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5 years ago

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

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

## 212 ~y

212 ~y ....................... ~y XlI T 0 0 ......... 0 T 0 V 0 ......... 0 0 ....................... 0 ¢ where ~/T is the transpose of the k x r matrix (~gab(0,0) ), and ~.c is the matrix:- 3y aVt~ (o,0) ~Yd I = 1 ..... k k+l ..... s. The rows of the submatrix diag {~T,~T ........ ~/T} can be regarded as vectors which span the tangent space to the GL(r) - orbits in M(k,r). Coronary 4.4 (i) There exist infinitesimally stable Morse families unfolding f(k) = ~2(k+l) if and only if s > max ( k, k+(k-r)r ). (ii) There exist generic 7/2-invariant Morse families which are not infinitesimally stable if and only if r < s. Proof (i) If rank V(0,0) = p < min (k,r) then the codimension of the orbit GL(r).~g(0,0) is (k-p)r. For stability we need ~ to be transversal to this, and so s-k > (k-p)r. The result follows on noting that the minimum value taken by (k-p)r, asp varies, is 0 if k r. (ii) This follows from the fact that there always exist generic Morse families which unfold ~ 2(s+l). By (i), these can not be stable ifr < s.

Theorem 4.5 213 (i) If r >_. s then generic Z2-invariant Morse families are infinitesimally stable and are equivalent to trivial extensions of the families :- k k ~.2(k+l) + ~yj~2j + ~ xj~2j-1 k < S. j=l j=l (ii) If s >- r -- 1 then the infinitesimally stable Morse families are equivalent to trivial extensions of the families :- k k-1 ~,2(k+l)+~ yj~.2J+ ~ ({Xj+Yk+j)Xl~2J+l +Xx~. 1 k

1O 9 I B I 7 I 6 5 4 I 3 2 1