on c1-complexes

Annals of Mathematics

On C1-ComplexesAuthor(s): J. H. C. WhiteheadReviewed work(s):Source: Annals of Mathematics, Second Series, Vol. 41, No. 4 (Oct., 1940), pp. 809-824Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968861 .Accessed: 14/10/2012 19:02

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AxNALs OF MATHBMA&ICS

Vol. 41, No. 4, October, 1940

ON C'-COMPLEXES

BY J. H. C. WHITEHEAD

(Received July 18, 1939)

1. This paper is supplementary to S. S. Cairns' work' on the triangulation of, and polyhedral approximations to manifolds of class C'. Its aim is to provide a foundation for theorems2 which involve both differential geometry and the theory of combinatorial equivalence.3 Theorem 8, for example, states that two Cl-triangulations of any manifold of class4 C' are combinatorially equivalent. Thus a manifold of class C' is like a recti-linear polyhedron in that it has a preferred class of combinatorially equivalent triangulations. This theorem depends on the definition of a C'-complex given in ?2 below, and does not apply, for example, to the algebraic complexes first considered by B. L. van der Waerden.5 For though two "algebraic triangulations" of the same space have a common algebraic sub-division, it is not certain that an algebraic triangulation of a recti-linear n-simplex is combinatorially equivalent to an n-simplex. The Cl-complexes, though more closely allied to, also differ essentially from the complexes considered by Cairns. Therefore we do not use Cairns' results, though the main ideas in many of our theorems are due to him.

We shall use Rn to stand for n-dimensional Euclidean space, and it is to be understood that Rn is Euclidean not only in its topology,but also in the sense of metric geometry. By a complex K we shall always mean a recti-linear, simplicial complex in Euclidean space, and K will stand for the mod 2 boundary of K. We shall denote a recti-linear, simplicial subdivision of K, but not necessarily a normal sub-division, by K', and if Ko is any sub-complex of K, then K' will be the sub-complex of K' covering Ko. By a simplex we shall always mean a closed simplex (i.e. a simplex with its boundary), and we shall use the letters A, B to stand for recti-linear simplexes. By an isomorphism t(KO) = K2 we shall mean a homeomorphism of K1 on K2 which maps each simplex of K, on the whole of one, and only one, simplex of K2, and which is linear throughout each simplex in K1. In ??2 and 3, theorem 5 excepted, K will always stand for a finite complex. We shall use the summation con-

1 Annals of Math., 35(1934), 579-87 (triangulation); 37(1936), 409-15 (approximations). 2 See, for example, J. H. C. Whitehead, Annals of Math. 41(1940), 825-832. 3Here we may take as a definition: two simplicial complexes K1 and K2, finite or

infinite, are combinatorially equivalent if, and only if, recti-linear models of K, and K2 have isomorphic recti-linear sub-divisions.

4See, for example, 0. Veblen and J. H. C. Whitehead, The Foundations of Differential Geometry, Cambridge (1932), Chap VI. By an n-dimensional manifold we shall always mean one which is covered by a countable set of open n-cells.

' Math. Ann., 102(1929), 337-62. 809

810 J. H. C. WHITEHEAD

vention in analytical formulae, with the additional convention that Roman indices take on the values 1, ... , n.

2. A map f(U) C R'4, of a region U C Rk (possibly U C Rk C Rm, where m > k), is said to be of class C1, or a C'-map, if and only if, it is given by equation of the form

(2.1) fi(xl ... , Xk) = ft(X),

where xl, ..., IX and yl, ..., y' are rectangular Cartesian coordinates for Rk

and for Ra, and the functions ft(x) have continuous derivatives at each point of U. The map f will be described as non-degenerate if, and only if, the Jacobian matrix of the transformation (2.1) is of rank k at each point of U. A map f(A) C Ra, of a k-simplex A C Rk will be described as of class C1 (non-degenerate) if, and only if, it can be extended throughout some open set in Rk, containing A, in which it is of class C1 (non-degenerate).

Let fa(A) C Rn (a = 1, 2) be two C'-maps of a simplex A, given by equations of the form (2.1). We shall describe f2 as an (e, p)-approximation to fi if, and only if,

11f2-fl|| e and ||df2-dfi I| -< p |I dfi I|

for each x e A and every vector dx, where

dft=af, dxX A=1,..* k)

and || Y2 Y Y = {(Y2 - Yi)(Y2 - Yi)J', I! y I = (yiyi)i When we are only interested in II df2 -dfi 11, or in 11 f2 - fi I, we may refer to f2 as an (co, p)-, or as an (e, X )-approximation to fi . Let a be the origin of the coordinates y and let pi and P2 be the extremities of the vectors df1 and df2, situated at a. If f2 is an (co, p)-approximation to fi, it follows from the geometry of the triangle aplp2 that ap2 ? ap1 + P1P2 ? (1 + p)api, and ap1 ? ap2 + p2pl < ap2 + pap, , whence (1 - p)api ? ap2 . Therefore, if p < ", we have ap, < 2ap2 and fi is an (co, 2p)-approximation to f2 . If f2 is an (cxa, pi)-approximation to fi, f3 an ( Ca, p2)-approximation to f2, and if ap3 is the vector df3, then,

PlP3 _ PlP2 + p2P3 < plapi + p2ap2 _ (Pi + P2 + plp2)api ,

since ap2 < (1 + pi)api. Therefore f3 is an ( Cx, p3)-approximation to fi, where P3 = P1 + P2 + P1P2 . Notice that p3 < 3P1 if p1 = P2 < 1. Combining these with the familiar relations for (e, co)-approximations, we see that, if f2 is an (el, pi)-approximation to fi, where Pi < 2, and f3 is an (E2, p2)-approximation to f2, then fi is an (El, 2pl)-approximation to f2 and f3 an (El + E2,

P1 + P2 + Plp2)-approximation to fi. Notice also that an (co, p)-approximation, f2, to a non-degenerate map, fi, is itself non-degenerate if p < 1. For to say that f is non-degenerate is to say that df # 0 if dx # 0, and II df2||

(1 - p)ljdfiII.

ON C'-COMPLEXES 81 1

LEMMA 1. If f2 is a non-degenerate (c, p)-approximation to a non-degenerate map f,, then the angle' between the vectors df1 and df2 does not exceed 7rp.

If p _ 1 or if p = 0 this is trivial. So we assume 0 < p < 1. Then, with the same notation as before, if the lengths ap1 and PlP2 are fixed, the angle 0 = angle piap2 is greatest when ap2 touches a circle of radius P1P2 and center Pi, in which case PlP2 is perpendicular to ap2 . Therefore sin 0 < plp2/apl < p, whence 0 g 'ir sin 0 < 7rp and the lemma is established.

By the radius r(A), of a simplex A C Rn we shall mean the distance from its centroid to its boundary. Let fa,(A) C Rn (a = 1, 2) be two linear, non- degenerate maps of a k-simplex A C Rk, let e be the maximum of I f2(x)-fi(x) I as x varies over A, and let r = r{fi(A)}.

LEMMA 2. Under these conditions f2(A) is an (E, 2E/r)-approximation to fi(A). By definition, f2(A) is an (E, co)-approximation to fM(A), and we have only

to prove that II df2 - df 1II < p II df1 11, where p = 2e/r. Since fi and f2 are linear, df2- df1 does not depend on the point x e A, but only on the vector dx. Therefore we may take x = x, the centroid of A, and we take x + dx e A. Then fi(x) is the centroid of fi(A), sincef1 is linear, andfi + df1 = fi(i + dx) eif,(A) whence II df1 II ? r. Therefore

df2-df 1II = I (f2+ df2) - (f1 + df1)- (f2f-) ||

= I! {fI2( + dx) - fi(t + dx)} - {f2(xt) -f')} |I

< Ilf2(: + dx) -fi(: + dx) Ij + IIf2(:f) -fi(.)! < 2e < -2I dfi r

and the lemma is established. Let A be a k-simplex in Rk and let f(A) C Rn be a non-degenerate Cl-map,

given by equations of the form (2.1). If b is any point in A these equations may be rewritten as

(2.2) y c' = a(xX- b) + -yt(x),

where c = f(b) and a' are the derivatives Oft/OxX, calculated for x = b. The image of A in the linear transformation Fb , given by

(2.3) y c = a(xX -b)

will be called the tangent simplex to f(A) at the point c. Since the derivatives aflaxX are continuous, and therefore uniformly continuous in the compact set A, it is an obvious consequence of lemma 2 that, given e, p > 0, there is a 6 > 0 such that Fb is an (e, p)-approximation to Fb'(A) provided I l b'-b ? 8. Since, at the point x = b', df = dFb' , we have:

LEMMA 3. Given e, p > 0, there is a positive 6 such that Fb is an (e, p)-approximation to f throughout the sub-set of A given by I x - b II < 6, for any b e A.

By the relative thickness,7 r(B), of a simplex B C Rm, we shall mean r/l,

6 By the angle between two vectors we mean the positive angle which does not exceed an I Cf. Cairns' definition of a 8-set (Triangulation, p. 583).


where r = r(B) and 1 is the diameter,8 l(B), of B, Let f(A) C R' be a non- degenerate C1-map of a k-simplex A C Rk, let bo, *I* , b, be any points in A and let B be the simplex bo ... b,. Let L(B) C R' be the linear map of B such that L(ba) = f(ba) (a = 0, ... , p).

LEMMA9 4. Given e, p, a > 0, there is a positive 5 such that, if r(B) ? a and l(B) ?< A, then L(B) is an (e, p)-approximation to f(B).

By lemma 3 there is a positive 61 such that Fb(B) is an (e/2, p/3)-approximation to f(B) for any b e By provided 1(B) ? S1 . Therefore, assuming, as we obviously may, that p < 1, the lemma will follow, with S = min (S1, S2), if there is a positive S2 such that L(B) is an (e/2, p/3)- approximation to Fb(B) for any b e By provided 1(B) ? 52. The transformations Fb(B) and L(B) are given by

y - c' = a -(ba b )ta

-i_ ci = a(b - bx)ta + yt(ba)ta (a = 0, P),

where ? t0 ?1, to + + tP = 1, bl X b are the coordinates of the vertex ba, and a' and yt(x) mean the same as in (2.2). Since the derivatives of y'(x) vanish when x = b there is a positive 6(vq), such that It Y(x) v II x-b II provided I x -b bII ? 6(), for a given v > 0. IfI = 1(B) = 6 and if b e B it follows that It y(ba) ? _ l. Since ta O and to + + tP = we have It y(ba)ta I< ql, whence

11 L(x) - Fb(X) I = | y(ba)ta ?1 _ rtq

for any point x = bata in B. It follows from lemma 2 that L(B) is an (llni, 2fllnl/rb)-approximation to Fb(B), where rb = r { Fb(B) }.

Now let x1, ..., Ik in (2.3), be rectangular coordinates for Rk D A, and let p1(b) be the smallest root of the equation

(2.4) - aX x, = 0

Then p1(b) is a continuous function of b, and is positive since f is non-degenerate. Therefore p1(b) has a positive lower bound, w 2 as b varies in the compact set A, and

I dFb = (afar dxX dx')' > w dx

for any b e A. Therefore rb > wr, where r = r(B), and i/rb ?_ 1/r < /Iwo, since r/i > o, whence

2flln'/rb _ 2-ni/Wo.

8 I(B) is the length of the longest side of B (P. Alexandroff and H. Hopf, Topologie, Berlin (1935), 607).

9 Cf. Cairns (Approximations, ?4).

ON C1-COMPLEXES 813

Also 1 _< 1(A) = 11, say. Therefore L(B) is an (e/2, p/3)-approximation to Fb(B) provided 1 < 52 = 6(v), where

= min (E/211nif, pwc/6n'),

and the lemma is established. By a (5, o)-subdivision of K CJ Rm we shall mean a subdivision K', such that

1(B) ? 5, r(B) > o-, where B is any simplex in K' and, as before, 1(B) and r(B) stand for the diameter and the relative thickness of B.

LEMMA10 5. There is a (5, o-)-subdivision of K for an arbitrary 5 > 0 and some o- > 0, which does not depend on the choice of 5.

Let the equations (2.3) now represent an arbitrary, non-singular, linear transformation F(Rk) C Rn. If ?l and k are the smallest and greatest roots of the equation (2.4) we have

P dx 11 _ 11 dy 11 _ rk1 dx 1

If B is any simplex in Rk and C = F(B) it follows that r(B) > KT(C), where K = (?,/k)i, and K > 0 since F is non-singular. Therefore, if K = F(Kl), where K, C RN and F is an isomorphism, and if K' is any subdivision of K1, there are constants wo, Ko, such that 1(B) < wol(B,), r(B) > Ko7r(B,), where B, is any simplex in K' and B = F(B,). Therefore we may replace K by an isomorphic complex in RN, and shall assume it to be a sub-complex of the simplex Al, whose vertices have rectangular Cartesian coordinates (0, * *, 0), (1, 0, ... , 0), ... , (0 ... * 0, 1), where N + 1 is the number of vertices in K. Let P be the polyhedral complex consisting of the convex cells into which RN is divided by the hyperplanes

ye = k 0a=O , N; k = O. i4- , 4-2, .**)

where yl, . *, yN are the coordinates for RN and yo = yl + * * + YN. Then P contains a sub-complex covering the simplex Aq, whose vertices are the points (0, * * *, 0), (q, 0, ... *, ), ... , (0, * * *, 0, q), for any integral value of q. The complex P also contains a sub-complex Q, which covers the hyper-cube given by 0 < yX ? 1 (X = 1, .<. , N), and each cell in P is congruent to a cell in Q under the group of translations. Let P' and Q' be the complexes obtained from P and Q by a normal subdivision, the new vertices being placed at the centroids of the corresponding cells. Then each simplex in P' is congruent to some simplex in Q' and its relative thickness is therefore at least o-, where a is the minimum relative thickness of the simplexes in Q'. Let Eq be the sub- complex of P' covering Ad and let E1 be the image of Eq in the transformation given by y" = yX/q. Then El is a subdivision of Al and so contains a sub- complex K', which is a subdivision of K. The relative thickness of each simplex in K' is at least o-, since the relative thickness is an invariant of the

10 Cf. Cairns (Triangulation, p. 585).


similarity group, and its diameter is less than ! N1. Taking q > 1- N1 the q .

lemma is established. By a C'-map, f(K) C R', or a map of class C', we shall mean a map which

is of class C' throughout each simplex in K. The map f(K) will be described as non-degenerate if, and only if, it is non-degenerate throughout each simplex. We shall also describe f(K) as a C'-complex, or a complex of class C', and as a non-degenerate complex if, and only if, the map f is non-degenerate. By an (e, p)-approximation to f(K), we shall mean a C'-map f'(K') C R , where K' is any sub-division of K, such that f'(A) is an (e, p)-approximation to f(A) throughout each simplex A C K'. We shall use Lf(K') to denote the map which is linear (possibly degenerate) throughout each simplex of K' and coincides with f at the vertices of K'. Notice that Lf (K') is not, in general, the image of K' in Lf(K).

THEOREM 1. Given a non-degenerate C'-complex f(K) C Rn, and e, p, o- > 0, there is a positive 5 such that Lf(K') is an (e, p)-approximation to f(K), where K' is any (5, -)-sub-division of K.

By lemma 4 there is a 5(A) > 0 such that, if K' is any {5(A), o}-subdivision of K, then Lf(A') is an (e, p)-approximation to f(A), where A' is the subcomplex of K' covering A C K. Taking 5 = min B(A), for any A C K, the theorem follows.

Let K1 be a sub-complex of a given complex K and let K be any subdivision of K1. By an extension of K' throughout K we shall mean a subdivision K' of K, which coincides with K' in K,. Let f(K) C Rn be a non-degenerate C'-map and let f,(K') C Rn be an (el, pi)-approximation to f(K1). By an (e, p)-extension of f,(K') throughout K we shall mean an (e, p)-approximation, f'(K') C Ra, to f(K), which coincides with fi in K', where K' is an extension of the subdivision K,'.

THEOREM 2. Given a non-degenerate C'-map, f(K) C Rn, a sub-complex K, C K and E, p > 0, there are positive numbers el, Pi, such that any (,E, pi)-

approximation to f(K,) has an (e, p)-extension throughout K. This will follow from an obvious induction on the number of simplexes in

Cl(K - K,), the closure of K - K,, when we have proved it in case K = A, a single simplex, and K, = A. Let a be the centroid of A, let 2co be the mid point of the segment ax1, for any xi e A, and let Ao be the simplex bounded by the locus of xo as xi describes A. Let A' be any subdivision of A and let P be the polyhedral complex consisting of the convex cells B X xoox, swept out by the segment xowx as xi varies over the simplexes B C A'. Let xt be the point on xoox such that xoxt:xtxl = t:(1 - t) (O < t ? 1) and let xI, * * *, 4, t be taken as coordinates for B X xoox,, where x1, * * k, 4 are Cartesian coordinates for any B C A'. Let A be the maximum and 5 the minimum attained by II df II for any xt e P and any vector (dxl, *.* , dX, dt) whose length is unity in terms of a Euclidean metric for A.

Now let f1(A') C Rn be an (El, pi)-approximation to f(A), and, treating f(xi)

ON C-COOMPLEXES 815

and f,(xi) as vectors in R', let 'y(x1) = fi(xl) - f(x,). Then f| ey el and, if dx1 is a unit vector, dy I I ? pi II df I _ pAz whence

(2.5) I4 I i I

_ II < C- ?

jIdy I _ IIdll ?< plA

where the index i refers to some rectangular Cartesian coordinate system for R'. Let f'(P) be the C'-map given, in vector notation, by

f'(xt) = f(xt) + ty(xl) (0 ? t < 1; xi e A').

Then

Ilf'-f11 = t IIy11 _ el and 11 df'-df 11 = 11 td'y + ydtII1

Taking (dxl dt) to be a unit vector, in which case I dt I is bounded, it follows from (2.5) and the continuity of the function 1I y 11 that there are positive numbers ei and pi such that II df' - df II _ p5 < p I df Il. If el < e we have also I If' - f I _ e and f'(P) is an (e, p)-approximation to f(P). Finally we take f' = f in Ao and extend the sub-division A' by starring Ao and each of the cells B X xox,, leaving A' untouched. The result is an (e, p)-extension of f, throughout A, and the theorem is established.

Let K2 be the complex consisting of all the simplexes in K which do not meet K, . As a corollary to theorem 2, replacing f,(K') by f2(K' + K2) with f2 = f, in K , f2 = f in K2, we have the addendum:

ADDENDUM. The extension f'(K'), referred to in theorem 2, may be chosen so that the subdivision K' leaves K2 unaltered and f' = f in K2 .

If B is any simplex in K we shall use N(B, K) to stand for the stellar neighbourhood of B in K, consisting of all the simplexes AB C K, where AB is the join" of A and B. If b is an internal point of B we shall also describe N(B, K) as the stellar neighbourhood,'2 N(b, K), of b. If f(K) C R n is any C'-complex, the recti-linear complex in Rn which consists of the tangent simplexes at f(b) to the simplexes in fIN(b, K) }, will be called the tangent star at f(b) to f(K). Thus the tangent star is the image of N(b, K) in a simplicial transformation Fb, which coincides with the transformation Fb(AB), defined by (2.3), throughout each of the simplexes AB, where b is internal to B. By a non-singular C'-complex, or map, f(K) we shall mean a C'-complex such that

1. f is (1 - 1) throughout K, 2. Fb is (1 - 1) throughout N(b, K) for each point b e K.

It follows from the second of these conditions that a non-singular map is non- degenerate.

THEOREM 3,. To any non-singular C'-complex f(K) C R n correspond positive numbers e, p, such that any (e, p)-approximation to f (K) is non-singular.

According to a previous observation, any (e, p)-approximation to f is non-

"1 Here we allow A to be 1, the empty simplex, in which case AB = B. 12 In general N(B, K) C but 0 N(b, K) if b e P.


degenerate if p < 1. Thus, taking p < 1, we may confine ourselves to non- degenerate approximations. On this understanding we shall prove a similar theorem with less restrictive hypotheses. A non-degenerate map f'(K') C R' will be called an I e, a I-approximation to f(K) if, and only if, I f - f I I< e

and the angle between df and df' is at most a, for each x E K' and non-zero vector dx, in any simplex of K'. Notice that this relation is symmetric between f and fi, and if fi is an I El, ai I-approximation to f and f2 an 1 E2, a2 l-approximation to fi, then f2 is an I q1 + E2, al + a2 I-approximation to f. By lemma 1 an (e, p)-approximation is an I E, 7rp I-approximation, but an example of the form y = x + e sin Xx (e, X > 0; 0 < x < 7r/2X) shows that an I E, 0 I-approximation need not be an (E, p)-approximation for any given p. Our theorem is:

THEOREM 3a . To any non-singular C'-complex f (K) C Rn correspond positive numbers E, a such that any I E, a I-approximation to f(K) is non-singular.

First consider the special case in which K = x1x0 + xOx2, where xoxx (X = 1, 2) are linear segments with no common point other than x0, and f(xlxO + xOx2) = Y1Y0 + YoY2 is linear throughout each of xoxx . In this case any (non-degenerate) I 0o, 0 I-approximation f'(K'), to f(K), is (1 - 1) provided 20 < angle YlYoY2 . For let R2 be the plane containing Yo, Yi and Y2, or any plane through these points if they are collinear, and let 1 C R2 be the external bisector of the angle YlYoY2, or the line Y1Y0Y2 if these points are collinear. Then the inclination to yoyx (X = 1 or 2) of any direction in Rn perpendicular to 1, is at least I angle YlYoY2* If 20 < angle YlYoY2 it follows that the vector df' is never perpendicular to 1, and its orthogonal projection on 1 points away from yo. Therefore the orthogonal projection of f'(K') on 1 is a non-singular image of K', whence f' is (1 - 1).

Let A1B and A2B be simplexes in Rm(A, # 1), let bo be the centroid of B, and let Oo(A1, A2, B) be the minimum attained by the angle a1bop for13 a, e Al , p e A2A, and let

O(A1, A2, B) = min {0o(Al, A2, B), 7r/2J.

It follows from a standard type of argument that Oo(A1, A2, B), and hence O(Ai, A2, B) vary continuously with the vertices of Al, A2 and B, provided the simplexes A1B and A2B remain non-degenerate (and under less stringent conditions). Let xx e AxB - B (X = 1, 2), let a, be the point in Al such that the simplex a1B contains x1, and let the line through x1 parallel to a1b0 meet B in b. Then the line through bo parallel to bx2 meets A2B in p, say, and angle x1bx2 = angle a1bop _ 0(A1, A2, B). Notice that the construction for b, given x1, is affine and so invariant under a linear transformation. We shall call b the A1-projection of x1 in B.

If A1B and A2B do not meet except in B we have 0 = 0(A1, A2, B) > 0, and I say that II xx - b 1 1 < x2 -x1 II cosec 0, where xx, and b mean the same

13 A2J = A2 if B is a 0-simplex.

ON C'-COMPLEXES 817

as before. For if angle x1bx2 > 7r/2 we have | xx - b || <| - X1 i<

x- xi1 I cosec 0, and if angle xibx2 < ir/2 we have

11 x- b 11 _< 11 X2 -xi II cosec (xibx2) < I x2 -xi II cosec o,

since x - b II does not exceed the diameter of the circle through x1, x2 and b. Now let AB C K, AxF # 1 (X = 1, 2; Al A2 = 0) and let

0(p) = 0{F,(Ai), Fp(A2), Fp(B)},

where p e B. Since f is non-singular, 0(p) is a positive, continuous function of

p e B and so attains a positive minimum. Let 7a be the least of these minima, calculated for every pair of simplexes in K which have a common point though neither is contained in the other, and let c be the greatest of the numbers cosec

0(A1, A2, B). Let f'(K') be an 0 o0, a l-approximation to f(K). We first show that the map F' is non-singular14 for each p e K. If F, were singular there would be two segments px1 and px2 in N(p, K') with the same image under FP. Therefore it is enough to show that F; (xi) # F' (x2) if x1 # x2, where x1 and x2

are arbitrarily near p. Let AxB be the simplex of K (not K') containing x, as an inner point, where A1B and A2B do not meet except in B. If one of AXB contains the other (i.e. if A, = 1 or A2 = 1) let b be the mid-point of the segment x1x2 ,if not let b be the A1-projection of x1 in B. Since I I xx - b I I < C I I x2-x , and by lemmas 3 and 1, we may suppose x1 and x2 to be so near p that:

1. xxb C N(p, K') (X = 1, 2), 2. F,(xib + bx2) is an i oo, a I-approximation to f(xib + bx2), 3. F,(xib + bx2) is an I oc, a I-approximation to f'(x1b + bx2).

Since f' is an 0 o0, a I-approximation to f it follows that F;(x'b + bx2) is al I 0o, 3a I-approximation to F,(xib + bx2). Since the angle between the segments F,(bxl) and F,(bx2) is at least 7a it follows from the special case of the

theorem already proved that F,(x1) # F,(x2). Therefore F, is (1 - 1). We now show that f' is locally (1 - 1). By a familiar theorem and lemmas 3

and 1, there is a a > 0 such that: 1. xx EN(b, K) (X = 1, 2),for some b eK, if I x2 - x1 I I , 2. Fb is an I oo, a I-approximation to f throughout the sub-set of K given by

11 X- b 11 < c~for any b E K. This being so, I say that f'(x1) 5 f'(x2) if 0 < I x2 - ?I < 6. For let 0 <

x2 -x 11 < a and let Al, A2, B and b mean the same as before. Since

-x b ?I < c II x2- xi I < cb, the map f(xib + bx2) is an oo, a I-approximation to Fb(xlb + bx2). Thereforef'(xib + bx2) is an I oo, 2a I-approximation to

Fb(xlb + bx2), and it follows from the special case of the theorem that f'(xi) #

f'(x2) . Finally we show that f'(K') is (1 - 1) throughout K' if it is an I (, a l-approxi-

mation to f(K), for a sufficiently small e > 0. The sub-set of the topological

14 If f' is assumed to be rectilinear this step is unnecessary.


product K2 = K X K, for which II x2- xi _ is compact, where 6 means the same as in the preceding paragraph. Therefore the continuous function II f(x2) - f(xz) II attains its minimum, say 3e, on this sub-set, and e > 0 since f is (1 - 1). Therefore, if x2 - xI i and if f'(K') is an I e, a I-approximation to f(K), we have

IIf'(x2) - f'(xI) I I> I If(x2) - f(xI) II - IIf(X2) - f'(X2) Il - I f'(xi) - f(xl) 11 > e > 0,

whence f' is non-singular and the proof is complete. As a corollary to lemma 5 and theorems 1 and 3 we have: THEOREM 4. Given a non-singular C'-complex f(K) C Ro, and e, p > O0

there i8 a 8ubdivision K', of K, such that Lf (K') i8 a norm-sngular, (e, p)-approximation to f(K).

3. Let Mn be an n-dimensional manifold of class C'. Without loss of gen- erality we assume Mn to be smoothly imbedded15 in R', and (e, p)-approximations to maps in kM will be measured in terms of the Euclidean metric for Rm. Let f(K) C U C M' be a non-singular Cl-complex, where U is the domain of an allowable coordinate system for Mn. Then theorem 4 is valid if the term linear is interpreted in terms of the coordinates for U, provided the sub-division K' is so fine that Lf (K') C U. For f(K) is compact, and the metric taken from Rn by the coordinates is continuous in terms of Rm, and the parallelism taken from Rn is a first approximation to the parallelism of Rm. If f(K) C M' is a Cl-complex such that f(A) C U(A) for each A C cl(K - K1), where U(A) is the domain of an allowable coordinate system for M", then the proof of theorem 2 applies to approximations in M', taking e to be so small that f'(A') C U(A).

By a C'-triangukation'6 of M', we shall mean a non-singular, locally finite"7 Cl-complex f(K) = M', which covers M'. By an (n-dimensional) unbounded, formal manifold we shall mean a simplicial complex K, such that the complement of each vertex is combinatorially equivalent to the boundary of an n-simplex.

THEOREM 5. If f(K) is a C0-triangulation of Mn, then K i8 an unbounded formal manifold.

Letf(K) = Mn be a Cl-triangulation. Then K, being a homeomorph of Mn, is n-dimensional and is a pseudomanifold'8 (i.e. each (n - 1)-simplex is on the boundary of precisely two n-simplexes). Therefore the complement, Kb, of any vertex, b, is a pseudo-manifold and hence a finite (n - 1)-cycle (mod 2). Now FbIN(b, K) = Fb(bKb) = f(b)Fb(Kb) C Ri, where Rn is the tangent flat nspace to Mn at f(b). Also Fb is an isomorphic map of bKb, since f is non-singular.

15 Hassler Whitney, Annals of Math., 3(1936), 645-80. 16 It follows very easily from theorems 4 and 2, by Cairns' piecemeal construction (cf.

lemma 7, below) that Mn has a Cl-triangulation. As we shall see this also follows from our theorem 6.

17 We recall this is the only passage in ??2 and 3 in which K may be infinite. 18 H. Seifert and W. Threlfall, Lehrbuch der Topologie, Leipzig (1934), 125.

ON C1-COMPLEXES 819

Therefore the radial projection of Fb(Kb) from f(b) in the boundary of an n-simplex A C R8, of which f(b) is an inner point, is a semi-linear, topological transformation, lrFb(Kb) C An. Since Kb is a finite (n - 1)-cycle it follows that lrFb(Kb) = An, and the theorem is established.

Two maps, f(K) and f*(K*), will be described as equivalent if, and only if, K* is the image of K in an isomorphism t, such that f = f*t. This is obviously an equivalence relation in the technical sense (i.e. it is symmetric and transitive) and we shall now identify any two Cl-complexes which are given by equivalent Cl-maps. Thus f1(K1) = f2(K2) will mean that the maps f1(K1) and f2(K2) are equivalent and the complexes f1(K1) and f2(K2) identical. If'9 K = K1 + *.. + K, we shall describe a non-singular Cl-complex f(K) as the non- singular union, f(K1) + *.. + f(KJ), of its sub-complexes f(K1), * *.. , f(Kq) . Conversely, a set of non-singular Cl-complexes f1(K1), ... , f,(K,) in Mn, will be said to have a non-singular union, f1(K1) + * * * + f,(K,), if, and only if, there is a non-singular Cl-complex f*(K*) = fi(K1) + * + f,(KJ), such that K* Ki* + * + K.' and each map f*(Kx) is equivalent to fx(Kx) (X =

1, * , q). Notice that, if K = K1 + * * + K, and if a given map f(K) is non-singular throughout each of K1, . . ., Kq, then f(K1), * * *, f(K,) may have a non-singular union even if f(K) is singular. But in this case f(K,) + * * * + f(K,) zx f(K). The following lemma is an obvious consequence of these definitions.

LEMMA 6. If f1(KI) and f(K) are non-singular Cl-complexes with a non- singular union, and if K = K2 + K3, then f1(K1) and f(K2) have a non-singular union and

f1(K1) + f(K) = {f1(K1) + f(K2) } + f(K3).

Two complexes f1(K1) and f2(K2) in Mn will be said to intersect in a common sub-complex if, and only if, their intersection, as point sets, coincides with f,(K1o) = f2(K20), where Ka0 is a sub-complex of Ka (a = 1, 2) and the map fi(Ko) is equivalent to the map f2(K20). Let this be the case and, without altering the notation, let us star each simplex Al, if there are any, belonging to Cl(K1 - Kio) but not to K1o and such that fi(Al) has the same vertices as a simplex f2(A2) C f2(K2). Then, replacing Ka by an isomorphic complex,20 if necessary, we may first separate K1 from K2 and then identify each simplex A C Kio with tA C K20, where t(K1o) = K20 is an isomorphism such that

fi = f2t in K1o. The result is a complex K* = K* + K , where K* is the image of Ka in an isomorphism ta such that t1 = t2t in K1o . Therefore f1(K,) and f2(K2) are sub-complexes of fP(K*), where f* = fata in K. If each of the mapsfa(Ka) is (1 - 1) so is f* (K*), since f1(K,) and f2(K2) do not meet except in f1(K1o). Let pi e K1o , P2 = tp1, p* = tapa and let Na = N(pa, Ka), N* =

19 Here addition is used as in the theory of sets. K) and K. may have simplexes in common and may even coincide.

20 e.g. a sub-complex of a k-simplex for an arbitrarily large k.


N(p*, K*). Then, subject to the above conditions, it is clear that f1(K1) and f2(K2) have a non-singular union, namely f*(K*), if, and only if,

1. fi(K1) and f2(K2) are non-singular, 2. f1(N1) and f2(N2) have a non-singuilar union, namely f* (N*), for each pi e Kio.

In general the tangent star at f*(p*) to f*(K*) may be singular for some point p* e t1Kio even if f1(K1) and f2(K2) are both non-singular.

Let K = Ko + E and let f(K) C Mn be a C'-complex such that f(Ko) and f(E) are non-singular. Also let f(E) C U, where U is the domain of an allowable coordinate system, which we regard as map, x(D) = U, of a region D CR'on U.

LEMMA 7. Under these conditions, given e, p > 0 there is an (e, p)-approximation, f'(K") C Mn, to f(K), such that f'(K"') and f'(E") are non-singular and have a non-singular union.

Let H C K be the sub-complex consisting of all the simplexes in K whose images in f meet f(E), let K1 = H Ko and let K2 = Cl(K - H). Then K = H + K2, H = E + K1, and Ko = K1 + K2. Without altering our notation we assume, after a suitable sub-division, that f(H) C U and also f(A) C U, where A C K is any simplex which meets H. By theorem 4, given e1, p1 > 0, there is an (e1 , pi)-approximation, f'(H') C U, to f(H), such that f' is "x-linear" throughout each simplex in H' (i.e. f'(A) = x(B) for each A C H', where B C D is recti-linear). We partially extendf' by writingf' = f throughout each simplex which does not meet H. Then, given 62, P2 > 0 and assuming ei and P1 to be sufficiently small, it follows from theorem 2 that the approximation f'(H') has an (62, P2)-extension, f'(K'), throughout K. Since f(E) and f(Ko) are non- singular, it follows from theorem 3 that f'(E') and f'(Ko) are non-singular if 62 and P2 are sufficiently small, which we assume to be the case. We also take E2 to be so small that f'(K2) does not meet f'(E'). Finally we take E2 -, P2 _ p, in which case f'(K") is an (e, p-approximation to f(K), where K" is any sub-division of K'.

The sub-set f'(H') C Mn, besides being the image of H' in f is the homeomorph, x(P), of a polyhedron P = F + Pi C D, where

F = x-f'(E'), P1 = xlf'(Ki)

x'f' being an isomorphism throughout each of E' and K1 since K1 C Ko and f'(E') and f'(K') are non-singular. Let P' = F' + P be a triangulation of P, F' and P being recti-linear sub-divisions of F and P1 which intersect in a common sub-complex. Since the map x-'f' (A) C P is non-degenerate for each simplex A C H', the triangulation P' determines a sub-division H" = E" + K7, of H', such that F' and P' are isomorphic in x'lf' to E" and K". There- fore x(F') = f'(E") and x(P') = f'(K1'). Let K" = H" + K2' = Ko + E" be an extension of the sub-division H" throughout K'.

Since x(P') = x(F') + x(P') = f'(E") + f'(K'), and f'(E") f'(K') = 0, the (non-singular) C'-complexes x(P') and f'(K)' intersect in a common sub- complex, namely f'(K' -K"'). Also any simplex A C P', such that x(A) meets

ON C1-COMPLEXES 821

f'(K2'), belongs to P' and not to F'. Therefore, if p e K -K2

and q-x-lf'(p), then N(q, P') = N(q, Pi). Since x(Pl) = f'(K1j) and f'(K") = f'(K') + f'(K2') it follows that x { N(q, P') } and f' { N(p, K"') } have a non-singular union, namely f' {N(p, Kg') }. Therefore x(P') and f'(K"2) have a non-singular union. But

x(P') = f'(E") + f'(K"')

and it follows from lemma 6 that

x(P') + f'(.K") = f'(E") + {f'(KI') + f'(K"') } = f'(E") + f'(Ko')

and the lemma is established. We now come to the main theorem. THEOREM 6. Given e, p > 0, and non-singular C'-complexes fx(Kx) C M'

(X = 1, *.. , q), there are (e, p)-approximations in Mn to fx(Kx), which have a non-singular union.

If q = 1 the theorem is trivial and we shall prove it by induction on the total number of simplexes in K2 , ***, K, after an initial sub-division such that fx(A) is in the domain of an allowable coordinate system, for each simplex A C Kx and each X = 2, . . ., q. Let K, = Kqo + A, where A is a principal simplex in K, and2' Kqo = Cl(K, - A), and let U be the domain of an allowable coordinate system, which contains f,(A). By the hypothesis of the induction, given 1 , p1 > 0 there are non-singular (6a, p1)-approximationsf' (K'), f,(K' ) C M' to fa(Ka) and fq(Kqo) (a = 1, , q - 1), such that f(K), * *, fq(Kqo) have a non-singular union. By theorem 2, given 62, P2 > 0 and provided El and p1 are sufficiently small, there is an (62, P2)-extension, f,(K'), of f,(K' ), where K' = K' + A' and fq(A') C U. We take 62 and P2 to be so small that fA(KX) is non-singular, according to theorem 3. We also take 61 - 62, P1 _ P2,

so that f,(Kx,) is an (62, P2)-approximation to fx(Kx) for each value of X = 1, * * *, q. Replacing f'(K') (X = 1, * * , q) by equivalent maps, if necessary, and taking care that no internal simplex of A' coincides with a simplex of22 K. (o = 1, , q - 1) we may, without altering our notation, represent fq(K') and the union fi(K') + + fq(K' ) as non-singular sub-complexes of aCl-complex g(K), where K = K' + + K' and g =fx in Kx. ThenK = Ko + A', where Ko = K' + * + K'0, and g(Ko) = fi(K') + + fq(K'0). Since g(A') C U it follows from lemma 7 that, given E3, p3 > 0, there is an (e3, p3)-approximation, g'(K'), to g(K), such that g'(Ko) and g'(A") have a non-singular union, where K' = KI + A" = Ki' + .* + Kq'. Then g'(Ko), and hence g'(K") (a = 1, * , q - 1), are non-singular, and since g'(K.') is an (e3, p3)-approximation to the non-singular complex g(K') (= fq(K')) we may

21 Kqo and fq(Kqo) are empty if K, = A. 22 This may require an internal sub-division of A' if there are internal simplexes with

all their vertices in the boundary. However, if the sub-division A' is given by the construction in the proof of theorem 2 there are no such simplexes.


take E3 and p3 to be so small that g'(K"') is also non-singular. This being so g' (Kq ) = g'(K") + g'(A"), and by lemma 6 we have

g'(KO) + g'(A") = {g'(K?') + + g'((K1o)} + g'(A")

= g'(Ki') + * + g'(K"). Finally g(K)) (= f)(K')) is an (E2, P2)-approximation to fx(Kx) and g'(Kx') is an (c3, p3)-approximation to g'(Kn). Therefore, taking E2 + E3 _ E, P2 + P3 + p2p3 _ p, the theorem is established.

Let V be any open sub-set of M" and let fx(Kxo) be the sub-complex consisting of all the simplexes in fx(Kx) which meet Cl(V). From the proof of lemma 7, and by adding to the hypotheses of the induction in theorem 6, we have the addendum:

ADDENDUM: If f1(Kio) , fq(Kqo) have a non-singular union the approximations in theorem 6 may be chosen so as not to disturb the part of this union which lies in V.

For in the proof of lemma 7 it is only necessary to sub-divide K, or to alter the map f, in those simplexes which meet H. If f(E) C Mn _ V no simplex f(A) CMwwhich meetsf(E) is contained in V. Iff(E) C Mn _ Cl(V) we may therefore assume, after an initial sub-division which leaves A unaltered if f(A) C V, thatf(H) CMn - V. Then A .H = 0 iff(A) C V.

We now require M" to be closed, a restriction which we remove later. THEOREM 7. There is a C'-triangulation of Mn. Since M" is closed it can be covered by the interiors of a finite set of non-

singular, n-dimensional C'-simplexes fi(Al), , fq(Aq). By theorem 6, given C > 0, there are (c, oo)-approximations, f)(A() (X = 1, ... , q), toffx(Ax), which have a non-singular union f(K) = fi(A') + ... + fq(Aq). It follows from well known theorems23 that, provided c is sufficiently small, each point of M" is internal to at least one of the cells fx(A'). Therefore f(K) covers M" and is a C'-triangulation.

THEOREM 8. If fi(Ki) and f2(K2) are two C'-triangulations of Mn, then K1 and K2 are combinatorially equivalent.

For, by theorem 6, there are non-singular approximations fx(K') C M" to fx(Kx) (X = 1, 2), which have a non-singular union. Since Kx is a pseudomanifold it is a cycle (mod 2). Therefore K( is a cycle (mod 2), and since the mapfx is topological it follows that M' is completely covered byfx(K,). There- fore fi(K') = f2(K'), since f(K') and f2(K') intersect in a common sub-complex, and K' is isomorphic to K.

With suitable restrictions, similar theorems to theorems 7 and 8 may be proved for a bounded manifold Mo C Mn. For example, let M" and also the frontier, Mn-', of Mo be manifolds of class C3, and let M" be given a Riemannian metric ds2 = gidxtdxj', where the functions gii are of class C2 in allowable coordinate systems for Mn. Then, for some 6 > 0, no two of the geodesic seg-

3 See, for example, Alexandroff and Hopf (loc. cit.), pp. 100 (theorem IV) and 459 (Rouchf's theorem).

ON C0-COMPLEXES 823

ments pq, of length 8, will meet each other, where pq C M' is normal at p X `

to Mn-1. If f(K) C Mn-1 is a Cl-triangulation of Mn-i the sub-set of M" covered by the segments pq is a non-singular Cl-image of the polyhedral complex K X (0, 1), which may be triangulated by a normal sub-division. It is now easy to show that, without disturbing f(K), some approximation to this triangulation may be extended throughout M'.

Assuming only that Mn and Mn-i are of class C', let f1(K1) and f2(K2) be two Cl-triangulations of Mn. By theorem 6, applied to the sub-complexes f1(K1o), f2(K20) covering M'l, by theorem 2, and since fx(Kxo) is the point-set frontier of fx(Kx), we may assume that f1(K1o) = f2(K20). By adding to the hypotheses of the induction in theorem 6 we see that, if the maps fx(Kx) (X = 1, * , q) are equivalent to each other throughout mutually isomorphic sub- complexes Kxo C Kx, then the approximations fx(Kx) C Mn, which have a non-singular union, may be chosen so thatfi(K'o) = ** = fq,(K'o) In the case of the triangulations fx(Kx) = Mn (X = 1, 2), with Kxo = Ax (mod 2), it follows that f(K') = f2(K2), whence K' is isomorphic to K.

4. We conclude by showing how many of these results can be extended to infinite complexes and open manifolds. An infinite complex f(K) C M" is to be such that only a finite number of simplexes f(A) (A C K) meet any compact sub-set of Mn. A manifold M" C Rm is to be a closed, but not necessarily compact, sub-set of Rm. An (e, p)-approximation to f(K) shall mean the same as before, except that e and p may now be any non-negative functions, e(p) and p(p), which are defined for each p e K, and (, p > 0 is to mean that e(p) and p(p) have positive lower limits in each compact sub-set of K. It is often convenient to define such a function in terms of a particular covering of K by compact sub-sets [F] (e.g. the simplexes or stellar neighbourhoods), only a finite number of which meet any one compact sub-set, and a positive function of sets, f(F), defined for each set in the covering. Then tj(p) may be defined as the minimum of j(F) for F containing p. Conversely, given tj(p), the function r(F) may be defined as the lower limit of tj(p) for p e F. For example, in the proof of theorem 2, with K finite or infinite, we may take e(p), p(p) to be defined in terms of given functions e(A), p(A), where A is any simplex in K, and s,(p), Pi(p) to be not greater than suitably chosen functions si(A), pi(A) if p eA. The theorem then follows by induction on the dimensionality of K - K, and the same construction as in the finite case. In proving theorem 3 we may define a in terms of a function &(N) > 0, where N is the stellar neighbourhood of any vertex in K. If as(N) is suitably chosen, any I so, a I-approximation to a given non-singular Cl-map is locally non-singular and it is easily shown, as in the finite case, that an I (, a 1-approximation is non-singular for a suitable 6(P) In the absence of lemma 5, which seems to be comparatively difficult if K

is infinite, we replace theorem 4 by the less explicit theorem: THEOREM 9. Given (, p > 0, there is a non-singular, rectilinear (e, p)-approxi-

nation to any non-singular Cl-complex f(K) C R .

g24 J. It. 0. WHITEREAD

This may be proved in the same way as the extension of theoren 6 to infinite sets of (possibly infinite) complexes fx(Kx) C M' (X = 1, 2, * * * ), only a finite number of which meet any one compact sub-set of M'. To prove this let M' be a closed set in R', referred to Cartesian coordinates y', .*, y'. Let Vr be the sub-set of M' for which II y II < r and, after a suitable sub-division, let each simplex of fx(K) which meets Cl(V,) lie in V,+1, for each X, ,u = 1, 2, .. By theorems 6 and 2 we may assume, after a suitable (E,, pr)-approximation to fx(Kx) (X = 1, 2, ... ; cr(p) < E(P), Pr(p) < p(p)), that the maximal sub- complexes of fA(K1), f2(K2), ... whose simplexes all meet Cl(Vr) have a non- singular union, for some24 r = 1, 2, * . . If follows from theorem 6 and its addendum that, by a suitable (e', p')-approximation to the first (Er, pr)-approximation, this condition can be maintained with r replaced by r + 1, without disturbing the part of the union, say g(Pr), which lies in Vr. The result will be an (Er+1, pr+i)-approximation to fx(Kx), for each X = 1, 2, * * , where Er+1-

Er + Er , Pr+1 = Pr + Pr + PrPrI. Since er < E2 Pr < p, we may choose Er, p' so that Er+1 < E, Pr+1 < p and the induction is complete. In the succeeding stages of the construction we may take Pr C Pr+1 C ... and the required union in g(P), where P = P1 + P2 ... This theorem carries with it theorems 7 and 8 for open manifolds.25

Finally, if M' is a manifold of class Ck (k = 2, *,co or w), a 0-complex f(K) C M' may be defined in the same way as a Cl-complex, and we have:

THEOREM 10. Given E, p > 0 and a non-degenerate Cl-complex f(K) C Mn, there is an (6, p)-approximation to f(K) which is of class Ck.

If M" be imbedded as a class Ck manifold in Rm, the flat (m - n)-spaces normal to M" form a system of class Ck-1. It is, however, possible26 to define a class Ck system of flat (m - n)-spaces approximately normal to Mn. By means of such spaces, we can project back into M" a recti-linear (e', p')-approximation to f(K), thus obtaining a proof of Theorem 10, provided (6', p') are chosen sufficiently small.

BALLIOL COLLEGE, OXFORD.

24 Notice that we only need theorem 6 for finite complexes and theorem 2 for finite K - K1 . We may take er(P), Pr(P) to be any constants less than the lower limits of e(p), p(p) for p e K-Cl(Vr+i), and Er(P) = Pr(P) = 0 for p e Kx - Kx Cl(Vr+i).

21 Here again Cairns' method leads to a more direct proof of the triangulation theorem. 26 Hassler Whitney, loc. cit., ?25.

on c1-complexes

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