5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

204 G general in Ep. It

204 G general in Ep. It follows that for genetic Morse families the organising centre f = F(.,0) is ~G finitely determined. If G is finite it follows from [R2, Prop.5.i] that f is actually finitely determined, and so by IS, Theorem 2.1] has an unfolding which is G-versal. More precisely, we have the following result. Proposition 2.6 G mp . Suppose G is f'mite and let f e Ep be finitely determined. Let U = j-~ with its induced action of G. Then there exists a G-invariant unfolding 5F: W*~BU ~ R such that for any representation V of G and G-invariant unfolding F : W*@V ~ [R with an organising centre F(.,O) which is ~G equivalent to f, there exists a G-equivariant map + germ ¢p : V,O ~ U,0 such that F(p,q) is 5~ G equivalent to ?(p,q~(q)). This may be regarded as a prenormal form for genetic G-invariant Morse families. We can also obtain a charaterisation of infinitesimally stable families in terms of this prenormal form. Proposition 2.7 + The Morse family ?(p,q)(q)) is infinitesimally 5~ G stable if and only if the map germ q) : V,0 ~ U,0 is infinitesimally 5~ G stable. Proof Using the Equivariant Preparation Theorem [ D,P ] it is easily shown that F(p,tp(q)) is G O + Oq mpq infinitesimally ~G stable if and only if the mapping ~ : G 0 "+ mqOq oG.F+ mqGEpqG ' induced by the mapping ~ ~ ~.F on Oq, G is surjective. Now note that mpq is O~.F + mq~r~ mpq isomorphic to U and so by the ordinary Preparation Theorem ....... is isomorphic to OrcF

Z (V,U) as an E q module. It follows that 205 G mpq ®re G is isomorphic to EG(V,U) as an Eq 8G(V,U) module and so the target of ~] is isomorphic to . Moreover the mapping mqGEG(V,U) itself corresponds to that induced from ~ ~ ~.¢p and so the surjectivity of ~ is equivalent to the 5~ G stability of ¢p. + A G-versal unfolding is always infinitesimally ~G stable. As a corollary of -i- Proposition 2.6 we obtain sufficient conditions on an infinitesimally ~G stable unfolding for it to be equivalent to a trivial extension of a G-versal unfolding. An obvious requirement is that the space U associated to the organising centre is isomorphic to an invariant subspace of V. The second condition is an upper bound on the dimension of V G. For each irreducible representation V i of G, let I.t(Vi,V) = dim~ HomG(Vi,V ). Define I~(V) to be the minimum of the non zero ~t(Vi,V). Corollary 2.8 + If F : W*@V,0 -~ ~ is an infinitesimally ~G centre f such that :- stable Morse family with organising nan (i) U = ~ is isomorphic to a G- invariant subspace of V (as representations of G), (ii) dim V G < dim U G + Iz(V), + then F is ~G equivalent to a trivial extension of a G-versal unfolding of f. Proof It is sufficient to suppose that F(p,q) = gr(p,q>(q)), where 9: W*~U -, ~ is a G- versal unfolding of f and q~ : V,0 ~ U,0 is 5~ G stable. The geometric criterion for 5~ G stability [ R2,Theorem 4.3 ] implies that the jet mapping:- jlq~lva : V G ~ U o •Hom (V,U) G is transversal to the orbits of the GL(V)G action on the target. Hence, if ~t denotes the

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