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