KENNETH C. MILLETT

F: (M,dM) x AS (N,3N) x AS

P P

1: A ^ A

such that

(i) the diagram is commutative, i.e., F is a fibered embedding,

(ii) F" (dN x A ) = BM x A , i.e., F is proper,

(iii) F|(K,K ) x AS = f x 1, i.e., F extends f,

(iv) for any simplex A linearly embedded in A , (N x A, F(M X A))

is a locally unknotted manifold pair, i.e., if B is a regular neighbor-

hood of x€F(MxA) in N x A then there is a proper piecewise linear

homeomorphism from (Bn+S, Bn+S n F(M X A)) to (Dn+S, Dm+S).

If f is not specified the complex is denoted by E(M,N).

1.2 DEFINITION. The complex of germs of proper embeddings of (M,BM)

into (N,BN) extending f is denoted by GE(M,N;f). It consists of the

simplices of the complexes of proper embeddings of neighborhoods (U,U ) of

(K,K ) into (N,3N) which extend f | K .

1.3 DEFINITION. The complex of proper concordances of (germs of)proper

embeddings extending a proper embedding f: (K,K ) - » (N,d) of a proper sub-

complex of (M,dM) is denoted by C(M,N; f)(GC(M,N;f)). It is the subcomplex

of E(l x M, I x N; 1 x f)(GE(l x M, I x N; 1 x f)) whose s-simplices F

satisfy

(i) F"1({0} x N) = {0} x M and

(ii) F"1({1} x N) = {1} x M.

l.k DEFINITION. The complex of (germs of) proper isotopies of proper

embeddings extending a proper embedding f: (K,K)-*(N,BN) of a proper