Semi-Coherence for Semianalytic Sets and Stratifications and Singularity Theory of Mappings on Stratifications

Journal of Singularities
volume 13 (2015), 42-56

Received 22 January 2014.

DOI: 10.5427/jsing.2015.13c

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Abstract:

We consider the conditions on a local stratification \cV which ensure that the local singularity theory in the sense of Thom-Mather, such as finite determinacy, versal unfolding, and classification theorems and their topological versions apply either to mappings on the stratified set \cV or for an equivalence of mappings which preserve \cV in source or target for any of the categories: complex analytic, real analytic, or smooth. For such a stratification \cV, it is sufficient that the equivalence group be a "geometric subgroup of A or K", and this reduces to the structure of the module Derlog(\cV) of germs of vector fields on the ambient space which are tangent to \cV. In the holomorphic or real analytic categories, with holomorphic, resp. real analytic stratifications, we show the necessary conditions are satisfied.

However, in the smooth category the general question is open for smooth stratifications. We introduce a restricted class of "semi-coherent" semianalytic stratifications (\cV, 0) and semianalytic set germs (V,0) (and their diffeomorphic images). This notion generalizes Malgrange's notion of "real coherence" for real analytic sets. It is defined in terms of both Derlog(\cV) and I(V) (the ideal of smooth function germs vanishing on (V, 0)) being finitely generated modulo infinitely flat vector fields, resp. functions. This class includes the special semianalytic stratifications and sets in [DGH], and semianalytic sets such as Maxwell sets, "medial axes/central sets", and the discriminants of C^\infty-stable germs in the nice dimensions. We further show that the equivalence groups in the smooth category for these stratifications are then geometric subgroups of A or K.

Keywords:

real coherent analytic sets, semianalytic stratifications, semicoherent sets and stratifications, singularity theory on stratified sets, geometric subgroups of A and K

Mathematical Subject Classification:

Primary: 57N80, 58K40, 58K60, Secondary: 32S05, 32S60, 58A35

Author(s) information:

James Damon
Department of Mathematics
University of North Carolina
Chapel Hill, NC 27599-3250 USA