PIECEWISE LINEAR CONCORDANCES AND ISOTOPIES 5

subcomplex of (M,3M) i s denoted by l(M,N; f) (Gl(M,N; f ) ) . I t i s the complex

of paths in E(M, N; f) (GE(M,N; f) ) or, equivalently, the subcomplex of

E(l x M, I x N; 1 x f)(GE(l x M, I x N; 1 x f) ) such t h a t , for any s-simplex

F, the diagram

F: I x M x AS I X N x AS

P P

s s

1: I x A ^ I x A

is commutative.

Let C(M,N) and l(M,N) denote the complexes when an f is not speci-

fied. Let i: l(M,N) C— C(M,N) be the inclusion of l(M,N) as a subcomplex

of C(M,N). One purpose of this research is to discover conditions under

which l(M,N) is a strong deformation retract of C(M,N). In the simplicial

category there are several equivalent statements of this property, e.g., i

is a homotopy equivalence, i^: n (l(M,N)) - » n (c(M,N)) is an isomorphism

for all s, and n (C(M,N),l(M,N)) = 0 for all s, cf. [21, Theorem 2.20].

s

Indeed, it is the last of these three which provides the access to the prob-

lem.

J. F. P. Hudson, in Concordance, Isotopy and Diffeotopy [391? proved a

more general version of the following theorem, cf. Theorem 1.1 [39]-

1.5 THEOREM (Hudson). Let f: (K,dM) - (N,dN) be a proper piecewise

linear embedding and F a 0 simplex of C(M,N; f). If n-m 3 there is

a proper piecewise linear ambient isotopy H, of I x N, fixed on

({0} x N) U F(I

X

K) U (I X dN) such that H^ = (l,F | ({0} x N)).

Theorem 1.5 implies that i^.:n (l(M,N;f)) - » n (c(M,N:f)) is onto if

n-m 3 or, equivalently, that n (C(M,N;f), l(M,N;f)) = 0. Furthermore