TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 170, August 1972
APPROXIMATINGEMBEDDINGSOF POLYHEDRA
IN CODIMENSIONTHREE(J)
BY
J. L. BRYANT
ABSTRACT.
Let F be a p-dimensional
polyhedron
and let Ç be a PL q-
manifold
without
boundary.
(Neither
is necessarily
compact.)
The purpose
of
this paper is to prove that, if q — p > 3, then any topological
embedding
of P
into Q can be pointwise
approximated
by PL embeddin-gs.
The proof of this
theorem uses the analogous
result for embeddings
of one PL manifold
into
another
obtained
by Cernavskii
and Miller.
Introduction.
embedding
embeddings
this
fact
izes
Cernavskii
of a PL zzz-manifold
PL
hedron
Recently
provided
to prove
into
that
and Miller
shown
M into a PL o-manifold
a topological
embedding
of Berkowitz
that any topological
Q can be approximated
q - m > 3- (See [5], [6], and [10].)
In this
of an arbitrary
Q (fl - p > 3) can be approximated
the results
have
by PL
[2] and Weber [13]-
paper
p-dimensional
embeddings.
Our theorem
This
by
we use
polygeneral-
can be stated
as
follows.
Theorem
polyhedron,
1. Suppose
that
0
that
P is a (not necessarily
is a PL q-manifold
f: P —* Q is a topological
P—» (0, 00) there exists
without
embedding.
compact)
boundary
Then for each
a PL embedding
g; P—»0
p-dimensional
(q — p > 3), and that
continuous
function
c
such that d(f(x),g(x))
< e(x)
for each x £ P.
Moreover,
choose
if R is a subpolyhedron
The
theorem
"moreover"
proved
space
shall
part
simplicial
f\R
is
PL,
always
When we say
assume
that
by the editors
that
November
and phrases.
PL
no further
then we may
to it.
we mean the underlying
space
as a closed
definition
subset
of an isotopy
reference
subset
of piecewise
R is a subpolyhedron
R is a closed
AMS1970 subject classifications.
Key words
embedded
an application
make
By a polyhedron
complex
We use the standard
manifold.
Received
1 is simply
[8], and we shall
and notations.
finite
(or manifold).
PL
of Theorem
by Connelly
Definitions
a locally
and
of P on which
g so that g\R=f\R.
of P.
of some
linear
(PL)
of a polyhedron
If /: P—» 0
of
euclidean
is a PL
map
P, we
map,
18, 1970.
Primary 57C05, 57C55, 57C35.
embedding,
PL
mapping,
PL approximation,
polyhedron,
PL manifold.
(1) Partially
supported by NSF Grants 11943 and 19964.
Copyright © 1972, American Mathematical Society
85
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86
J- L- BRYANT
then
S, = Cl \x\f~
euclidean
¿ x\
«-space
En
metric
spaces
Given
mappings
l f(x)
shall
is denoted
X and
denote
the singular
by Bn;
I = [0, 1].
Y, a continuous
f, g: X —►Y, we define
x £ X. (We ambiguously
use
[August
d(f,
set of /.
function
g) < e to mean
d to denote
the metric
The unit ball
in
e: X—► (0, <x>),and two
d(f(x),
g (x)) < e(x)
of any space
under
tot each
considera-
tion.)
If K and
K collapses
L are locally
finite
to L in the sense
of K to L determine
a proper
meaning
images
that
inverse
simplicial
of [12].
complexes,
In particular,
deformation
retraction
then
we require
that
of \K\ onto
|L|
of compact
sets
of K, then we use the notion
of the trail
of H, tr H, under
defined
7 of [16].
byZeemanin
Chapter
are compact).
The important
K N L means
that
the collapse
("proper"
If H is a subcomplex
the collapse
properties
K>» L as
of tr H ate
(i) dim tr H < 1 + dim H,
(ii) dim L n tr H < dim H, and
(iii) K\LutrHNL
as proved in Lemmas 44 and 45 of [16J.
If e: \K\ —►(0, <») is continuous,
e-collapse,
onto
written
|L|.
K \
Observe
subdivision
that
where
and the dimension
K and
/Or'
4 of [16]).
X \
complexes
This
to L^T' can be arranged
an
of |K|
derived
so that
of the simplexes
that
to a polyhedron
L such
that
X ¿Si Y and
so that
K €"S L.
a subcomplex
H such
it is
of K — L
|K| = X,
Vu
however,
trZ
can be seen
\
that
Then
that
|L| = Y, and
there
V is an wze-collapse
remarks
each
K collapses
derived
Chapter
sub-
1, Lemma
upon the triangulaof the collapses
(for some
together
are locally
of X. Let
is an rth
\H\ = Z (see
tr Z depends
in any event
from our above
Y if there
Z is a subpolyhedron
tr Z = jtr H\ C X. Of course,
It follows,
Y u tr Z and
above).
of K
retraction
and if K^r> is an rth
only on the diameter
K and
as above
containing
Define
tion involved.
to an ¿-deformation
(e-collapses)
to L. Suppose
L be chosen
division
the collapse
m depends
X collapses
simplicial
(e-collapses)
rise
of K to L is called
of K - L.
A polyhedron
finite
L, if it gives
the collapse
if K n* L is an f-collapse
of K, then
an we-collapse,
then
m as described
with the proof
at
Lemma 45 of [l6].
Preliminary
1 is stated
Theorem
a respectively
Then
lemmas.
as follows.
for each
2. Suppose
with
The approximation
that
M and
Q are
q — m > 3, and that
continuous
theorem
we require
to prove
Theorem
of dimensions
m and
(See [5], [6] and [l0].)
PL
manifolds
f: M —* Q is a topological
e: M—* (0, <x>)there
exists
such that d(f, g) < e.
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a PL
embedding.
embedding
g: M—* Q
1972]
APPROXIMATING EMBEDDINGS OF POLYHEDRA
I am deeply
helping
sion
indebted
me simplify
of this
Edwards,
Richard
many of the proofs
Miller,
and the referee
for
appearing
in the original
ver-
paper.
Lemma
/|lntB
to Robert
and correct
87
1. Suppose
is
PL.
Then
that
there
f; B
exists
—»Int Q (q — k > 3) is an embedding
an extension
of f to an embedding
such
that
F:
Bq—> Int 0 such that F\Bq - Bd B k in PL.
Proof.
This
The next
is essentially
lemma
a sequence
of applications
is a straightforward
application
of Theorem
of the Tietze
9 of [15].
extension
theorem.
Lemma
borhood
there
2. Suppose
of /(Int
exist
Bk)
that
f; B
in Q and let
a neighborhood
—►Int 0 is an embedding.
e. [Q - /(BdBfe)]
V of /(Int
Let
H be a neigh-
—»(0, oo) be continuous.
B ) in Q and a continuous
Then
8: Int B
—►
(0, oo) such that, if g: Bk —»V u f(Bk)
is an embedding within 8 of f on Int Bk
that agrees
exists
with
f on BdB
, then
there
U U f(Bk) such that h
= identity,
retraction
onto g(Bk).
of V U f(Bk)
Lemma
there
exist
3. Suppose
that
a neighborhood
(0, oo) and -q: [Q -/(Bd
f: B —►IntO,
V of /(int
Bk)]-*(0,
that
r¡ collapses
h : V U f(B
) —►
U and
e are as in Lemma
B ) in Q and continuous
2. Then
8: IntB—►
oo) such that, if g: Bfe-> V Uf(Bk) is a PL
embedding within 8 of f on IntB
a polyhedron
an e-homotopy
h \g (Bk) = identity for all t £ I, and h l is a
that agrees with f on BdB
to g (Int B ), 7ie?2 there
exists
and if X C V is
an e-homotopy
h :
V U /(Bd Bk)—» U U /(Bd Bk) such that hQ = identity, b \X = identity for all
t £ I, and h
Sketch
spond
is a retraction
of proof.
to e/2
and
Assume
of V U f(Bk)
initially
U as in Lemma
where g: B —»V is PL on IntB
small,
second-derived
(e/2)-homotopy
h'
neighborhood
(t e 7) given
onto X U g(Bk).
that
8 and
V ate chosen
so as to corre-
2. Suppose
X C V 77-collapses
to g (Int B ),
and within
8 of / on Int B . Let TVbe a
of X in V. Let
by Lemma
2. Think
h' be the 1-level
ping cylinder of a map Bd TV—♦X given by the collapse
I—*Nbe
such
that
a(p,
of an
of N as the topological
map-
TVN» X, and let a: Bd TVx
0) = p, cx|Bd TVx [O, 1) is a homeomorphism
onto
TV- X,
a(Bd TVx Si!) = X, a(p x /) is small for p £ Bd TV(say, less than r¡(p)). Let
ß : X—»X (7 £ I) be a strong
ßQ = 1 and
ß,:
rj-deformation
X—►g (Int B ) a retraction.
retraction
For
of X onto
¿3 and
g (Int B ), with
77 sufficiently
small,
maps ¿'|Bd TV:Bd /V-»g(Int Bk) and ß xa( - , 1): Bd TV-* g (int Bk) will be
(c/2)-homotopic
ß^(p,
via a homotopy
y : Bd TV—»g (Int B ) (t £ I) with yQ(p) =
1) and yAp) = h'(p).
Define h" : TV-» X by
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the
88
J.L.BRYANT
[August
Íy1_2t(a(p,0)), 0<t<y2
(p £ Bd N),
ß2_2t(a(p,
1)),
lA<t<l
and h; V —>X by
p £ V - N,
h(p)
p £ N.
Then
h is a retraction
of V onto
h will be the 1-level
via the identity,
X, and if 8 and
of the appropriate
homotopy
r¡ ate sufficiently
h
(t £ I) (extended
be a neighborhood
Then
functions
then
to /(Bd
B )
of course).
Lemma 4. Suppose that f: B —»Int 0 is an embedding
tinuous.
small,
(q - k > 4). Let U
of /(Int Bk) in Q and let e: [Q - /(Bd Bk)]—> (0, <*>)be con-
there
exist
a neighborhood
V of /(int
B ) in Q and continuous
8: Int Bk —»(0, <*>)and r¡: [0 - /(Bd Bk)] —»(0, °°) satisfying
the follow-
ing conditions:
If g: B —►
V Cl f(B ) is an embedding that is PL and within 8 of f on
Int B
and agrees
with
f on Bd B , if Y is a polyhedron
closed
PL subset of Q —/(Bd S )) with q - dim Y > 4, if W is a neighborhood
of g (Int S ) in Q, and if X is a polyhedron
XN
XÍ1 g (int B ), then there
exists
in V (embedded
as a
in W such that a - dim X > 3 and
an isotopy
h
(t £ I) of 0 such
that
(i) h = identity,
(ii) h = identity
on g(B ) U X and outside
of U,
(iii) hx(W) 3 Y,
(iv) d(h
identity) < e on Q - /(Bd Bk) for each t £ I.
This
lemma
with standard
and [14]
is proved
engulfing
by using
arguments.
for the engulfing
techniques
Lemma 5. Suppose that f:B
be a neighborhood
uous.
Then
there
homotopies
obtained
The reader
is referred
required
from Lemma
2 together
particularly
to [l],
[ll],
here.
—►
Int 0 is an embedding
(q —k > 4). Let U
of /(Int Bk) in 0 and let e: [0 - /(Bd Bk)} —»(0, °°) be continexist
a neighborhood
V of /(int
B ) and continuous
i}- [Q - /(Bd Bk)] —»(0, °°) and 8: Int Bk —»(0, °°) satisfying
functions
the following con-
ditions :
If g: B —»V is a PL embedding within 8 of f that is PL on Int B
agrees
with
f on Bd B , if X is a polyhedron
in V (q — dim X > 3) that
to g (int B ) via an -q-collapse,
and if Y is a polyhedron
then
Z in U such
there
exists
a polyhedron
that
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and
collapses
in V (q — dim Y > 4),
1972]
APPROXIMATING EMBEDDINGS OF POLYHEDRA
89
(i) Z D X U Y,
(ii) Z N g (Int Bk), and
(iii) dim(Z -X) < dim Y + 1.
Proof. Suppose that /: Bk-* Int Q, U D /(Int Bk), and <r:[0 - /(Bd Bfe)]~»
(0, oo) are given
sponding
to U and
tion of /.
Let
Let
rth
e/3
72= e/3-
W be a small,
derived
we can
as in the hypothesis
as in Lemma
3, and let
Suppose
X and
that
second-derived
subdivision
obtain
of the lemma.
g: B
Y satisfy
V and
the conditions
of X in V such
to X.
the conclusion
h (t £ I) that is fixed
From
¿5 corre-
—» Int g be a ¿¡-approxima-
neighborhood
of W 77-collapses
an isotopy
Choose
in the lemma.
that every
of Lemma
on g (B ) U X and outside
3
U
such that h AW) D Y and d(h{, identity) < e/3 on Q - /(Bd Bk) for each t £ I.
Let Yj = h~l(Y), and let Zj = tr(YjU X) under the collapse W >. X >.
g (Int Bk). Then Z, = X u tr Y1 and Z jN g (Int Bfe) is a 2r/-collapse.
Let Z =
AjiZj) = X UÄjdx Yj) (since
dim(Z- X) <dimY + 1.
Main lemmas.
Suppose
h ^X = identity).
that
Then Z ^
/ is a complex
and that
g (int B^) and
ff is a simplex
of /.
Let K = St(ff; /), L = Bd a * LK(er; ]), K = \K\ - \L\, and ff = Inter. Denote by
ff the barycenter
of cr. There
is a triangulation
K.
of |K| - Bdcr such
that
K.>» H. , where 77, C K. and \H. | = <j. Thus we have tr Z defined for a subpolyhedron
Z of |K. |. Set
Given
/, ff, K, and
x e |/C|, let
consider
and let
Let
L as above,
[zj, x] denote
Then
PL
joining
there
that
exists
with
cone
j = dim a),
structure.
notation
of Lemma
f(d(z)))
be left for the reader.
e: a—* (0, 00) is con-
satisfying
by applying
The next
that
x, y £ a and
< e(z) for each z £ [3, x], where 6:
homeomorphism
6 is obtained
and
8: a —* (0, 00) such
8(a)
= a and
the continuity
two lemmas
0(x) = y.
of / and
are successive
/""
generaliza-
tions of Lemma 6.
Lemma
7. Given
f: a~*Q
and
e:o—*(0,
<x) as in Lemma
8, T]-.o—*(Q, 00) such that if gQ, g ■a —»O are mappings
tj(x)
for each
d(g0(z),
If
let us
g (a) = 0. We use this
f: a —*Q is an embedding
a continuous
[ff, *1 —* [ff, y] is the linear
and shall
a * \L\.
Similarly,
0 * Bd B' (where
y £ Bq to 0 in this
homeomorphism
d(f(x), f(y)) < 8(x) imply d(f(z),
The proof
\K\ as the cone
a and x in |K|.
lemmas.
Lemma 6. Suppose
tinuous.
joining
0 * Bd Bq, with subcone
the segment
g: a —»B7 be a fixed
in the following
q - k > 3 throughtout.
let us consider
the segment
Bq as the cone
[0, y] denote
k = dim K and assume
x £ a, then
gA^(z)))
<e(z)
x, y £ a and
d(g
6, there
with d(g.(x),
(x), g Ay)) < 8(x)
for each z £ [5, x].
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imply
exist
f(x)) <
90
J. L. BRYANT
Lemma
subcone,
uous.
8. Suppose
that
Then
following
that
A and
B are cones
f: o —*Q is an embedding,
there
exist
[August
8: [A - Bdo|—»
over
and that
a each
containing
a as a
e: [A — Bda] —* (0,°o)
(0, °o) and
is contin-
-q: a —>(0, oo) satisfying
the
properties:
If G0: A —»0 and G.; B —»O are mappings with d(G .\a, f\a) < r] (i = 0, 1),
then
there
exist
neighborhoods
U of a in A and
y £ V, and d(G Q(x), G {(y)) < 8(x)
V of o in B such
imply d(GQ(z),
G x($(z))) <((z)
that
x £ U,
for each z £
[d,x\.
°
°
o
Lemma 9. Suppose f: K~* Q is a closed embedding with f\K - a PL. Then
o
for each
o
o
e: K —» (0, oo) and each
neighborhood
o
N of a in K, there
exists
a PL
map G; K-* Q such that
(9.1) G\K-N = f\°K-N,
(9.2) G|o" is a PL embedding,
(9.3) d(f, G)<e,
(9.4) S- CAÍ, and
(9.5)
G is in general
Proof.
embedding
This
position.
is a straightforward
application
f\a: a —*Q) and general
of Theorem
o
e: 0 —* (0, °o) there
~~
exist
to the
position.
°
Lemma 10. Suppose that j: K—►0 is a closed
Given
2 (as applied
o
a neighborhood
o
embedding with f\K —a PL.
°
N of a in K and continuous
o
functions
n: 0 —►(0, °°) and
8: K —*(0, oo) satisfying
If G: K-* 0 is a PL map satisfying
placing
\e, N\ and if X.
(10.1) SGu oCX1
and
= trXj
Z
the following
(9.1)-(9.5)
are polyhedra
such
conditions:
of Lemma 9 with \8, N\ rethat
C N,
(10.2) Yj = G(Xj) CZj -r/N, G (a), and
(10.3) dim[(Zj- Yj) nG(K)]<k-j
(j > 4),
then
there
exist
polyhedra
X? and
Z?
such
that
(10.4) Xj CX2 = trX2 C °K,
(10.5) Y2 = G(X2) C Z2 ^. G la), and
(10.6) dim[(Z2 - Y2) n G(°K)] < k - j - 1.
Proof.
Lemma
Given
5. Choose
e: 0 —+ (0, oo), let
~
N to be a regular
U = N (f(a)),
neighborhood
and f(N) C V. Assume that G: K —>Q satisfies
replacing
and choose
c
°
8, ri, and
of a in K such
that
V as in
N = tt N
(9.1)—(9.5) of Lemma 9 with 5
the e of Lemma 9 and that G(N) C V.
Suppose that XjCN and Z^CQ satisfy (10.1)—(10.3)- Let X2 =
tt(G-l(Zx)),
and let Y2 = G(X2). Then dim(X2 - X j) = dim(Y2 - YJ <k-j+
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1
1972]
APPROXIMATING EMBEDDINGS OF POLYHEDRA
91
( <q - 6). Observe that if ¿5 and 27 are sufficiently
small, then G~ (Z ) C N, and
so Y2 C V. Now apply Lemma 5 with Z
X and Cl (Y - Y j) replacing
Y. This
gives
us a polyhedron
(i) Zj uci(y2
(ii)
Z2 f\.
G(ff), and
Z
we have (since
satisfying
- y,)cz2,
(iii) dim(Z2-Zj)
Putting
Z.CO
replacing
<dim(Y2-
in general
Yj) + 1 < *■- 7 + 2.
position
with respect
to G (K) (keeping
Z
\j Y - fixed),
Zj H G(°K) C Y2)
dim[(Z2 - Y2) n G(K)] <dim[(Z2
- Zj) O G(K)]
< (k - j + 2) + k - q < k'- ; —1,
o
.
,°
o
Lemma 11. Suppose that f: K —»Q is a closed embedding with f\K —a PL.
Given
o
e: O—*(0, 00) and a neighborhood
°
TV of a in K, there exists
o
°
8: K—►(0, 00)
such that, if G: K—>Q is a PL map satisfying (9.1)—(9-5) of Lemma 9 with
ÍS, N\ replacing \e, TV},then there exist polyhedra X C TVand Z CO satisfying
(11.1) ff USGCX
= trX CN,
(11.2) G(X) = ZD G(K), and
(11.3) Z N, G(a).
Proof. Apply Lemma 10 inductively.
The next
sult
lemma
is the key lemma
of J. Cobb announced
to consider
of this
The specific
paper.
Its proof
form of Cobb's
depends
theorem
upon
a re-
we wish
is the following.
Theorem
sional
in [7].
3 (Cobb [7]).
complex
K and that
Suppose
that
L is a subcomplex
f is an embedding
of \K\
of a finite
k-dimen-
into a PL q-manifold
Q (q - k > 3) such that f\ \L\ and f\ \K\ - \L\ are PL. Then for each e > 0 there
exists
an isotopy
h; 0 x I—►Q x / such
that
(i) h. = identity,
(ii) hj: \K\->Q is PL,
(iii) hj\ \L\ =f\ \L\,and
(iv) 73|(Ox/)-(/(|L|)x!0!)
is PL.
Lemma 12. Suppose that f: \K\ —»0 is an embedding with f\K - a PL. Then
for each
O
e: K —» (O, 00) and each
~
neighborhood
o
o
TV of a in K, there
exists
an em-
bedding /': |K|—♦ Q such that
(i)/'|jL|
U(|K| - TV)=/| \L\ u(|K|-/V),
(ii) f'\K is PL, and
(iii) d(f(x), f'(x)) <e(x) for x £ °K.
Proof.
Given
£: K—» (0, 00), choose
8: K—* (0, 00) and
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77: a —»(0, 00) as in
92
h L. BRYANT
[August
o
Lemma
8 corresponding
from Lemma
to e = £. Let
11 with
d(f,
G: K—» (Q - f(L))
G) < r¡ on a and let
X and
be a PL
map obtained
Z be the associated
o
poly-
°
hedra having the property that a \J Sr C X = trX C N, G (X) = Z Cl G(K), and
o
Z\t,G(o).
°
Triangulate
cial, and let
U
and
Z in 0 - f(\L\),
Assume
V
be small,
respectively,
that
so that G: K—>Q - f(\L\)
second-derived
neighborhoods
is simpli-
of X in K and
such that G(U') = V' D G(°K).
5 is chosen
we shall still call
°
K and Q-/(|L|)
so that
G) that agrees
G extends
with /on
Let H: Bq—►0 be an embedding
to a map of |K|
into
Q (which
|L|.
such that
H\(Bq-BàBj)
is PL (/ = dimo-),
H(0) = G (a), and
Hg = G.
Let
o
U be the closure
K lying
(in
in the closure
second-derived
|K|) of a small,
of N in |K|,
neighborhood
fy the conclusion
assumption
of Int B1 in Bq — Bd B',
of Lemma
that
second-derived
8 with
U is starlike
neighborhood
and let V be the closure
Ç, replacing
in K from a,
in B
both chosen
e. We shall
V is starlike
make
of a in
of a small,
so as to satisthe further
in Bq from
0, and
H(V) O G(|K|) CG(A/u a).
Since
X = tr X N a, there
Fj|K
is PL,
exists
a small
homeomorphism
F.:
\K\ —►\K\ such
that
F j| |L| U (|K| - N) u a = identity,
Fj(l/)n
It r¡ is sufficiently
that d(G(x),
and
K= U'.
small
GFj(x))
and if X is suitably
< S(x)/2
Similarly,
there
F2|/(|L|)
U G(a) = identity,
F2\Q-f(\L\)
exists
chosen,
then we can guarantee
for x £ K.
a small
homeomorphism
F2: 0 —►0
such
that
is PL,
F2H(V)-f(\L\)
= V', and
d(F2G(x), G(x)) < C(x)/3 for x £°K.
Thus, GFj(Bd 17)= G(|K|) OF2/i(Bd V) is homeomorphic to |L|. Let L' =
H'^-HGilKDn
F2H(Bd V)) C Bd V, and let K' = L' * 0 C V C Bq. Observe
that U = a * Bd U and GF j(Bd U) = F HlL').
Define F3: (7-» V by Fj(x) = H~^F~ lGF%1x) if x e Bd (7 and F |[5, x] =
Ö: [â, x]—» [0, F Ax)]
is the linear
homeomorphism
described
fine G': |K|-> Q by
G'(x)
= F2HFA,x)
G'(x) = GFj(x)
it x £ U,
if*
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$17.
previously.
Now de-
93
APPROXIMATING EMBEDDINGS OF POLYHEDRA
1972]
Then,
assuming
Let
r¡ < £,, G
is an embedding
that
H ■ V—* Bq be a homeomorphism
closely
such
that
approximates
/.
H Aß1 = identity
and
Hr\Bq - Bd By is PL. Then G ^ U -» Bq, defined by G , = H1H-1F~1G',
proper embedding
of the cone â * Bd 77 into Bq = 0 * Bd Bq such that
is a
G.|c7 is
PL and G X\U- a is PL.
Consider GjBd U: Bd U -» Bd Bq . GjBd Bj is PL and GjBd U - Bd Bj
is
PL.
Hence,
suchthat
GjBd
by Theorem
zeQ= identity,
3, there
¿jGjÍBd
a small
isotopy
Í7) is a subpolyhedron
k
(t £ I) of Bd Bq
of BdBq,
&1G,|Bda
=
a, and ¿|(Bd Bq x /) - (Bd B> x \0\) is PL. Copy this isotopy in a small
PL collar
W of Bd Bq in B«. Let U , be (7 minus a small collar of Bd 77 in (7
and obtain a PL embedding
Extend
G ■.U . —* Bq - W by coning over k jG , (Bd U) C Bd W.
G2:U1^Bq-W
to G : (7 -» Bq by sending
&(Bd 77 x /) C W in a natural
G j| 77- Bd ff is PL,
Gj.
exists
Thus if G
way.
G,|Bd
Then
G,:
(7 - U r to
U —* Bq is a proper
77 = G ,|Bd 77, and G, is a closed
is sufficiently
embedding,
approximation
close to G,, then the embedding
to
/ : \K\—> 0 de-
fined by
/'(*)=
F2HH~lGAx)
/'U)
is
PL
/
of Theorem
be a triangulation
of /
to obtain
inductively
of P.
/J
using
Appendix.
dimension
as yet,
we shall
1 in dimension
in [7l.) We shall
wards
that
a proof
outline
q.
require
idea
Isotopy
Theorem
bedding
such
that
of Cobb's
|L|u
(|K| - TV).
embedding.
p-dimensional
approximates
Let
simplexes
/ and satisfies
— |/p-1|)
the
is PL. Now proceed
12 we applied
to get Lemma
theorem
Theorem
12 in dimension
[7] (Theorem
3 in dimension
to be essentially
theorem
3 in
q. In
3) has not appeared
q, assuming
the same
and an engulfing
Theorem
as that
theorem
indicated
due to Ed-
[3], respectively.
[9]).
K and that
f\ \L\
of Lemma
of Theorem
appears
(Edwards
complex
2 to the open
that
polyhedron
an isotopy
[9] and Bryant-Seebeck
k-dimensional
P —►0
in the proof
a proof
(The
/on
12.
q — 1 to a compact
of the fact that
Theorem
/.:
with
/: P—► Q is a topological
= f\ \JP~ l\ and /1| (|/|
Lemma
Observe
that
Apply
\Jp~l\
if x ^ 7/
of / on K, and agrees
1. Suppose
an embedding
two properties:
view
= G'(x)
on K, is an approximation
Proof
if x £ 77,
is
PL.
Suppose
that
L is a subcomplex
of a finite
f: \K\ —►0 (q — k > 3) is a topological
Then,
given
e > 0, there
exists
8 > 0 such
emthat
if g .: \K\ —*0 (i' = 1, 2) is a PL embedding such that g \ \L\ = f\ \L\ and
dig-,
f) < 8, then
is fixed outside
g.
and
TV(/(|K|
g2 are
PL e-ambient
isotopic
in 0
- |L|)).
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via an isotopy
that
94
J. L. BRYANT
Engulfing
dimensional
locally
Theorem
(Bryant-Seebeck
[August
[3]).
Suppose
that
X is a compact
k-
ANR in Q (a — k > 3, q > 5) such that Q - X is 1-ULC (uniformly
simply
connected)
and
A C X is closed
in X.
Then,
given
e > 0, there
ex-
ists 8 > 0 such that if g: X—* 0 is an embedding with d(g, 1) < 8 and g\A = 1
and if U is an open set
in Q containing
g(X),
then there
exists
a PL
e-isotopy
h (t £ I) of 0 that is fixed outside N (X - A) such that hQ = 1 and h A\V)3 X.
Moreover,
chosen
if X is a polyhedron
so as to fix
(This
to the author
to prove
is slightly
stronger
is contained
PL
then
ht (t £ I) can be
that the above
therein.
refinement
Indeed,
of Theorem
of Theorem
it has
2.1 of
been pointed
2.1 of [3] is actually
of [3]. The "moreover"
part
out
needed
was obtained
by
in [5]-)
Proof
Suppose
is
than the statement
implicitly
some of the theorems
Cernavskir
g: X—* 0
g (X).
statement
[3I, but its proof
and
of Theorem
that
3 in dimension
L is a subcomplex
a, assuming
of a finite
Theorem
^-complex
1 in dimension
K and that
a.
/: \K\ —» 0
(a - k > 3) is an embedding such that f\ \L\ is PL and f\ \K\ - \L\ is PL.
(We shall
assume
in dimension
that
q > 5, since
a = 4 [4].)
Let
both Theorem
N , N-, • ••
of \L\ in \K\ such that A'.,, C Int/V. and
\L\ UC1(|K| -N),
Now proceed
mation
/':|K|—»
Engulfing
i= 1, 2, ...
just
Theorem
agrees
to construct
0°°-,
of PL
3 are well
known
neighborhoods
N .= \L\, and let P.=
.
as in the proof
0 of f that
1 and Theorem
be a sequence
of Theorem
with
2 of [3].
Start
with a PL
/ on P . Use the Isotopy
sequences
of small
|G;.!°°=j of (0, f(\K\)) satisfying
(1) G = lim .
G . ' • • G , and G = lim .
PL
pushes
Theorem
{G .P°
approxiand
and
G. * • • G, are small pushes
of 0,
(2) Gf = G'f ,
(3) g....Gj/|p.
(4) the
PL
on G . • • • G j/(|X|
phism
G'G~
= g; ...g;/'|p.,
isotopy
of G . + , (respectively,
- N .) (respectively,
is isotopic
to the identity
G^+j)
to the identity
G! • • • G'.f'(\K\
is fixed
—N .)). The homeomor-
via the appropriate
isotopy
of Q.
REFERENCES
1. R. H. Bing,
State Univ.,
pp. 1-18.
Radial
E. Lansing,
engulfing,
Conference
Mich., 1967), Prindle,
on the Topology
Weber and Schmidt,
of Manifolds
Boston,
(Michigan
Mass.,
1968,
MR 38 #6560.
2. H. W. Berkowitz,
Piecewise
linear
approximations
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
of homeomorphisms
(preprint).
1972]
APPROXIMATING EMBEDDINGS OF POLYHEDRA
3. J. L. Bryant and C. L. Seebeck
III, Locally
J. Math. Oxford Ser. (2) 19 (1968), 257-274.
4.
J. C. Cantrell,
574-578.
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in euclidean
k-space,
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Amer.
Math. Soc.
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MR 29 #1627.
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Topological
embeddings
of manifolds,
Dokl. Akad. Nauk SSSR
187 (1969), 1247-1250= Soviet Math. Dokl. 10 (1969), 1037-1040.
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Piecewise
linear
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MR 41 #4547.
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371. Abstract #68T-241.
two,
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Unknotting
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R. J. Miller,
11.
C. L. Seebeck
The equivalence
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Math. Soc. 148 (1970), 63-68.
12. A. Scott, Infinite
of close
codimension
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in codimension
Ann Arbor, Mich., 1969-
piecewise-linear
three
embeddings
embeddings,
an (n — l)-manifold
greater
Ann.
in an n-manifold,
than
(to appear).
of Math,
Trans.
(to appear).
Amer.
MR 41 #2692.
regular neighbourhoods,
J. London Math. Soc. 42 (1967), 245—
253- MR 35 #3672.
13- C. Weber,
Plongements
Helv. 42 (1967), 1-27.
de polyèdres
dans
le domaine
métastable,
Comment
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MR 38 #6606.
14. P. Wright, Radial engulfing in codimension
three, Duke Math. J. 38 (1971), 295-
298.
15- E. C. Zeeman,
Unknotting combinatorial
balls,
Ann. of Math. (2) 78 (1963), 501-
526. MR 28 #3432.
16. -,
(mimeograph
Seminar
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Hautes
Etudes
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1963
notes).
DEPARTMENT OF MATHEMATICS, FLORIDA STATE UNIVERSITY, TALLAHASSEE, FLORIDA
32306
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