The name is absent

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 ⅞f^r-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 J^r(∕)(x). If we let J^r(n,p) denote the
space of r-jets of 2f^r-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
J^r(y,M). If we choose local coordinates in V and M, then J^r(V,M) be-
comes a fibre bundle with fibre J^r(n_tpj and group L^r(n,p) (see [8]).

If/: V → M is of class at least ⅞f^r, the r extension of f is defined by:
J^r(∕); V→ J∖V,M), where x → J^r(∕)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 J^r(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 J^r(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 J^r(n,p) which is invariant under the group

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