on embeddings of 1-dimensional compacta in a ... - Project Euclid

P AC I F I C JOU R N AL O F M AT H E M AT I C S Vol. 33, N o . 3, 1970

ON EMBEDDINGS OF 1 DIMENSIONAL COMPACTA 4 IN A HYPERPLAN E IN £ J . L.

BRYAN T AN D D . W.

SU M N ERS

In this note a proof of the fallowing theorem is given. THEOREM 1. Suppose that X is a 1 dimensional compactum in a 3 dimensional hyperplane Ez in euclidean 4 space E4, that ε > 0, and that / : X + Ez is an embedding such that d(x,f(x)) < e for each xeX. Then there exists an ε push h of (E4, X) such that h\ X = / .

The proof of Theorem 1 is based on a technique exploited by the first author in [3]. This method requires that one be able to push X off of the 2 skeleton of an arbitrary triangulation of E* using a small push of E\ This could be done very easily if it were possible to push X off of the 1 skeleton of a given triangulation of E 3 via a small push of E\ U nfortunately, this cannot be accomplished unless X has some additional property (such as local contractibility) as de monstrated by the examples of Bo the [2] and McMillan and Row [9]. However, we are able to overcome this difficulty by using a property of twisted spun knots obtained by Zeeman [10]. In the following theorem let B* denote the unit ball in E\ Bz the intersection of B* with the 3 plane x4 = 0, and D2 the intersection of B* with the 2 plane xλ = x2 = 0. 2. Let X be a 1 dimensional compactum in B3 such 2 that XΠ Bd D = 0 . Then there exists an isotopy ht: B* +B* (t e [0,1]) such that ( i ) h0 = identity, (ii) ht I Bd B* = identity for each t e [0, 1], and (iii) hί(X)ΠD2= 0. THEOREM

Proof. Let I = D2 Π B3. Since X does not separate B3, there exists a polygonal arc J in B3 —X joining one endpoint of I to the other. We may assume, by applying an appropriate isotopy of Z?4, that J+ , the intersection of J with the half space x3 ^ 0 is contained in I. Let F be a 3 cell in B3 such that F Π J = J+ and Ff)X = 0 , and let J_ be the intersection of J with the half space x3 ^ 0. Now spin the arc J_ about the plane x3 = x4 = 0, twisting once, so that at time t = π, J_ lies in F. (See Zeeman [10] for the details of this construc tion.) Observe that the boundary of the 2 cell C traced out by J_ is the same as Bd D2. 555

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I t follows from [10, Corollary 2] t h a t t h e pair (B\ C) is equivalent 2 4 to t h e pair {B\ D ) by an isotopy t h a t keeps Bd B fixed. Such an isotopy, of course, will push X off of D2. TH EOREM 3. L et X be a 1 dimensional compactum in a 3 plane 3 4 4 E in E . Then for each 2 complex K in E and each ε > 0, there 4 exists an e push h of (E , X) such that h(X) f] K — 0.

Proof. G iven a 2 complex K and ε > 0, we m ay assume first of 3 all t h a t none of t h e vertices of K lies in E . Also, we m ay move t h e 1 simplexes of K slightly so t h a t th ey do n ot meet X. Let σ be a 2 simplex of K such t h a t σ [\ X Φ 0. By moving X an arbitrarily small amount, keeping it in E\ we can ensure t h a t each component of σ Π X n ot only lies in I n t σ> bu t h as diameter less t h an ε. H ence, we can get σ f] X into a finite number of m utually exclusive line segments I 19 , I n in I n t σ n E 3, each of which having diameter less t h an ε. Let B19 , Bn be a collection of m utually exclusive 4 cells in E 4, each of diameter less th an ε, such t h a t each triple (Bό, Bβ Π E 3, Bj Π o) is equivalent to t h e triple (B4, B3, D2) (as defined above) an d such t h a t Bj Π a Π E z = I jm N ow apply Theorem 2 to each of t h e Bό(j = 1, . . . , r a ) . Suppose that Xc E z c E 4 and f: X —>E 3 are as in the statement of Theorem 1 with d(x,f(x)) < ε for each xeX. Then for 4 each δ > 0 there exists an e push h of (E > X) such that d(h(x)), f(x)) < δ for each x e X. LEM M A.

Proof. Apply t h e proof of Lemma 2 of [3] with p = 2 and q = 1. The proof of Theorem 1 is now obtained by applying t h e technique employed in t h e proof of Theorem 4.4 of [7]. The only additional observation t h a t should be made is t h a t if X is a compactum in E* satisfying t h e conclusion of Theorem 3 and if g is a homeomorphism of E 4, th en g(X) also satisfies t h e conclusion of Theorem 3 with respect to 2 complexes in t h e piece wise linear structure on E 4 induced by g. L et X be a 1 dimensional compactum in a S hyper plane in E . Then for each ε > 0 there exists a neighborhood of X in E 4 that ε collapses to a 1 dimensional polyhedron. COROLLARY. 4

This follows from t h e fact t h a t every 1 dimensional compactum can be embedded in E 3 so as to have th is property in E 3. Bothe [2] and McMillan and Row [9] have examples which show 3 t h a t n ot every embedding of t h e M enger universal curve in E h as

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small neighborhoods with 1 spines. REMARK 1. Notice that Theorem 1 is a consequence of a special case of a theorem of Bing and Kister [1] if X is either a 1 dimensional polyhedron or a O dimensional compactum. If X is a 2 dimensional polyhedron, then Theorem 1 is false in general as pointed out by Gillman [6]. It would be interesting to known for what 2 dimensional compacta Theorem 1 holds. For example, this theorem is true if X is a compact 2 manifold [5].

2. One of the important properties of a compactum X in a hyperplane in E n is that E n X is 1 ALG (see [8]). If n dim X ^ 3, n this is equivalent to saying that E — X is 1 ULC. In [3] and [4] it n is shown that any two such embeddings of X into E (regardless of whether they lie in a hyperplane) are equivalent, provided n ^ 5 and 2 dim X t 2 ^ n. Although there is no hope of improving this theorem by lowering the codimension of the embedding (at least for arbitrary compacta), Theorem 1 lends credence to the conjecture that this result holds when n = 4. REMARK

REFERENCES 1. R. H . Bing and J. M. Kister, Taming complexes in hyperplanes, D uke M ath. J. 3 1 (1964), 491 512. 2. H . G. Bothe, Ein eindimensionales Kompactum im E 3, das sich nicht lagetreu im die Mengersche Universalkurυe einbetten lάsst, F un d. M ath. 5 4 (1964), 251 258. 3. J. L. Bryan t, On embeddings of compacta in euclidean space, P roc. Amer. M ath. Soc. 2 3 (1969), 46 51. 4. , On embeddings of 1 dimensional compacta in E\ D uke M ath. J. (to appear) 4 5. D. G illman, Unknotting 2 manifolds in 3 hyperplanes in E , D uke M ath. J. 33 (1966), 229 245. 6. , The spinning and twisting of a complex in a hyperplane, Ann. of M ath. (2) 85 (1967), 32 41. 7. H . G luck, Embeddings in the trivial range, Ann. of M ath. (2) 8 1 (1965), 195 210. 8. 0. G. H arrold, Jr., Euclidean domains with uniformly abelian local fundamental groups, Tran s. Amer. M ath. Soc. 6 7 (1949), 120 129. 9. D. R. McMillan, J r. and H . Row, Tangled embeddings of one dimensional continua, P roc. Amer. M ath. Soc. 22 (1969), 378 385. 10. E. C. Zeeman, Twisting spun knots, Tran s. Amer. M ath. Soc. 115 (1965), 471 495. Received N ovember 5, 1969. GP 11943. F LORID A STATE U N IVERSITY

This research was supported in part by N SF gran t