On right-equivalence. (English) Zbl 0593.58014
Study of G-equivalence of germs (in particular for \(G=A\), C, H, L, R) is the basis of singularity theory. A key invariant is the locus of instability. Equivalence of germs implies equivalence of the corresponding loci, and also equivalence of the restriction of the germs to their instability loci, so that a natural approach to classification is to classify possible types of restriction to a given locus. This point of view is taken up in this paper for R; the instability locus in this case is the critical set.
A \(C^{\tau}\) map-germ f: (N,x\({}_ 0)\to (P,y_ 0)\) has non- degenerate critical set if (i) \(J(f)=\{\phi \in C_ N^{\tau}|\) \(\phi\) vanishes on \(\Sigma\) (f\(\}\), (ii) \(\Sigma\) (f) has dimension p-1 in N. Theorem. Let f: (N,x\({}_ 0)\to (P,y_ 0)\) be a map-germ with non-degenerate critical set. Let g: (N,x\({}_ 0)\to (P,y_ 0)\) be a map-germ satisfying: (\(\alpha)\) \(\Sigma\) (f)\(\subset \Sigma (g)\); (\(\beta)\) \(f| \Sigma (f)=g| \Sigma (f)\); (\(\gamma)\) Im Tg\({}_ x\subset Im Tf_ x\) for x in some dense subset of \(\Sigma\) (f). (i) Suppose that f has cokernel rank \(>1\), or that f has cokernel rank 1 and zero full second intrinsic derivative. Then f,g are \(R_{J(f)}\)- equivalent. (ii) Suppose that f has cokernel rank 1 and non-zero intrinsic derivative. Then the following are equivalent: (a) f,g are \(R_{J(f)}\)-equivalent. (b) \(j^ 2f\), \(j^ 2g\) are R-equivalent. (c) \(\Sigma (f)=\Sigma (g)\), g has non-degenerate critical set, and in the \(C^{\infty}\) and \({\mathbb{R}}\)-analytic cases the Hessians H(f), H(g) have the same index. (d) g has cokernel rank 1 and the Hessians H(f), H(g) have the same rank, and, in the \(C^{\infty}\) and \({\mathbb{R}}\)-analytic cases, the same index. If f is also generically finite-to-one on its critical set, then (\(\gamma)\) is redundant, being implied by (\(\alpha)\), (\(\beta)\).
The converses to the above \(R_{J(f)}\)-equivalence results also hold; for, as is easily seen, if map-germs f, g are \(R_{I(\Sigma (f))}\)- equivalent then f,g satisfy (\(\alpha)\), (\(\beta)\), (\(\gamma)\) (indeed with equality in (\(\alpha)\), (\(\gamma)\)).
A generalization of this theorem is given for arbitrary smooth or analytic map-germs.
A \(C^{\tau}\) map-germ f: (N,x\({}_ 0)\to (P,y_ 0)\) has non- degenerate critical set if (i) \(J(f)=\{\phi \in C_ N^{\tau}|\) \(\phi\) vanishes on \(\Sigma\) (f\(\}\), (ii) \(\Sigma\) (f) has dimension p-1 in N. Theorem. Let f: (N,x\({}_ 0)\to (P,y_ 0)\) be a map-germ with non-degenerate critical set. Let g: (N,x\({}_ 0)\to (P,y_ 0)\) be a map-germ satisfying: (\(\alpha)\) \(\Sigma\) (f)\(\subset \Sigma (g)\); (\(\beta)\) \(f| \Sigma (f)=g| \Sigma (f)\); (\(\gamma)\) Im Tg\({}_ x\subset Im Tf_ x\) for x in some dense subset of \(\Sigma\) (f). (i) Suppose that f has cokernel rank \(>1\), or that f has cokernel rank 1 and zero full second intrinsic derivative. Then f,g are \(R_{J(f)}\)- equivalent. (ii) Suppose that f has cokernel rank 1 and non-zero intrinsic derivative. Then the following are equivalent: (a) f,g are \(R_{J(f)}\)-equivalent. (b) \(j^ 2f\), \(j^ 2g\) are R-equivalent. (c) \(\Sigma (f)=\Sigma (g)\), g has non-degenerate critical set, and in the \(C^{\infty}\) and \({\mathbb{R}}\)-analytic cases the Hessians H(f), H(g) have the same index. (d) g has cokernel rank 1 and the Hessians H(f), H(g) have the same rank, and, in the \(C^{\infty}\) and \({\mathbb{R}}\)-analytic cases, the same index. If f is also generically finite-to-one on its critical set, then (\(\gamma)\) is redundant, being implied by (\(\alpha)\), (\(\beta)\).
The converses to the above \(R_{J(f)}\)-equivalence results also hold; for, as is easily seen, if map-germs f, g are \(R_{I(\Sigma (f))}\)- equivalent then f,g satisfy (\(\alpha)\), (\(\beta)\), (\(\gamma)\) (indeed with equality in (\(\alpha)\), (\(\gamma)\)).
A generalization of this theorem is given for arbitrary smooth or analytic map-germs.
MSC:
| 58C25 | Differentiable maps on manifolds |
| 58K99 | Theory of singularities and catastrophe theory |
| 57R45 | Singularities of differentiable mappings in differential topology |
| 32S05 | Local complex singularities |
Keywords:
equivalence; right-equivalence; map-germs; singularity theory; locus of instability; non-degenerate critical set; smooth or analytic map-germsReferences:
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