PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume
72, Number
1, October
1978
TRIANGULATIONOF STRATIFIED OBJECTS1
R. MARK GORESKY
Abstract.
outlined.
A procedure for triangulating an abstract prestratified set is
1. Introduction. The purpose of this article is to give a short accessible
outline of how to construct, for any abstract prestratified set W (see Mather,
[7]), a simplicial complex K cRN and a homeomorphism/: \K\ -» W which
is locally a smooth triangulation of each stratum. In particular, if X is a
subanalytic set, it admits the structure of a Whitney stratified object, and
therefore of an abstract prestratified set. Therefore X can be triangulated
(Hardt, [2]).
Triangulation theorems for stratified objects have been obtained independently by Hendricks [3] (unpublished), Johnson [5] (unpublished), and Kato
[6] (in Japanese). I am grateful to R. McPherson for helping to clarify the
ideas in the proof. I also wish to thank the Editor, R. Schultz, and an
anonymous referee for several valuable suggestions and for pointing out an
error in the original version of this paper.
2. Families of lines. Let W be an abstract (or Thorn-Mather) stratified
object (Thom [10], Mather [7]). For each stratum X, let Tx denote the
"tubular neighborhood" of A1in W,mx: Tx -* X the retraction map, and px:
Tx -» [0, oo) the "tubular distance function." Let
Sx(e)={pETx\px(p)
= e),
Tx(e)={pETx\Px(p)<e},
T,(£)= U { Tx (e) | dim(*) < i},
Wip) = \j{X\àim(X)
Xe° = X - Ti_x(e)
< p),
where dim(A) = i,
Sf(X) = XE°n (U {SY(e) | dim(T) < />}).
Adjusting px by a positive scale factor /: X -» R by setting p'x(x) =
f(irx(x))px(x) we may assume Sx (e) is a Thom-Mather stratified object (with
tubular projections ffrn5t(e) = 7Ty|T n Sx(e) etc.). Note that mx extends
continuously to the closure Sx (e)-> X.
Define a family of lines on IF to be a number 8 > 0 together with a system
Received by the editors December 6, 1976 and, in revised form, October 11, 1977.
AMS (MOS) subjectclassifications(1970).Primary57C99,57D05.
1 Research supported by NSF MCS 7606323.
© American
193
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Mathematical
Society
1978
R. M. GORESKY
194
of radial projections for each stratum X,
rx(e):Tx-X^Sx(e)
defined whenever 0 < e < S, which satisfy the following commutation
relations: If X < Y (i.e., iflcf)
then
(1) rx(s) o rY(e') = rY(e') ° rx(e) G Sx(e) n SY(e') for all e',
(2) Px ° rr(£) = Px>
(3) Py ° rx(£) = Pr>
(4) mx o rY(e) = irx,
(5) if 0 < e < e' < o then rx(e') ° />(e) = />(£'),
(6) 7TX° ^(e)
= irx,
0) rx(E)\Tx(e) n y: F^e) n Y-> Sx(e) n Y is smooth.
For each stratum X we define a stratum preserving homeomorphism
hx: Tx - X^>SX (e) X (0, oo)
by
M/0 = ('"x(£)(f)>Px(f))/¡^ identifies ^(e)
Proposition
lines.
with the mapping cylinder of Sx(e) -^ X.
1. Every Thorn-Mather stratified object W admits a family of
Proof. rx(e) is constructed by increasing induction on the dimension of the
stratum X, with the case dim(A') = -1 trivial. Suppose rY(e) has been
defined for each stratum Y of dimension < / — 1 and suppose X is a stratum
of dimension /'. Mather [5] constructs for e > 0 sufficiently small, a projection
Tx — X —>Sx(e) which is a candidate for rx(e) but which must be altered so
as to satisfy condition (1) with respect to all strata Y < X. Suppose inductively that rx(e) has been defined and satisfies condition (1) with respect to
each stratum Z where q < dim(Z) < / — 1 and let y be a stratum of
dimension q. Fix e' > 0. Redefine rx in the region TY(e') n Tx by setting
r'x(e)(p)
= hYx |ry(
|
)rx(e)hYx ^
Z- )(/>), *(py (/>))), Py(p)
where
hY\ TY - Y^SY(e'/2)
X (0, oo),
hY(p) = (rY(e'/2)(p),pY(p))
and where d>:R -» R is a smooth nondecreasing function with
<p(x)= e'/2
if x < e'/2
d>(x) = x
if x > | e'.
and
It is now clear that r'x(e) satisfies the necessary commutation relations with
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195
TRIANGULATION OF STRATIFIED OBJECTS
irY, pY, and rY in the region Tx(e) n TY(e'/2) and that it continues to satisfy
conditions (2)-(5) with respect to all other strata. Furthermore, if Y < Z < X
then r'xrz = rzr'x concluding the nested induction. Q.E.D.
(This procedure is essentially described in Hendricks [3].)
3. Triangulations. By "polyhedron" we mean "Euclidean polyhedron" as in
Hudson [4]. However if K and L are polyhedra and J c K is a subpolyhedron and if /: J -* L is a P.L. embedding then we can define a new
polyhedron K uy L by finding triangulations A, B, C of J, K and L such that
/4 is a full subcomplex of B and C and then re-embedding the abstract
simplicial complex B u A C into Euclidean space.
Definition. A smooth triangulation of a manifold X is a polyhedral pair
(Tí, L) and a homeomorphism /: K - L -» X such that there exists a
simplicial pair L4, 5) with \A\ = K, \B\ = L such that for each simplex
a E A and for each point p E b~— \B\ there is a neighborhood U of p (in the
plane containing a) and a smooth embedding /: U —>X which extends
/l(tf n 5).
Definition. A triangulation of a stratified object H7is a polyhedron_7f and
a homeomorphism /: 7v —>W7such that, for each stratum X, f~x(X) is a
subpolyhedron of K, and/- '(A') -* X is a smooth triangulation.
For the remainder of this paper we will assume W is a Thom-Mather
stratified object with a family of lines and d > 0 is sufficiently small that, for
each stratum Y, the intersection of any collection of link bundles Y n Sx(d)
n • • • n S^(rf) is transverse to the intersection of any disjoint collection
Y n Sz (d) n • • • n 5Z (¿/). Thus, Y¿ is a manifold with corners.
Remark. In order to triangulate a stratified object W, we will first triangulate the "interior" X¿ of each stratum A" and then glue on the mapping
cylinders of the projections SY(d) -» Y.
Definition. An interior ¿/-triangulation of an «-dimensional stratified
object W" is a collection of disjoint polyhedra {Kx) indexed by the strata X
of W, together with an embedding
/: K u (J Kx -» W
x
such that, for each stratum X (say din^X) = /'),
(l)f(Kx) = X2 C X,
(2)/!^ is smooth,
(3)f~x(Sx(d)) = f-\Sx(d)
- T¡_,(</))is a subpolyhedronof Tí",
(4)/"* ° *x °/: r'Wrf))->/-'(*)
is P.L.
_
(It follows that/ '(S* (d)) ->Sx (d) is an interior ¿-triangulation of Sx (d).)
A partial (d,/?)-triangulation of W" is an embedding/: K^>W satisfying
(2), (3), and (4) as above, but with (1) replaced by
if dim(A') < nS£(A-)
ifdimiÀ") -«.
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1,
R. M. GORESKY
196
If /: K -» W" is a partial (d, /z)-triangulation and if d' < d we define the
d'-extension of (K,f) with respect to A1 to be the following embedding g:
L^W:
Let J = f-\Sx(d))
and define/,: 7 x [d', d] -+Tx(d)-
Tx(d') by/,(/>, r)
= rx(tXf(j>)).
LetL = K L)JX{d]J X [¿/', ¿/]andsetg|/i: = /andg|(L
- tf) =/,.
Taking ¿/'-extensions with respect to all strata X (where din^X) < zz - 1)
results in an embedding h: H -*W (independent of the order in which the X
were chosen because of the commutation relations). Then if Z is the ndimensional stratum of W, h~x(W(n~1) ij Sg.(Z))-> W is called the partial
(d', /z)-triangulation induced by (K,f).
Theorem.
Every stratified object has an interior d-triangulation for d
sufficientlysmall.
Proof. Let Bn and C be the propositions stated below. We shall prove
these propositions in the following order:
°n-i>
C„fl', CnX; •
• ,Cnn_,;
B„; Cn+10;
.
Proposition Bn. Let W" be an n-dimensional stratified object. Suppose f:
Ä"—»W("_1) is an interior d-triangulation. Let d' < d. Then there is an interior
d'-triangulation g: L —»Wsuch that K is a subpolyhedron of L andg\K = /.
Proposition
Cnj). Let W" be an n-dimensional stratified object and suppose
f:K—>W is a partial (d, p — l)-triangulation. Let d' < d. Let j: J -* W be the
(d',p — \)-partial triangulation induced from (K,f). Then there is a partial
(d', p)-triangulation h: H —>W so that J is a subpolyhedron of H andj\J = j.
Proof
of Proposition
B„. Given an interior ¿/-triangulation /: K-^>
W(n~x\ let d' < d" < d and apply Proposition Cnfi through C„„_, to obtain
a partial (d", n — l)-triangulation g: L^>W such that K c L and g|L = /
Let (L, g') be the ¿/'-extension of (L, g) taken with respect to all strata of
dimension < zz- 1. Suppose Z is the /z-dimensional stratum. Then g'(L') d
Zd. - Zd„. Choose a smooth embedding ß: N -> Z of a simplicial complex N
so that Zd„ c ß(N) c Zd,. Then, as in Munkres [8] there is a refinement N'
of N and an approximation ß' of ß so that (N', ß') and (L, g') fit together in
a triangulation of Zd.. Thus, gluing N' and L' along their common intersection gives the desired ¿/'-triangulation.
Proof of Proposition Cnj). Given a partial (d,p - 1)-triangulation
/: K->W, let L0=f-l(Sx(d))
where dim(X) = p. Then (Lo-/|L0) is a
partial (d,p —2)-triangulation of Sx (d). Let d' < d" < d and let a,: Kx -»
W be the ¿/"-extension of (K,f) taken with respect to all strata Y where
Y 7^ X. Let Z be the zz-dimensional
stratum.
Then L, =
<xx-x(Sx(d)C\(W("-X) u S$,rl(Z))) is a subpolyhedron of Kx and
a\X ° mx ° a\- Lx -* Kx is piecewise linear.
■'*
Since (L,, a,) is the (d",p - 2)-partial triangulation of Sx(d) induced by
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TRIANGULATIONOF STRATIFIED OBJECTS
197
(L0, f\L0) we may apply Proposition C„_Xj)_x through C„_,„_2 to find a
partial (d", n — 2)-triangulation g: M—>Sx(d) such that L, is a sub-
polyhedron of M, and g\Lx = ax.
Note that a," ' ° trx ° g: M -> a," '(AT)is P.L., for if F > X then 77^can be
locally factored:
iy
Sx(d) n SyGf")
il y
-►
Sx(d) n y-►
X
g
g-x(sx(d)nsY(d"))
*cx-1(Sx(d)nY)^^cx-\X)
where iy and Hx are P.L.
We now triangulate the rest of Z n Sx(d) so as to make ttx piecewise
linear, where Z is the «-dimensional stratum in W.
Let (M2, g2) be the ¿/'-extension of (M, g) taken with respect to all strata in
Sx(d). Thus g2(M2)D (Zf, - Zl) n Sx(d). By Putz [9] there is a
polyhedron N and a smooth embedding ß: N ^>Z n 5^(i/) so that
ß(N)D
Zj¡. andafx
° itx ° ß: N -+ ax-x(X)is P.L.
(*)
According to Putz, (N, ß) can be approximated by an embedding (N2, ß2)
so that (*) continues to hold and so that (M2, g2) and (N2, ß2) intersect in a
subpolyhedron P, i.e.,
P = g2-,(ß2(N2))
Take L = N2uPM2
= ß2-l(g2(M2)).
and let y: L -+SX (d') by
y\M2 = rx(d')
°g2;
y\N2 = rx(d')
° ß2.
Finally, let (K2, a2) be the partial (d',p — 1)-triangulation of W induced by
(K, f). The commutation relations guarantee that (K2, a2) and (L, y) intersect
in a subpolyhedron Q, i.e.,
Q = Y-'( W)
n (**"-» u 5T1 (Z))).
Thus, 7^2U g L -> W is the desired partial (d', />)-triangulation.
4. Triangulation of mapping cylinders. Let /: K -» L be a simplicial map
between simplicial complexes. Let L' be a barycentric subdivision of L.
Choose a barycentric subdivision K' of K so that Tí' -» U is simplicial. (The
barycenter of a simplex a is denoted â.)
Let SK be the simplicial mapping cylinder of/ in the sense of Cohen [1] and
let S'Kbe the subdivided mapping cylinder, i.e.,
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R. M. GORESKY
198
SK = L U {A * ô0âx ■ • ■ âp\A E L,o0 < o, < • • ■ < op E K,
and A </(a0)},
SK".L'\J
{f0f, ■ ■ • Tqô0- ■ ■ ôp\r0 < t, < • • • <rqE
K,
tq < /K)> a0 < a, < • • • < op E L).
It is good to think of SK as consisting of the pieces [A * f~x(D(A))\A
E
L) where D(A) denotes the dual of A in L.
The simplicial retraction i: S'k —>L' is given by 7t(t0 • • • Tqô0■ ■ • ôp) =
^o ' ' ' fqfiôo)' ' ' fiôp)- Thus the corresponding retraction \SK\ -*\L\ is
piecewise linear.
Let MK be the topological mapping cylinder of F, MK = |ÄT|X [0, 1] u
|L|/(x, 0) ~ fix) and let it: MK -» \L\ be the retraction ir(x, t) = f(x).
We will define a continuous H: \SK\-*MK such that for each simplex
o E K, H takes \S„\ homeomorphically to Ma and such that H\ \SK — L\ is
smooth. For fixed o E K suppose the vertices of t = f(o) are denoted
v0, vx, . . ., v„. For any y E o there is a unique decomposition y = S^r/V,'
such that f(y'i) = t>,for 0 < z < n. This determines zz + 1 smooth projection
maps P¡: a - f~x(v0 . . . v,, . . . vn) -*f~\v¡) by P¡(y) = y¡.
Now let y4 < t (say, ^4 = v0vx ■■- vk) and let B = â0â| .. . a. where
A < f(o0) and o0 < ox < • • ■ < ap < o. Each x E A * B may be written
x = ty + (I - t)z where t £ [0, 1], y E B and z = 2*=0è,u, £ A. Define
H':A * B -> a X [0, 1]by
#'(*)-L +(l - 0 Í VzOO.')Then /z" is smooth provided f(y) £ U f=o<uo • • • *>,• • • u„> which is
guaranteed by the property f(y) E D(A). Composing with the projection
|a| x [0, 1]-» A/0 we obtain the desired map H. H commutes with the
retraction to \L\ and //|(|A^| u |L|) is the identity.
Lemma. H is a homeomorphism.
Proof. In the case a = t and / = identity, H is linear. A typical simplex
ô0â, • • • ôp * âpôp+ x ■ ■ ■ ôn in 5„' is mapped onto the region in Ma given by
I ¿ <*A,
A £ ¿o" ' K X[0. 1]k + ö, + • ' • + Vi < *- '
and ¿z0+ ¿z, + ■ • • + ap > 1 - / >,
Using this fact, H ~ ' is easily found.
In general, if /: a -+ t is simplicial, let 5T be the simplicial mapping cylinder
of the identity map r —>t and let A/Tbe the topological mapping cylinder. Let
H7: \ST\-» A/Tbe the above homeomorphism.
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199
TRIANGULATION OF STRATIFIED OBJECTS
We obtain a commuting diagram
H
where F (A * â0 • • ■ôp) = A */(â0 ■ ■ ■âp) and G(x, t) = (f(x), t).
Suppose the vertices of t are % vx, . . ., vn.
For each y G/"'(f) define an embedding gy: r->a
2^0è,F,(y).
This induces sections Fy of F and G of G by
by gy(2"=0¿,t),)=
7? (/I *f0- • • f,) = A *gy(r0- ■ ■ iq)
whenever/I < t0 . . . . < tq < t, and Gy(x, t) = (gy(x), t) whenever x G t.
These sections, for different y G /~'(f) decompose the interiors of |5„| and
of Ma. Furthermore, H ° F = Gy ° HT so H takes each section in \Sa\
homeomorphically to the corresponding section in Ma. Thus H is a
homeomorphism.
5. Proposition.
Every stratified object W can be triangulated.
Proof. Let /: Tí -» W be an interior ¿/-triangulation. We prove by descending induction on / that there is a polyhedron J¡ and an embedding g¡:
Jt -* W so that K - /„ c /„_, C • • • C 7„ ft|Ä-= f, g\Jp = gp, g,(/,) =
f(K) u (W — T¡_x(d)) and for each stratum X, g¡~l(X) is a subpolyhedron
and gi~x(X)^> X is smooth. We also assume that if dim(Y) < /' - 1 then
g~x(SY(d)) is a subpolyhedron and gr\sY(d))-+g,rx(Y)
is P.L. Then g0
will be a triangulation. We start with gn = f.
To construct (•/,_,, g,_,) from (/„ g¡) suppose X is the (i - l)-dimensional
stratum. Choose simplicial complexes A and B such that \A\ = g¡~^(Sx(d)),
1731
= g~x(X) and A -> 73is simplicial. We also demand that if y < X then
8i~X(Sx(d) n y) and g,~'(^(^)n
Sy(¿/)) are full subcomplexes. Choose
barycenters for 73, associated barycenters for A and let 5 be the simplicial
mapping cylinder of A -» B. Let M be the topological mapping cylinder of
\A\ —>1731and let F: |5| —>M be the homeomorphism from §4. Define G:
M^ W by G(p,t) = rx(td)(p). Let F = |yl| u |T3| and define /,_, = J¡
U, |S|,&_,| |S| = G o F and g,_,|/, s g¡. One checks that (/,_„£,_,)
satisfy the induction hypotheses. Then g0: J0 -^ W is the desired triangulation.
References
1. M. Cohen, Sinyilicial structures and transverse cellularity, Ann. of Math. (2) 85 (1967),
218-245.
2. R. Hardt, Triangulation of subanafytic sets and proper light subanafytic maps, Invent. Math.
(3) 38 (1977),207-217.
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200
R. M. GORESKY
3. E. Hendricks,Ph.D. Thesis,M.I.T., Cambridge,Mass.,(1973).
4. J. Hudson, Piecewise linear topology, Benjamin, New York, 1969.
5. F. Johnson, Thesis, Univ. of Liverpool, 1972.
6. M. Kato, Elementary topology of analytic sets, Sûgaku 25 (1973), 38-51. (Japanese)
7. J. Mather, Notes on topological stability, Mimeographed Notes, Harvard Univ., Cambridge,
Mass., 1970.
8. J. Munkres, Elementary differential topology, Ann. of Math. Studies, no. 54, Princeton Univ.
Press, Princeton, N. J., 1966.
9. H. Putz, Triangulation
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10. R. Thorn, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240-284.
Department of Mathematics, Massachusetts
Massachusetts 02139
Institute
of Technology,
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Cambridge,