PIECEWISE LINEAR CONCORDANCES AND ISOTOPIES
3
SECTION 1. DEFINITIONS AND STATEMENTS OF RESULTS
All of the definitions and results concern either the simplicial category
or the piecevd.se linear category. For definitions and properties of the
simplicial category see Curtis [21] or, for a different point of view, Rourke-
Sanderson [66]. All simplicial complexes will satisfy the Kan extension con-
dition. In the main, the basic terminologies, definitions, and notations for
the piecewise linear category follow those employed by Zeeman [73] ancl- Hudson
[38,39l» Many of these concepts are extended to the fibered situation in this
paper. Such extensions will be signaled by the word "fibered" preceding the
term. The abuse of the term simplicial complex given for both geometric com-
plexes and the semisimplicial complexes should cause little difficulty as the
meaning is always clear from the context.
Let M and N denote compact piecewise linear manifolds of dimensions
m and n and boundaries dM and 3N, respectively. Let A denote the
standard s dimensional simplex, R = {(x ,.. .,x ) |x. £ R},
R+n=
[ x a
n
| x
n
0},
R_n=
{x e
Rn
I xn 0}, I = [0,1], rJ = [-r,r],
rn i-r,n -r^n
^n-1 __n „n-l „n-l ^ _ n „n-l
rD = rJ x X rJ, ID = D , £ = 3D , £
+
= £ fl R+ , £_
£ f]B , and let B denote any piecewise linear homeomorph of D . A
subspace (K,K ) of a space (L,L ) is said to be proper if K\K C L\L .
A piecewise linear map f: (K,K ) - (L,L ) is proper if (f(K), f(K )) is a
proper subcomplex of (L,L ).
1.1 DEFINITION. Given a proper piecewise linear embedding of a proper
subcomplex of (M,M) in (N,dN), f: (K,K ) (N,BN), let E(M,N; f) de-
note the simplicial complex of locally unknotted proper piecewise linear
embeddings of (M,3M) into (N,3N) extending f. That is the simplicial
complex whose s simplices are proper piecewise linear embeddings
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