56
RICE UNIVERSITY STUDIES
3. Thom TransversaHty Theory
The following discussion of Thom Transversality Theory is taken from
the notes of Levine [8].
Let V and M be manifolds of real dimensions n and p respectively. We
define the r-jet from V to M with source x and target y of a ⅞fr-map
ft V → M as the equivalence class of all ⅞?-maps from V to M which take
χ into y, all of whose partial derivatives at x of orders ≤ r are equal to
those of f.
Wc denote the r-jet of f at x by Jr(∕)(x). If we let Jr(n,p) denote the
space of r-jets of 2fr-maps f:V → M with /(x) = y, then J'(n,p) becomes
a euclidean space if we take the values of the partial derivatives at x as
the coordinates of a jet. The set of all r-jets from V to M is denoted by
Jr(y,M). If we choose local coordinates in V and M, then Jr(V,M) be-
comes a fibre bundle with fibre Jr(ntpj and group Lr(n,p) (see [8]).
If/: V → M is of class at least ⅞fr, the r extension of f is defined by:
Jr(∕); V→ J∖V,M), where x → Jr(∕)W∙
Let S be a submanifold of codimension q in M, and let/: V→ M be a
differentiable mapping. / is said to be transversal to the submanifold S
at a point xe Lif either:
(i) f(x)*S, or
(ii) /(x)∈S and the image under df of the tangent space to V at x and
the tangent space to S at/(x) = y span the tangent space to M at y.
If/ is transversal to S at every point хе V, we say that / is transversal
to S. In this case one can prove that ∕~*(S) is a regular submanifold of V
of codimension q in V, or void.
Let L(V,M,s) denote the set of all s-times Continuouslydifferentiable
maps from Lto M. On L(L, M,s) we put the topology of compact conver-
gence of all partial derivatives of orders ≤ s.
Assume s > r ≥ O and let N be an (s—r) differentiable regular submani-
fold of Jr(V,M) where V and M are at least l2,s differentiable paracompact
manifolds of dimensions n and p respectively. Suppose that the codimension
of N in Jr(K,M) is q.
Theorem 3.1. (Tranversality Theorem of Thom). The set of maps in
L(V,M,s) whose r extensions are transversal to N on V is everywhere
dense if (s-f)>max(n-q,0'). In addition if V is compact, this dense
set is open.
For any pair of positive integers (n,p) a singularity manifold of order r
is a regular submanifold of Jr(n,p) which is invariant under the group