3^honotopy equivalences

9

°t((xl'***'V* =

(x1/..-/ x. y t,x.+1,..., xk) if x. t

(xlf---# x^) if x. t

DEFINITION 3.3. Let W be a space and f : W - * M a hcmotopy.

We say that f is a (p,j,6)-hanotopy if d(pf ,(£* pfQ) 6 for

every t € I . Furthermore, if W C S C M , we say that S contains

a (p,j,6)-hcmotopy of W if there exists a (p,j ,6)-hcmotopy

f : W - S with f = Id .

Let K be a subocmplex of M , we say that mesh(K) p (3)

if for every simplex a e K , diamp ( | a |) 6 .

Let L c K be complexes. Let s denote the siraplicial

collapse s E K = KQ V ^ \ •..\ K, = L (the symbol s includes the

given ordering of the elementary simplicial collapses) . As in

[17] or [6], associated with the collapse s = K V » L we have a PL

strong deformation retraction s : K - * K .

DEFINITION 3.4. Let K be a subocmplex of M . We say that

K is a (P/jf 6)-collapse if there exists a suboamplex L of K and

a simplicial collapse s = K ^ L such that: a) mesh(K) p (6) ,

and b) s : K - K c M is a (p,j,6)-hcmotopy.

It is not very difficult to see that: Given e 0 there exists

6 0 such that if p is a surjective map, K is a (p,j,6)-col-

lapse and K1 is a subdivision of K , then there exists a subdivi-

sion K" of K1 which is a (p,j,e) collapse.