisation is not yet in sight. In Mather's articles the emphasis was on C^-stability, and the question of density of stable maps. For the local case

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1981

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