On Generic Composites of Maps

Abstract

We show that (in the nice dimensions) if a stable map is restricted to a generic submanifold of the source manifold, then the resulting map will also be stable. At the same time, the analogous result for versal unfoldings is proved. A few examples of applications to the extrinsic geometry of submanifolds of Euclidean space are discussed. 1. Preliminaries and statements of results Throughout, X, Y and Z are manifolds (without boundary), and # is any of the standard group actions on C(7, Z) (etc.) arising in singularity theory: ^2, ^?, 5£, *$, $0 and Jf. These induce actions on the (multi-)jet spaces ^(Y,Z). We say that the family of maps F:YxU >Z (y.u)* >fu(z) is locally &-versal if for every (y,u)eYx U, the germ of at (y,u) is a #-versal unfolding of the germ of fu at y. We say, moreover, that is ^-versal if for every we C/and every finite subset S J(Y, Z) to the ^-orbits (for r sufficiently large), ^-versality can be stated in a similar way in terms of transversality in multi-jet space (also for s large enough: any s ^p + £+ will suffice). More precisely, the infinitesimal condition for to be ^-versal at (y, u) is Te^-fu + U{F'i} = 6(fu) (1.1) at y, where F = dF/du^i^ 1,...,/, are the initial speeds of the unfolding, 6(f) denotes the C(7)j/-module of vector fields along the germ/, that is, of smooth sections of/* TZ, and, for the various ^, = tf[6(Y)), Te<£-f=f*mv-6(J), The tangent spaces in the jet spaces are obtained from these by factoring out by the maximal ideal to the appropriate power. For definitions and properties of versality, see, for example, Damon [1], Martinet [5] or Wall [11]. Received 3 July 1989. 1980 Mathematics Subject Classification 58C27. Research supported by the SERC. Bull. London Math. Soc. 23 (1991) 81-85