embeddings of polyhedral manifolds

Annals of Mathematics

Embeddings of Polyhedral ManifoldsAuthor(s): M. C. IrwinSource: Annals of Mathematics, Second Series, Vol. 82, No. 1 (Jul., 1965), pp. 1-14Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970560 .

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Embeddings of Polyhedral Manifolds* By M. C. IRWIN

1. Introduction

The results of this paper resemble embedding theorems obtained by A. Haefliger [1] for differentiable manifolds. We work in the polyhedral category (see [8, p. 62] and ? 2 below), and, to simplify proofs, we deal only with compact spaces, but indicate where this condition may be relaxed. Let M be an m-manifold with boundary OM, and Q be a q-manifold with boundary OQ. Given a map f: M-u Q, let S(f ) denote the singular set of f, that is to say the closure in M of {x e M; f -lf(x) contains at least two points}. The main theorem is

THEOREM 1.1. Suppose M, Q are respectively (2m - q)- and (2m - q + 1)- connected and m _ q-3. Let f: M,y M, int M- Q,-Q, int Q be a map such that S(f) c int M. Then f is homotopic rel OM to an embedding.

Remarks. 1. OM may be empty, in which case OQ may also be empty. 2. The theorem is true even if Q is not compact. 3. By using the fact that M and Q have collars if OM and OQ are non-empty

[5, p. 326], it is quite easy to remove from the theorem the conditions (i) that f takes int M into int Q, (ii) that S(f ) c int M.

Of course f must still embed SM. As special cases of the theorem we have

COROLLARY 1.2. Any element of W7m(Q), where Q is (2m - q + l)-connected and m ? q - 3, may be represented by an embedding of an m-sphere in Q.

COROLLARY 1.3. Any reconnected M without boundary, r ? m - 3, may be embedded in euclidean (2m - r)-space.

Theorem 1.1 may be regarded as a generalisation of an embedding theorem of Penrose, Whitehead, and Zeeman [4] which emerges above as Corollary 1.3. We do not obtain their results for embedding manifolds with boundary in euclidean space, for all our theorems deal in homotopies rel OM, whereas their methods involve movements of the boundary.

In ? 2 we describe the polyhedral category and quote some results, includ- ing Zeeman's engulfing theorem, which we need for the proof of Theorem 1.1 (in ? 3). In ? 4 we extend Zeeman's theorem so that we can make better use of it in manifolds with boundary. We can then prove, in ? 5, embedding theorems

* Research for this paper was supported by a D.S.I.R. grant.



2 M. C. IRWIN

like Theorem 1.1 when f does not take OM into OQ. Finally, in ? 6 we set down counter-examples which, to some extent, justify our connectivity conditions in the embedding theorems. Counter-example 3 proves that what would be a reasonable statement of the sphere theorem [6] for 4-dimensional manifolds is, in fact, false. Counter-example 4 does the same for Dehn's lemma.

I should like to express my sincere gratitude to E. C. Zeeman for suggesting this problem, and for the great help and encouragement he gave me during my research on it. In particular he has considerably shortened this paper by supplying (? 4) the replacement for a very cumbersome proof of the relative version of his engulfing theorem.

2. Terminology

A triangulation t of a compact topological space X is a homeomorphism onto, t: I K I - X, where K is a finite simplicial complex. A polyhedral structure Qx on X is a set of triangulations of X such that

(i) if t: I K I \ X is in Qx and f: L K is a piecewise linear homeomorphism onto, then tf: I L I - X is in Qx,

(ii) if s, t e Qx then t's is a piecewise linear homeomorphism onto. A (compact) polyhedral space X is a compact topological space X, together with a polyhedral structure Qx (for the definition of polyhedral space in the non-compact case, see [8, p. 61]). If K is a finite simplicial complex, then I K I has a natural polyhedral structure given by all piecewise linear homeomor- .phisms onto K. A polyhedral space M is a polyhedral m-manifold if, for one, and hence for any, t: I K I - M in QM, K is a combinatorial m-manifold. A polyhedral m-ball (resp. m-sphere) has a triangulation by an m-simplex (resp. the boundary of an (m + 1)-simplex). Also, f: X Y is a polyhedral map of X into Yif, for all s: I K I - X in Qx and all t: I L I Yin Q7, t-rfs is piecewise linear. And X is a polyhedral subspace of Y if it is a topological subspace and the inclusion is a polyhedral map. This implies that we can triangulate Y by a complex so that a subcomplex triangulates X. The set of all polyhedral spaces and maps is the polyhedral category.

In the rest of the paper we drop the adjective polyhedral and all objects are polyhedral unless otherwise stated. When it does not lead to confusion, we shall use the same letter to denote

(i) both a space and a chosen complex triangulating it, and (ii) both a map of one space into another and the piecewise linear map in-

duced on complexes triangulating them. If Y is a subspace of X, there is an elementary collapse from X to Y if X = Y U B7 and Y n B7= B-', where B7 is an n-ball and B"- is an (n - 1)-ball in



POLYHEDRAL MANIFOLDS 3

the boundary of Be. If we can get from X to Y by a finite series of such elementary collapses (with various n) we say that X collapses to Y, written X\Y. If Y is a point, then X is collapsible, written X \ 0, and X then collapses to any point of itself. If X is a subspace of a manifold M, with boundary OM, then we say X collapses to Y admissibly in M if X\ Y U (X n aM) \ Y.

Suppose X is a subspace of an m-manifold M. We may triangulate M so that X is a subcomplex. Let N(X, M") be the closed simplicial neighbourhood of X in the barycentric second derived M" of the complex M. Then Whitehead proved [5, p. 293]

THEOREM 2.1. If X\ 0, then N(X, M") is an m-ball.

COROLLARY 2.2. If X\ 0 admissibly in M, then N(X, M") is an m-ball which meets OM in an (m - 1)-ball.

The corollary is immediate, as N(X, M") nf M = N(X nf M, aM") and x n SM\ O.

We are now in a position to quote Zeeman's engulfing theorem [9, 11 Ch. 7].

THEOREM 2.3. Let M be a k-connected m-manifold, k _ m - 3, and X and K be subspaces in int M such that X \ 0 and K is k-dimensional. Then there exists a (k + l)-dimensional subspace L with Kc L c int M such that Xu L 0.

In other words we may enlarge the collapsible subspace until it engulfs K, using a dimension only one higher than that of K. As a special case when X is a point we have

COROLLARY 2.4. Let M and K be as in the theorem. Then there exists a (k + 1)-dimensional collapsible subspace L with Kc L c int M.

Also, from Theorem 2.1,

COROLLARY 2.5. Let M, X and K be as in the theorem. Then there is an in-ball A in t M with X U Kc int A.

The m-ball is a second derived neighbourhood of X U L. In the theorem and corollaries M need not be compact.

We shall also use, in ? 4, simplicial collapses (elementary contractions in Whitehead's terminology [5, p. 247]). Let L be a subcomplex of a complex K. There is a simplicial collapse from K to L if K= L + a + z and L n f = xoz-) where a = xz is an n-simplex and x is one of its vertices. We shall label the collapse (v, z) and call n its order. We can rearrange any finite sequence of such collapses into one whose collapses have non-increasing order [5, p. 248].



4 M. C. IRWIN

If X \ Y admissibly in M, there is a triangulation of M such that X and Y are subcomplexes and X goes to Y by a sequence of admissible simplicial collapses [11, Ch. 3]. A simplicial collapse is admissible in the combinatorial manifold M if either

(2.1) intI a IU int Iz- Iint IMI,

or

(2.2) a and z e OM.

Let A and C be balls and f: CA - OC be any map. We can choose a map g: A > C, such that g I &A = f, as follows: Choose triangulations s: I a I A and t: I -I C, where a and z are simplexes. Choose points x E int I a I and

yeintI z1. Let h:I a IIz Isend x to y, be t-lfs on IOa1, and map the rest of

I a I = x I a I by extending linearly. Then g is ths-1. Note that if f embeds SA, then g embeds A. We call g a linear extension of f.

Throughout the paper I is the unit interval.

3. Proof of Theorem 1.1

The theorem will be proved if we can find an m-ball A c int M and a q-ball C c int Q such that S(f ) c int A and f maps M - A, OA, int A into Q - C, SC, int C respectively. We can then extend f: OA S &C linearly to an embedding g: A > C and this, together with g I M - int A = f, gives the required embedding.

The existence of such balls A and C is guaranteed if there are collapsible subspaces D c int M and E c int Q such that S(f ) c D and f -'(E) = D. For we may triangulate M and Q so that D and E are subcomplexes and f is simplicial. By Theorem 2.1, A = N(D, M") and C = N(E, Q") are balls. S(f) cint A because D c int M, and f -'(E) = D, f simplicial, ensures that f takes M - A, OA, int A into Q - C, SC, int C. We try to construct collapsible subspaces D and E with the required properties.

We may deform f rel OM into general position [11, Ch. 6] in Q, so that dim S(f ) ? 2m - q. Notice that we may apply Corollary 2.4 in M to (2m -q)- dimensional subspaces, and in Q to (2m - q + 1)-dimensional subspaces. The- manifolds have the correct connectivities, and q ? m - 3 implies that 2m - q ? m - 3 and 2m - q + 1 < q - 3. Now if we had, in the hypothesis of the theorem, the stronger inequality 3m < 2q - 2, m > 1, it would be very easy to construct D and E. Corollary 2.4 in M gives that S(f ) c Do c int M, where Do \O and dim Do = 2m - q + 1, and in Q gives that f(Do) c EO c int Q, where EO \ 0 and dim EO = 2m - q + 2. When we have moved E0, relf(DO), into general position with respect to f (M), we have that f (M) n EO = f (DO), as




(2m - q + 2) + m < q. Thus f-'(E0) = D0, as the singular set S(f ) is contained in D, We can take Do as D, E, as E.

With the weaker inequality m ? q - 3, however, we can only say that F, = closure (f-'(E) - DJ), the subspace which we think of as the obstruction to Do and E, being D and E, has dimension < (2m - q + 2) + m - q 3m - 2q + 2. But m ? q - 3 implies that this is no greater than 2m - q -1, which is an improvement on the original inequality for S(f). We now, using Theorem 2.3, enlarge Do and E, to D1 and E1, D1 to contain F0 and E1 to contain

f(Dj), and obtain another obstruction F1. We continue this process. For each enlargement we add a subspace of dimension one less than the time before. As a result, the obstruction dwindles to nothing.

More precisely, we formulate an inductive hypothesis: 4>(i). There exist collapsible subspaces Di and Ei such that

S(f)cDic int M, f(Di)cEic intQ,

and Fi closure (f-'(Ei) - Di) has dimension ?2m - q - i-1.

Di

f(s(f))E

Figure 1

The case i = 0 has been proved above. We assume D(j - 1), j ; 1, and prove 4F(j).

Fi-, has dimension ? 2m - q - j. By Theorem 2.3 in M, with Dj-, the collapsible subspace, there exists G, with Fj-, c G c int M, such that dim G = 2m - q-j + 1 and D - D - U G \ O. Now apply Theorem 2.3 in Q to f(G) and the collapsible subspace Ej,. There exists H, with f(G) c Hc int Q, such that dim H = 2m-q-j + 2 and Ej = Ej-, U H\ O. As f(G)c H and f(Dj_1) c Ej~, f (Dj) c Ej. Move H, rel f (G) U Ej~, into general position with respect to f(M). Then

Ej n f (M) -f (Dj) c- H n f (M) -f (Dj),

and has dimension ?(2m-q-j -- 2) + m-q ? 2m-q-j-1 (as q _ m + 3), and so has its counter-image under f, as f is an embedding outside S(f ) c Dj.



6 M. C. IRWIN

This completes the proof of q)(j), and the induction. q1(2m -q) tells us that F2mq has dimension ? -1, i.e., is empty, so f'-(E2mq) D2mq. We take D2m_- as D and E2m.q as E.

4. The generalisation of Zeeman's theorem

We shall need two lemmas.

LEMMA 4.1. Let F \ Fo admissibly in an m-manifold M, and Go be ax admissibly collapsible subspace of M containing F0. Then there exists anz admissibly collapsible subspace G of M containing F such that dim (G - F) ?

dim (Go - Fo). PROOF. Triangulate M so that F, Fo are subcomplexes and F goes to Fo by

admissible simplicial collapses. We may perform these in the order, first, all of type (2.1), and then all of type (2.2). It is sufficient to prove the result for one simplicial collapse, and the lemma follows by induction.

ar

Figure 2 Go U D collapsing to Go

Suppose F goes to F. by the simplicial collapse (a, z-) where a xz. Let r be the barycentre of z and aM be a subdivision of M such that both G. and the straight line segment x: are subcomplexes of aM. Choose a point y in int x: sufficiently near x for D xy8z- to contain no vertices of au interior to a. D consists of a number of truncated simplexes of au, and Go U D \ Go admissibly in M. The elementary collapses are of non-increasing dimension, and a typical one removes the interior of a truncated simplex and the interior of its face in yiz- (see Figure 2). Go U D \G. is admissible, for either (Go U D) n AM- Go n aM, if (v, z) is of type (2.1), or (Go U D) n AM- Go U D, if (a, z) is of type (2.2). Since Go \ 0 admissibly, Go U D \ 0 admissibly.

It is easy to determine a homeomorphism h of the m-ball zLk(z, M) onto itself which is the identity on (Oz)Lk(z, M) and takes D onto a. We define h = 1 outside the m-ball. Then h takes Fo U D onto F. The required G is h(Go U D). Dim (G - F) ? dim (Go - Fo), for G - F is homeomorphic to Go U D - Fo U D, and the latter is contained in G - F,.




h

Figure 3

LEMMA 4.2. Let B1,, B, B3 be a nest of (m - 1)-balls (i.e., each contained in the interior of its predecessor). Let t e int I. Then there is a homeomorphism of B1 x I onto itself which is the identity on (OB1 x I) U (B1 x 1) and takes (B3 x I) U (B1 x [t, 1]) onto (B2 x I) U (B1 x [t, 1]) (see Figure 4).

B B3 B, I

0 t I 0 tI

Figure 4

PROOF. This follows easily from a regular neighbourhood theorem of Hudson and Zeeman [2].

We now state and prove our version of Zeeman's theorem.

THEOREM 4.3. Let M be a k-connected m-manifold, k < m - 3, and T be a (k - 1)-connected (m - 1)-submanifold of OM. Let X and K be subspaces in int M U int T such that X\ 0 admissibly in M, dim K= k and dim K n OM= k - 1. Then there exists a (k + 1)-dimensional subspace L with Kc L c int MU int T such that X U L \ 0 admissibly in M.

As a special case when X is a point of int T, we have

COROLLARY 4.4. Let M, T, and K be as in the theorem. Then there exists a (k + 1)-dimensional admissibly collapsible subspace L with

KczLc(intMU intT.

Both theorem and corollary are true if M and T are not compact. PROOF OF THEOREM 4.3. This is in two steps. We prove (i) that the theorem holds if M is an m-ball and T is an (m - 1)-ball; (ii) that, with the hypotheses of the theorem, M contains an m-ball A such

that A nf M = B is an (m - 1)-ball in int T and K U X c int A U int B.



8 M. C. IRWIN

Obviously (i) and (ii) together give the theorem. (Sketch proof (i). We enlarge X to a subspace F, in int M U int T, which

contains K. F may not be admissibly collapsible, but we can choose it so that it collapses admissibly to a k-dimensional subspace Fo such that Fo n T is collapsible. We put a (k + 1)-dimensional degenerate cone Go on Fo. This is admissibly collapsible. Finally we use Lemma 4.1 to blow up Go into G containing F. Then G is X U L.)

PROOF (i). We may suppose without loss of generality that M and T are an m-simplex and an (m - 1)-face, for they are homeomorphic to such a pair. By Theorem 2.3 in T there is a k-dimensional subspace D with K n T c D c int T such that D U (X nT) \ O. Then DU X\ D U (XfnT)\O, so DU X\0 admissibly. Let aM be a subdivision of the simplex M such that K and D U X are subcomplexes of aM, and D U X goes to a point by a sequence of admissible simplicial collapses with non-increasing order. Let E be the subcomplex of D U X left after all collapses of order ? k + 1 have been performed. Let pp 1 < i < n, be those k-simplexes of K such that pi = z for some simplicial collapse (a, z) of D U X to E. Then the complex F = D U X + K collapses to Fo = E + (K - n> pi) by exactly those simplicial collapses which take D U X to E. Certainly, then the polyspace F collapses admissibly to the polyspace Fo.

Fo has dimension k, and there is an admissibly collapsible (k + 1)-dimensional subspace Go of M containing F0, namely, the degenerate cone with base Fo and vertex any point x of int M (that is, the set {z; z e xy, some y e Fol) Go \, x(Fo n T) by collapses towards the vertex x. x(Fo n T) is a proper cone. Fo n T collapses to a point, p say, in T by just those simplicial collapses of D U (X n T) which have order _ k. For each such simplicial collapse (a, z-) we perform the elementary collapse (xa, xz). This gives us

x(Fo n T) \ xp U (Fo n T).

Now xp \, p. Thus x(Fo n T) \ Fo n T \ 0. We have proved that

Go\GonaM= Fon T\,0

so Go is admissibly collapsible. By Lemma 4.1, there exists an admissibly collapsible subspace G of M con-

taining F (and hence X and K), such that dim (G - F) ? dim (Go - Fo) _ k + 1. Furthermore F - X c D U K and so has dimension _ k. We define L to be K U closure (G - X), so that X U L = G (we can enlarge L slightly if its dimension is < k + 1).

(Sketch proof (ii). The manifold M has a collar. Let X0, Ko be the pieces of X, K in the collar and X1, K1 the closed complements of X0, Ko in X, K, respectively. We use Zeeman's theorem to put X1 and K1 in an m-ball in int M,




join this ball to the boundary by a small tube, and then use Lemma 4.2 to enlarge the tube to cover XO and Ko.)

PROOF (ii). By Corollary 2.5 applied in T to the (k - 1)-dimensional subspace K n AM and the collapsible subspace X nf M, there exists an (m - 1)-ball B1 in int T with (X U K) n AM c int B1. Triangulate M so that X, K and B1 are subcomplexes and X goes to a point by admissible simplicial collapses. Let N = N(&M, M") be the closed simplicial neighbourhood of OM in the second derived of the complex M. Then [5, p. 326] there is a homeomorphism h of OM x I onto N such that h(x, 0) = x for all x e OM. (This is what we mean when we say M has a collar.) Let X.= XnN, K. = KnN, X1,= closure (X- X.) and K1,= closure (K - K,). Let C = h(B1 x 1). Whitehead's explicit construction of h is such that XO U Ko c h(int B1 x I).

If (a, z) is a typical simplicial collapse of X to X nf M, and ,1 = a n xl, Z' = n x1, then replacing each (a, z-) by the elementary collapse which removes int a1 U int z-1 gives a collapse X1 U C \ C, and C, as it is a ball, is collapsible. Apply Theorem 2.3 in M to the k-dimensional subspace K1 and the collapsible subspace X1 U C. There exists a (k + 1)-dimensional subspace D in int M containing K, and such that X1 U C U D \ 0. (If we now took a second derived neighbourhood of X1 U C U D, we would get the m-ball in int M men- tioned in the sketch proof.) Choose a point x e int B1 and let E be h(x x I). If we move D into general position rel X1 U C U K1, it will not hit E, except pos- sibly at h(x x 1). Thus

G = X1 U C U D U E\ E\ x.

We have, above, chosen a triangulation of M and made a barycentric second derived subdivision to get M". Let aM" be a further subdivision such that G is a subcomplex. Then, by Theorem 2.1, AO = N(G, (aM")") is an m-ball, and Bo = AO n O&M, being N(x, (aM")"), is an (m - 1)-ball in int B. (In the sketch proof AO is the ball in int M with the little tube to the boundary added.) Ao is a topological neighbourhood of G, and G contains E h(x x I) and C - h(B1 x 1). Choose an (m - 1)-ball B3 in int Bo and a number t, 0 < t < 1, such that

h((B3 x I) U (B1 x [t,1])) c int AO U int Bo.

Recall that KO U Xoczh(int B1 x I). Choose an (m - 1)-ball B2 in int B1 such that B3 c int B2 and Ko U XO c h(int B2 x I). Now apply Lemma 4.2 with B1, B2, B3 as here chosen. Let k: B1 x I - B1 x I be the homeomorphism of that lemma. Define a homeomorphism j of M onto itself by j I h(B1 x I) = hkh-1, j = 1 else- where. Then the pair A, B that we are looking for is j(Ao), j(Bo) (see Figure 5).



10 M. C. IRWIN

0 ~A'

BB, B0,B B

Figure 5

5. Further embedding theorems

LEMMA 5.1. Let A be an m-ball, and B be an (m - 1)-ball in OA. Let C be a q-ball, m ? q - 2, and f: OA C be an embedding, taking int B and A- int B into int C and SC respectively. Then we may extend f to an

embedding g: A )C. PROOF. Zeeman has proved [10] that ball-pairs with codimension ?3 are

unknotted. Let h be a homeomorphism of the (q, m - 1)-ball-pair (C, f(B)) onto a standard pair (,q-m+lAm-l, Am-l). Join hf(A - int B) to a point of Am-l

and map the resulting m-ball back by h-1. Its image is an m-ball D spanning f(&A) in C. Now let g: Am D be a linear extension of f: OA OD. Then g is the required embedding.

As before we suppose that M and Q are respectively an m-manifold and a q-manifold, and that f: M-u Q embeds OM in Q (but no longer necessarily in &Q). We further suppose that R = f-'(&Q) is an (m - 1)-submanifold of OM. Let T = closure (M -R). Then we have the further embedding theorems.

THEOREM 5.2. If M is (2m - q)-connected, T is (2m - q - 1)-connected, Q is (2m - q + 1)-connected and m ? q - 3, and if S(f) c int M U int T, then f is homotopic rel OM to an embedding.

THEOREM 5.3. If M is (2m - q)-connected, T is (3m - 2q + 1)-connected, Q is (2m - q + 1)-connected and m ? q - 3, and if S(f ) c int M, then f is homotopic rel OM to an embedding.

Remarks. 1. If f(M) c int Q, T = OM. 2. Q need not be compact. 3. By using a collaring argument, we may dispense with the condition

that f -1(&Q) c OM. Re-defining R as aM nflf(aQ), we need not even insist that R is a manifold. For the theorems to remain true we must in their state-




ment replace T and int T by OM - R. 4. If q = 2m, Theorem 5.2 needs Q simply connected. As long as f(OM) ? &Q,.

a "piping" argument like that in Lemma 2.7 of [4] produces an embedding even if Q is not simply connected.

PROOF OF THEOREM 5.2. We modify that of Theorem 1.1. We aim at con- structing

(i) an m-ball A in M such that A nf M= B is an (m - 1)-ball and S(f ) c int A U int B,

(ii) a q-ball C in Q such that f maps int A U int B, OA-int B, M-A into int C, SC, Q - C, respectively. Once we have these balls we can, by Lemma 5.1, extend f: OA B to an em-- bedding g: A m B, and this, together with g I (M - A) U OA = f gives the required embedding of M in Q (although B contains points of S(f ), it is of course embedded by f; the whole of OM is, by hypothesis).

By Theorem 2.1 and Corollary 2.2, it is sufficient to find an admissibly collapsible subspace D in int M U int T and a collapsible subspace E in int Q with S(f ) c D and f -(E) = D. We move f into general position rel OM so that dim S(f ) ? 2m - q and dim (S(f ) n aM) ? 2m - q - 1, and then, as in Theorem 1.1, we make an inductive hypothesis:

T(i). There exist admissibly collapsible Di and collapsible Ei such that- S(f ) c Di c int M U int T, f (Di) c Ei c int Q. f -1(Ei) - Di has dimension ? 2m - q - i - 1, and (f-1(Ei) - Di) nf M has dimension ? 2m - q - i -2

The proof of P(i) is obvious, given that of ?1(i) in Theorem 1.1. We just use Theorem 4.3 instead of Theorem 2.3 in M.

PROOF OF THEOREM 5.4. This is almost identical with that of Theorem 5.3. As S(f ) c int M, we may use Corollary 2.4 instead of Corollary 4.4 in the first step of the proof of T(i), which is altered to read (f t(Ei) - Di) nf M has dimension ? 3m - 2q - i + 1.

6. Counter-examples

There are counter-examples which show the necessity for some conditions on M and Q in Theorem 1.1. Throughout Br denotes an r-ball and Sr an r- sphere.

Counter-example 1. M is real projective space of dimension m = 2r and. Q is Sq, 2r + 3 ? q < 2.2r. We can map M to a point of Q, but it is well known that there is no embedding of M in Q. M is not (2m - q)-connected.

Counter-example 2. Hudson's non-embedding theorem deals with the case of an insufficiently connected M with boundary [3].

Counter-example 3. The sphere theorem for 4-dimensional manifolds.



12 M. C. IRWIN

would take the form: If Q is an orientable 4-manifold with w2(Q) # 0, then some non-zero element of w2(Q) can be represented by an embedding of S2 in Q. This is false, for we here prove, for each m ? 2, that there exists a 2m- manifold Q with Wm(Q) # 0, such that no non-zero element of Wm(Q) can be represented by an embedding of Sm in Q. Q is not simply connected, so- it does not satisfy the connectivity requirement for Corollary 1.2. I am grateful to D. B. A. Epstein for suggesting this Q to me some time ago as a candidate for such a counter-example.

Consider P = &Im+- x Im. Make the identification

(0, X2S ...

X2m+1)

(1, Xm+2, ***, X2m+l, X2, ***, Xmi) , m odd, (1, Xm+3, Xm+2, Xm+4, * * X2m+1, X2, * * Xm+l) , m even.

Let Q be the resulting orientable 2m-manifold, and i: P Q be the identification map. Essentially Q is the neighbourhood in R2m of an m-sphere which cuts itself orthogonally in one point. Q is not simply connected, a generator of its fundamental group being represented, for instance, by the straight line path

i(OY 2,9 i, . .

.*, 21) i(O9 0. 21, . . .,

2) M 9l 0 , 2, *-- *2

2 .9,1,1 *** 21) =i(09 2, 1, 2 * , 2)

Let Q be the universal covering space of Q and p: Q > Q be the projection. Then Q is the image of P*, a countable number of disjoint copies Pr of P, under the identification

(0, X2, . . *, X2m+1)r

= (1, Xm+2, ***, X2m+1, X2, *.*, Xm+l)r+i , m odd, = (1, Xm+3, Xm+2, Xm+4, . . *, X2m+1, X2, ..., Xm+l)r+l, m even,

for all r, where (x1, ..., X2m.+)r are the coordinates of a point of Pr. Let i*: * P Q be the identification map. Q is not compact.

Ho(Q)- Z Hr(Q) = O0 0 < r < m, and Hm(Q) is a free abelian group on a countable number of generators jr which may be represented by 1r, the spheres i*((x1, X2,

X ,m+, 1, * * , ).2 for all (x1, - - *, xm+i) C &Im+1.

Let j be the homeomorphism of P* onto itself which takes a point of Pr+T to the corresponding point of Pr, for all r. Suppose a is a non-zero element of Zm(Q, a) where a is a point whose counter-image under i is a single point of P, and let S be an m sphere embedded in Q which respresents a. Then p-'(a) is a set {ar} of points, with ar in i *(Pr) and ar =i*ji*-1(ar+1), and p-'(S) is a set {Sr} of disjoint m-spheres, with ar in Sr and Sr = i*ji*-1(Sr+l). Now p induces an isomorphism between Wm(Q, ao) and Wm(Q, a), so SO represents a non-zero element d e 7w(Q, a0). m(Q, ao) -Hm(Q) by the Hurewicz isomorphism, and so SO,




which may be considered as a spherical cycle, represents a non-zero element

f6m G Hm(Q). Let .m = E c=, rcyr where cup c, # 0. The homeomorphism i*ji *-' sends Yr to Yr-, and hence induces an automor-

phism of Hm(Q) which takes -yr to yr-l Therefore Su-,-, represents an element

Er2uv1 Cr+V+1-U7r of Hm(Q). Now the intersection numbers of the ele- ments of Hm(Q) are given by

yr.r+l +1 all r.

brags =0 P s > r +1.

It follows that .m /*m,. = + cvcu # 0. Thus Su-v_1 and SO intersect. But we have already pointed out that they must be disjoint. This is a contradiction.

Counter-example 4. Dehn's lemma for 4-dimensional manifolds would state: If f: B2 - Q4 is such that S(f) c int B2, then there exists an embedding

g: B2 - Q with f(QB2) = g(QB2). It is certainly true if f(OB2) ? OQ, by Remark 4 following Theorem 5.3, so our counter-example must have f(&B2) c &Q.

For m 2, let M = Bm, and Q = S' x B2m-', and f: OM-)&Q be an embedding such that the sphere f(&M) links itself once homologically around the S' of &Q (see Figure 6). We can extend this embedding to a map f: M a Q such that S(f ) c int M (by collaring). Suppose there were an embedding g: M - Q such that g(&M) = f(&M) = S, say. Let B = g(M). Let Q be the universal

covering space of Q and p: Q - Q be the projection. Then Q = R' x B2m-1

where R' is the real line, p-'(S) is a countable set {Sr} of disjoint (m - 1)- spheres in R' x &B2m' such that Sr and Sr+1 are homologically once linked, and

p'1(B) is a set {Br} of disjoint m-balls in Q with Br spanning Sr. Let a, b e be such that B0 U B, c [a, b] x B2m-1 = B2m, a 2m-polyball. Then Lemma 6.1 below applied to B2m gives a contradiction.

Figure 6

LEMMA 6.1. If Sip S2 are homologically once linked s-spheres in &B28+2, then there do not exist disjoint (s + 1)-balls B1, B2 spanning Si, S2 in B28+2.



14 M. C. IRWIN

PROOF. Suppose that such a pair of balls exists. Then, without loss of generality, we may assume their interiors lie in int B28+2 (by collaring). If we glue on an isomorphic copy to B28-2 by corresponding points on the boundary, we get a (2s + 2)-sphere X containing an (s + 1)-sphere S which is the two copies of B2 glued together. S and S, are homologically once linked in E; because S2 and S, are homologically once linked in OB28+2. Therefore B1 cuts S, and hence B2, in at least one point. This is a contradiction.

UNIVERSITY OF LIVERPOOL

REFERENCES

1. A. HAEFLIGER, Differentiable imbeddings, Bull. Amer. Math. Soc., 67 (1961), 109-112. 2. J. F. P. HUDSON and E. C. ZEEMAN, On regular neighbourhoods, Proc. Lond. Math. Soc.,

(3) 14 (1964), 719-745. 3. J. F. P. HUDSON, A non-embedding theorem, Topology, 2 (1963), 123-128. 4. R. PENROSE, J. H. C. WHITEHEAD and E. C. ZEEMAN, Imbedding of manifolds in euclidean

space, Ann. of Math., 73 (1961), 613-623.

5. J. H. C. WHITEHEAD, Simplicial spaces, nuclei and m-groups, Proc. Lond. Math. Soc., 45 (1939), 243-327.

6. , On 2-spheres in 3-manifolds, Bull. Amer. Math. Soc., 64 (1958), 161-166. 7. E. C. ZEEMAN, Knotting manifolds, Bull. Amer. Math. Soc., 67 (1961), 117-119. 8. , Polyhedral n-manifolds, Topology of 3-manifolds and related topics, (Ed. K.

Fort), Prentice-Hall (1961). 9. , The Poincare conjecture, n>5, Topology of 3-manifolds and related topics, (Ed.

K. Fort), Prentice-Hall (1961). 10. , Unknotting combinatorial balls, Ann. of Math., 78 (1963), 501-526. 11. , Seminar on combinatorial topology, Institut des Hautes Etudes Scientifiques,

Paris, (mimeographed, 1963).

(Received April 13, 1963) (Revised August 31, 1964)



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