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