26 DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES 193

that u(z) = 1 for all z e cp(U) and w(z) = 0 for all z $ <p(V ). Show that for each e > 0
there exists a vector b e R« such that: (1) ||b|| < e; (2) the function

takes its values in i//(T) for x e V ; (3) the mapping g = g\» : M -> N defined by #0) =
«A~"1('A(/W) + w(<pW)b) for * s W, #U) = f(x) for x e M - V is transversal over Z at
the points of g~l(Z) n 0; (4) the mapping g is transversal over Z at the points of
ff~1(Z) n A. (To satisfy (3), use Sard's theorem (16.23,1) for the mapping

of <p(V) into Rq. To satisfy (4), it is sufficient that it should be satisfied at th e points
of g~l(z) n K, where K — A n (V — U). For this, consider the mapping

(x, b) h-» (gb(x), D(pr2 o i/r o #b o <p- l)(<p(x)))

of K x R9 into N x R/*n ; observe that it is continuous and transforms K x {0} into a
subset of the open set ((N - A) x R«") u (A x Mq) of N x R«n, in the notation of
Section 16.23, Problem 4.)

(b) Let M, N be differential manifolds of dimensions m, n respectively, let/: M ~>N
be a C°° -mapping, let Z be a submanifold of N of dimension p — q, and let A be a
closed subset of M such that /is transversal over Z at all points of An/~*(Z).
Finally let d be a distance defining the topology of N, and let 8 be a continuous func-
tion on M, everywhere >0. Show that there exists a C°°-mapping g : M ~>N which is
transversal over Z, coincides with /on A, and is such that d(f(x),g(x)) ^ S(x) for all
x e M (Thorn's transversality theorem). (There exists an open neighborhood S of A
such that /is transversal over Z at the points of /" !(Z) n S. Consider a denumerable
locally finite open covering (T*)fcao of N, where T0 = = N — Z and the Tk (k ;> 1) are
domains of definition of charts ifjk of N, such that ^(T*) = Hfc x lk , where Hfc is open
in Rp~« and lk is open in Ra, and ^(T^ n Z) — Hfc. Then take a locally finite open
covering (Wfc) of M, for which the Wfc are domains of definition of charts of M, and
which refines the covering formed by the intersections of the open setsf~l(Tk) with
M — A and S. Construct g by induction on k, using (a).)

26. DIFFERENTIABLE HOMOTOPIES AND ISOTOPIES

(16.26.1) The notion of homotopy, defined in (9.6) for paths, extends to
arbitrary continuous mappings. Given two continuous mappings /, g of a
topological space X into a topological space Y, a homotopy off into g is a
continuous mapping <p of X x [a, fi] (where a < fi in R) into Y such that
cp(x, a) =/(*) and <p(x, /?) = g(x) for all x e X. The mapping g is said to be
homotopic to/if there exists a homotopy of/into g. Just as in (9.6) it is im-
mediately shown that if g is homotopic to/, then/is homotopic to g; and that
if h is homotopic to g, and g is homotopic to/, then h is homotopic to/; in
other words, the relation "/is homotopic to g" is an equivalence relation on
the set of continuous mappings of X into Y.nd are contained in either T or M — N. Con-