ON STABILITY OF О MAPPINGS OF MANIFOLDS WITH BOUNDARY 201

a , let e^ denote the distance from (Po,(K^) to V^-L^. Then г^>0 for each a e Л. We define O^ to be the set of all ^ e r^(7r?;Tl/) such that U^xj)\\ <e„ for all xe(p^(LJ and all tel. We write Ôy^OyOP'^inifUl For each ol e A, each X e K^ and each £, e Ou, there exists a unique curve [0; e] -> I^, ^ -^ Га(фаМ.О such that 7„(фДх),0) = ф^(х) and

27а ( <Ра ( Д0 . / / /ЧАЛ

- - - - - - ^^— == ЦуЛя>Лх1г)^).

If с G Ô(^, we can obviously choose c= 1, since the curve needs more than unit time to get out of (p^(LJ. We can then define

for ^ G Ô(;, (x,f) £ K^xL That this definition has the properties stated in the proposition is proved in the same way as Lemma 2 in § 7 of [1].

4 . Proof of the main result.

Now we will prove Theorem 2.3. As the procedure of this proof is just the same as that in § 7 of [1], we will restrict ourselves to point out the differences from this proof.

The assumption is that

ß = (co(/|M),fa|M),r^(tP),r-(tM),r-((/|M)*tP))

is surjective. From Mather we have an isomorphism of Г'^(ТР)^С°^(Рх1) with Г'^(прТР). By restriction we get a bijection of Г'^(ТР)®С'^(Рх1) with r°^(npfP). As fP is the union of the vector bundles TQ over all strata Q of P, it is easy to see that this bijection is also an isomorphism. It follows that i (ß) is isomorphic to the mixed homomorphism

i { ßy = (а,)9,Г-(я|?7Р),Г-(7гйт),Г-((а|М)оям)*ГР))

where

a = а)'иопм)\Г^(прТР) and ß = t'(fonj^)\r^(n^fM).

Instead of the parameter space X we shall use

X = {geX\ yze M(g(zx/)c:str (/(z)))} ,

but the base point is still Xo=J o%^. We let J g C^(M x I,P) denote the germ of the inclusion of X into C^iMxI^P). And we let r^(npfP) denote the set of germs u at Xq of mappings X -^ C^iMxI, fP) having the following property: for suitable representatives пр of the germ кр and й of u, we have жр(х)^