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.
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