Hr(X1) + Hr(X2) - Hr(Xl n X2) <- H,+,(Xi U X2),

Transcription

1 ~~~~~~~~A' A PROBLEM OF BING* BY R. L. WILDER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR Communicated July 8, In a recent paper,' R. H. Bing utilized (and proved) a lemma to the effect that if {Ai; i = 1, 2,..., m} and {Bj; j = 1, 2,..., ti are finite collections of disjoint disks in E3, and p, q are points separated by UA U UBj, then there exist integers h and k, 1. h. m, 1 < k < t, such that Ah U Bk separates p and q. He raised the question whether this remains true if the Ai's and B/s are contractible continua. It is the purpose of the present paper to show that not only is the answer affirmative, but that certain general separation and linking theorems are related thereto. THEOREM In Sn, n > 2, let I{As; i = 1, 2,.. ml and I Bj; j = 1, 2,,.... tj be finite collections of disjoint, nonempty, (n - r - 2)-acyclic closed sets, where r is a fixed integer such that 0. r < n -2. If a cycle Zr links UA i u UBj, then there exist integers h and k such that Zr links Ah U Bk-2 Proof: Since they are finite, we may reduce the collections { A i4, { BjJ so as to be irreducible with respect to the linking hypothesized. After so doing, we suppose there are at least two elements in one of the collections; say, m > 1. Let B = U.= Bj, F1 = A, UB F2 = Um=2Ai U B. Then Zr links neither F1 nor F2. Let K denote a carrier of Zr, and let K1, K2 be closed sets containing K which carry homologies Z, - 0 in Sn - F1 and Sn - F2, respectively. The exact (Mayer-Vietoris) sequence ill Al Hr(K1) + Hr(K2) - H,(K1 n K2) - Hr+l(K1 U K2) shows that some element [Zr+1] of Hr+1(K1 U K2) must be an antecedent of the element [Zr] of H,(K1 n K2) containing Zr. Let Zr+1 E [Zr+1 ] Since r < n - 2 and the sets Bj are (n - r - 2)-acyclic, B is (n - r - 2)-acyclic and Sn- B has a closed subset M containing K1 U K2 which carries a homology Zr+i 0. There exist open sets U1, U2 such that - M n F1 c UaC U1 C Sn- F2 -K1 K2, M n F2 C U2 C U2 C Sn - F1 -K1 A K2 - U1. Let Xi = K1 U (M - Ul) X2 = K2 U (M - U2). Note that XI U X2 = M, and X1nX2 C Sn-(F1 U F2) =Sn-Um Ai U B. In the diagram Hr(K1) + Hr(K2) < Hr(K1 n K2) < Hr+1(K1 U K2) VIIi I i A Hr(X1) + Hr(X2) - Hr(Xl n X2) <- H,+,(Xi U X2), ja' [Zr+±] = Ai[Zr+1] = 0 since i[zr+1] = 0. Consequently, j[z,] = 0, implying Zr 0 in Sn - (F1 U F2), contradicting the fact that Z,+ O in Sn = Um.Ai U B. 683

2 684 MATHEMATICS: R. L. WILDER PROC. N. A. S. COROLLARY In sn, n > 2, let {As} and {Bj} be finite collections of disjoint, closed, (n - 2)-acyclic sets, and p and q points separated by UAi U UBp. Then there exist indices h and k such that Ah U Bk separates p and q. Remark: Since a contractible continuum is acyclic in all dimensions, Corollary 1.1 gives the affirmative, answer to Bing's question. Example 1.1: The necessity for assuming n > 2 is shown by a circle J in S2. Let a, b, c, and d be points in a cyclical order 0 on J, let Al and A2 be the arcs ab and cd, respectively, in the order 0 on J, and B1, B2 the arcs bc, da. These arcs form sets { Ai}, { Bj} satisfying the conditions in the hypothesis of Theorem 1.1 with n = 2 and r = 0, but a 0-cycle links J without linking any Ai U Bj. Example 1.2: The necessity for assuming the sets (n - r - 2)-acyclic is shown by the example in S3 of a cylindrical surface closed at the ends; let the lateral surface be A, and the respective bases B1, B2, and again r = 0. Example 1.3: The necessity for r < n - 2 is shown by the four vertical faces of a cube in S3; let Ai, A2 be two of these faces that are opposite, and B1, B2 the other two. With r = 1, a 1-cycle links U2=1Ai U U =,Bj but no Ah U Bk. Clearly the difficulty with the case r = n - 2 is that since n - r - 2 = 0, the (n - r - 2)- acyclicity does not extend to U=,Bj for t > 1. COROLLARY 1.2. If 5n, n > 1, is the union of two finite collections { A i}, {Bj} of disjoint closed sets which do not separate s5, then for some pair i, j, Sn = Ai U B,. Proof: With Sn C Sn+1, r = 0 and "n" replaced by "n + 1," Corollary 1.1 applies. More generally, one can state: COROLLARY 1.3. If MO, n > 1, is an orientable n-manifold which is the union of two finite collections {As, { Bj} of disjoint closed, (n - 1)-acyclic sets, then for some pair i, j, Mn = Ai U B,. Proof: With Mn C S' for suitable m, some Zm-n-i links Mn; apply Theorem Since the euclidean dualities extend to nonclosed sets,3 the question arises as to whether the above theorems cannot be extended to nonclosed collections. If M is any (closed or not) subset of S", then a compact cycle Z, 0 in M, if for every open set U containing M, Z, - 0 on some compact subset of U.4 And if M is any point set such that for every compact r-cycle Zr of M, Z, 0 - on a compact subset of M, then we call M strongly r-acyclic. Presumably the following lemmas are known, but we include them for lack of specific citations: LEMMA 2.1. Let {Ma} be a collection, indexed by some simply ordered set A, of closed subsets of a compact setm in 5", such that if a, < a2, then Mal, D Ma,, and each of which is linked by some fixed cycle Z, of Sn - M. Then Z, links flama. LEMMA 2.2. If a closed subset M of SO is linked by a cycle Z,, then Z, links some closed subset M' of M irreducibly. If r < n-1, then M' is connected. Since a Z, linking a closed subset M of sn is linked with a compact cycle Zn.-,- of M, application of Lemma 2.2 gives LEMMA 2.3. If a closed subset M of S' is linked by a cycle Z, then there exists a cycle Zn-r-l of M linked with Z, and having an irreducible (closed) carrier in M. LEMMA 2.4. If { Ai} is a countable collection of separated subsets of 5n, r < n-1, and there exists a compact cycle Zn-r-l of UAi = M which is linked by a cycle Z, of S5 - M, then Zr links a compact cycle of some Ai.

3 VOL. 54, 1965 MATHEMATICS: R. L. WILDER 685 Proof: Let K be a compact carrier of Zn-,-, in M. Since Z, links K, it links irreducibly a closed, connected subset K' of K by Lemma 2.2. Then K' c A, for some i. For the sets K' n A are closed, since the Ai are separated, and hence compact, and no continuum is the union of a countable number, at least two, of disjoint nonempty compact sets.5 THEOREM 2.1. In Sn, n > 2, let IAsi; i = 1, 2,.., ml and I{Bj; j = 1, 2,.,t} be finite collections of disjoint, nonempty, separated, strongly (n - r - 2)-acyclic sets, where r is a fixed integer such that 0. r < n - 2. Let Z, be a compact cycle of Sn - (A U B) (where A = Uf,- A, B = U=,Bj) such that Z, d 0 in Sn-(A U B). Then Z, is linked with a compact cycle of some Ai U Bj. Proof: We may suppose that the omission of any Ai or B, would allow Zr 0 in the complement of the remainder of A U B. Suppose, as in the proof of Theorem 1.1, that m > 1. Then Z. - 0 in the complement of each Ai, Bj, and Ai U B,. By duality, Z, is linked with a compact cycle Zn-,-l of A U B, and by Lemma 2.2 we may suppose Zn-,-l is carried by a closed set F which Z. links irreducibly. By Lemma 2.4, F cannot lie in A or in B, since Z, links no Ai or B,. And since Al and U'=2Ai are separated, we can write F - B = [F n (Al - B)] U [F n (U=2Ai- B)] separated. Since Sf is completely normal, there exist disjoint open sets U and V such that U D F fl (Al-B), V Fn (U{-2A1-B). The set F- (U U V) = F1 is closed and F1 c: B; also F' s 0 since F is connected by Lemma 2.2. Note, too, that the sets F, =(FnU)u F', F2 = (FnV) UF1 are closed, with intersection F1 c B. The set B is strongly (n - r - 2)-acyclic inasmuch as the Bj are, and consequently compact (n - r - 2)-cycles of F1 bound on compact subsets of B. This allows a "splitting." Zn -r 'Zn-r- + Z2 n-r- on A U B, where Z,rlisonF, U B C A, U B anda-r lon F2 U B C Un 2Ai U B. Since Zr is linked with Znri, it is linked with either Zn-r- or n -r-1; but in either case we have a contradiction of the fact that Zr links neither Al U B nor UX'=2Ai U B. Definition: A point set A in Sn separates a pair of points p, q of Sn- A if some compact subset of A separates p and q. It is easy to show that for A to separate points p and q of S - A, it is necessary and sufficient that Zo 0 0 in S - A where Zo is the nontrivial 0-cycle carried by p and q. COROLLARY 2.1. In s8, n > 2, if {A 4, { Bj} are finite collections of disjoint, separated, strongly (n - 2)-acyclic sets such that UAi U UBj separates a pair of points p and q, then some Ai U B1 separates p and q. Remark: Applications to the case of orientable manifolds as in Corollaries are obvious. 3. Are there analogues of the above theorems for infinite collections? THEOREM 3.1. Let A = U =,Ai be a countable union of separated sets A1 in Sn, n > 1, and B a closed (n - r - 2)-acyclic subset of Sn, r < n - 1. If Zr is a compact

4 686 MATHEMATICS: R. L. WILDER PROC. N. A. S. cycle of Sn - (A U B) such that Z, # 0 in Sn - (A U B), then Z, is linked with a compact (t - r - 1)-cycle of some A, U B. Proof: Suppose A X, B, Z, as in the hypothesis, but that no A1 U B is linked by ZT. Then Zr links no B and does not link B; and by Lemma 2.4, Z, links no compact cycle of U 7=,(Ai- B) since the sets A -B are separated. By duality, Zr is linked with a compact cycle of A U B whose carrier, F, constitutes a compact subset of A U B linked by Zr. Using the method employed in proving the well-known Brouwer Induction Theorem (and Lemma 2.1), it can be shown that there exists a compact set H U B c F U B such that H C Sn- B and no proper subset H' of H has the property that H' U B is a compact set linked by Z,. For each i, let Hi = H n A i; by our supposition there exist at least two nonempty Hi's. Then there exist indices j, k, j 5 k, points x>, Xk of Hj, Kk, respectively, and disjoint open sets Uj, Uk containing x;, Xk, respectively, and lying in Sn- B, such that no Hi meets both Uj and Uk. For if this were not the case, every point of each Hi would be a limit point of the union of the other Hi's-a situation easily shown impossible by classical methods based on the "Cantor Product Theorem" (as in proving every perfect subset of the real line uncountable, for example). Since Zr cannot link (H U B) - Uj, there exists a chain6 Cr+1 such that cr+l = Zr in Sn- [(H U B) - Uj]. Similarly, there exists a chain Cr±1 such that 6Ck~= Zr ins"- [(H U B) - Uk]. Since B is (n - r - 2)-acyclic, the cycle Ci+1 + Ck+i cannot link B, and consequently there exists a chain Cr+2 such that BCr+2 = Cr+i + Cr in S" - B. Now the intersection of H U B and ICr+211 has a component, L, meeting both Ck and +.7 But Cr-2 is in 8- B, so that L must be a subset of H. And L can meet C11+11 only in Uj, and Ck+JJI only in Uk. Consequently L is a subcontinuum of H which, by the choice of Uj and Uk, cannot lie in a single Hi. But this implies that L is the union of the countable collection (at least two in number) of disjoint closed sets L n Hi, in contradiction to the Sierpinski theorem cited above.5 We conclude that some A, U B must be linked by Zr. COROLLARY 3.1. Let A = U 7= Ai be a countable union of disjoint, closed, (n - r - 2)-acyclic subsets Ai of sn, n > 2, r < n - 2, and B a countable union U7=1Bj of separated, strongly (n - r - 2)-acyclic sets Bj such that B is closed. If Zr is a cycle of Sn - (A U B) such that Zr 6 0 in Sn - (A U B), then Z, links some Ai U B;. Proof: The set B is (n - r - 2)-acyclic since r < n - 2. By Theorem 3.1, Zr links some A1 U B, say A1 U B. But A1 is closed and (n - r - 2)-acyclic, so that we can apply Theorem 3.1 to A1 U B and conclude that Z, links some A1 U B,. Remark: Corollary 3.1 is obviously the generalization of Theorem 1.1 to infinite collections. That the hypothesis of the closure of B in Theorem 3.1 is necessary is shown by the following example: In the coordinate plane, let

5 VOL. 54, 1965 MATHEMATICS: R. L. WILDER 687 M = {(x,y) y = sin, < x- and N={ (xy) Ix = O- 1 < y < 1}. On N, let A, denote the half-ofen interval 0 < y < 1; let A2 denote the open interval -I/2 < y < 0; and for n > 2, let An denote the open interval - - < y < Let B consist of M and (1) all points x,, ); (2) arcs Tn of semicircles having the intervals bxn as diameters, where b = (0, -1) and which meet M U N only at b and x.; (3) an arc from b to (-, 0) lying in the fourth quadrant of E2 and meeting M U N U UT,, only at b and (1, (). Let p and q be points of the fourth quadrant separated by M U N, and Zo the nontrivial 0-cycle carried by p and q. From Corollary 3.1 we can get the analogue for infinite collections of Corollary 1.3 ĊOROLLARY 3.2. If Mn, n > 1, is a closed n-manifoldl which is the union of two countable collections A = I A i4, B = { Bj}, where the elements of A are disjoint, closed, (n - 1)-acyclic sets and the elements of B are separated, strongly (n - 1)-acyclic sets whose union is closed. Then for some pair of integers i, j, Mn = A i U B j. * This research was supported by the Air Force Office of Scientific Research. lbing, R. H., "Approximating surfaces from the side," Ann. Math., 77, (1963). 2 The homology employed in this paper is the modulo 2 theory. For definitions of "linking" and other such terms not defined herein, the reader may refer to Wilder, R. L., "Topology of manifolds," Am. Math. Soc. Colloq. Pub., 32 (1949). 3 See Wilder, R. L., op. cit., Theorem 9.1, p See Wilder, R. L., op. cit., Theorem 4.6, p Sierpinski, W., "Un the*orbme sur les continus," T6hoku Math. J., 13, (1918). 6 Without loss of generality we may assume all chains and cycles (including Zr) used here to be based on subdivisions of S as described in Wilder, R. L., op. cit., chap. II (see especially sections 5.1, 5.2, and 5.12 thereof). 7If C is a chain, then by ljc1 we denote the set of all points in cells of the chain. The existence of L follows from Lemma 1 of my paper "Generalized manifolds in n-space," Ann. Math., 35, (1934). 8 Every closed n-manifold is orientable, modulo 2. One could replace "closed n-manifold" here by "locally orientable generalized closed n-manifold" (see Wilder, R. L., op. cit.); indeed, more general continua could also be included here-irreducible carriers of n-cycles, for example.