http://link.springer.com/content/pdf/10.1007/978-94-010-1289-8_4.pdf

Generic Equivariant Maps

1. Stability. Let X D Y be real analytic (or, more generally, closed semianalytic) subsets of R with dim X < n, and let M C TV be submodules of (C°°(R)) obtained (as modules of global sections) on tensoring by C°°(R) coherent real analytic subsheaves M C N of (0(R)), where 0(R) denotes the sheaf of real analytic functions on R. Let M(Yt X) (similarly for N) be the space of m-tuples 0 of Taylor fields on X flat on Y such that at each point x G X, <t>x is in the formal completion Mx of M at x. Let r: N(Y, R ) —• N(Yy X)/M(Y, X) = P(Yt X) denote the restriction. THEOREM 1. There is a continuous E: P(Yf X) —• N(Y, R ) such that rE = 1.

/pdf/spaces-of-smooth-functions-on-analytic-sets-1hb40cq8yi.pdf

Spaces of smooth functions on analytic sets

Differentiable mappings between foliated manifolds

Extension theorems are proved for smooth functions on a coherent real analytic space for which local defining functions exist which are finitely determined in the sense of J Mather, (1968), and for smooth functions invariant under the action of a compact lie group G thus providing the main step in the proof that smooth infinitesimal stability implies smooth stability in the appropriate categories In addition the remaining step for the category of CxG-manifolds of finite orbit type is filled in

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Extension theorems for smooth functions on real analytic spaces and quotients by Lie groups and smooth stability

John Mather has proved that infinitesimal stability implies stability for proper maps in the category of smooth manifolds. This result gives a computable algebraic criterion for stability. In this paper it is shown that there is an extension of Mather's result when the range is only assumed to be a compact semianalytic set of some real Euclidean space—this class of spaces is an obvious maximal candidate for which computations can be carried out using only classical polynomial algebra. The proof depends on a splitting theorem for the restriction map from the smooth functions on a Euclidean space to those on a closed subset and is proved by an algebraic-geometric method derived from the work of B. Malgrange. No knowledge of functional analysis is assumed although an alternative analytic method for proving the main result is also indicated. Only simple applications are given (mostly to functions defined locally in the neighbourhood of an isolated hypersurface singularity of the type studied by J. Mi...

A geometric splitting and the c∞ stability of maps from semianalytic sets

/pdf/stability-of-equivariant-smooth-maps-bizfbrux9l.pdf

V. Poénaru

Papers

Stability of equivariant smooth maps