Singular manifolds

Extendibility and transversality

“…and q be positive integers. It is possible to assign to each tuple (a 19 , a n ) of nonnegative integers, with a λ >p -q and a x > > a n , a submanifold Z(a u -,a n ) of J n (p, q) in such a way that i , a m )(q -p + a,) -Σf=2 P(^ > ajifli-i -^) For a proof, see [2] or [8]. It is possible to define complex submanifolds CZ(a l9…”

“…This would, however, take space. The point we are trying to make here is that the singular types constructed in [8] give rise to singular types of maps of real manifolds into complex manifolds.…”

“…But from the point of view of [8] the conditions mentioned above can be interpreted as restrictions on the (n + &)-jet of the map /: M -• C n , where / is the inclusion of M m C n . Careful examination of the restrictions on the jet of / reveals that "most" (n + &)-jets satisfy these restrictions; so, therefore, do "most" maps in C m topology, for m large enough (verifying Bishop's remark).…”

“…In § 2 the notation and some of the main ideas of [8] are reviewed with special attention to the situation considered here. Computations comparing jets of maps and Lie brackets are done in § 3.…”