AFOSR TN 60-1062 THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Note EXTENSION OF LOCAL AND MEDIAL PROPERTIES TO COMPACTIFICATIONS WITH AN APPLICATION TO CECH MANIFOLDS R. L. Wilder UMRI Project 03000 and 03597 under contract with~ MATHEMATICAL SCIENCES DIRECTORATE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH CONTRACT NOS~ AF 49(638)-104 AsD AF 49(638)-774 WASHINGTON, Do Co administered by: THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR September 1960

To Eduard. echo In Memoriam

ABSTRACT Of central importance in topology and its applications have been the manifolds of various dimensions. In 1936, in a paper published in the Proceedings of the National Academy of Sciences, Eduard Cech proposed for study a type of manifold which embodied a condition theretofore not used, to wit, that every point have a neighborhood whose one-point compactification is an orientable closed manifold. In the present paper, the chief question studied relates to the implication of this condition if applied to arbitrarily small neighborhoods of a point. This necessitates a search for conditions under which a locally compact space, which has a given type of local connectedness, will preserve this under compactifications. Necessary and sufficient conditions are obtained which apply to both the one-point and to the Freudenthal compactifications. In particular, it is found that if a manifold satisfies the Cech condition for arbitrarily small neighborhoods of a point x, then x has arbitrarily small neighborhoods that are r-acyclic (in terms of compact homology) in all dimensions r. The question:. which then obviously arises, whether all manifolds of the type in current use have such neighborhoods, is answered by providing an example of one which does not. iii

INTRODUCTI ON This work was originally inspired by a paper of E. &ech [2] in which he proposed a definition of generalized closed manifold("absolute n-manifold") according to the following procedure: (1) One first defines the concept of an orientable n-dimensional generalized closed manifold; (2) the n-dimensional generalized closed manifold, orientable or nonorientable, is then defined as a compact space in which each point has a neighborhood whose one-point compactification is an orientable n-dimensional generalized closed manifold. In considering this mode of definition, one notes that condition (2) does not state that each point is to have arbitrarily small neighborhoods of the type described, so that in the case of the orientable closed manifolds, the entire manifold may be taken as the required neighborhood. This raises the question whether one could replace (2) by the following- (2.) the n-dimensional generalized closed manifold is a compact space in which each point has arbitrarily small neighborhoods whose one-point compactifications are orientable n-dimensional generalized closed manifolds. Now manifolds are locally connected in all dimensions and simple examples show that the one-point compactifications of locally connected, locally compact spaces are not generally locally connectedo For example, the subspace of the coordinate plane constituted by the set of points ((x,y) x a positive integer, y 2 O}((x,O) x > 0} is a connected, 1-le space, but its one-point compactification is not 1-lc. We shall show that the requirement in condition (2') would imply the existence for every point of arbitrarily small neighborhoods that are r-acyclic for all r (in terms of homology with compact carriers)o Since it is well known that spaces which are lcn, n > 0, do not generally have such acyclic neighbrohoods, the question arises whether manifolds must possess 1

themo We give an example of a manifold in which such neighborhoods do not exist for a certain point. Consequently, since the co-struction given can. yield a manifold either orientable or -nonorientable, to use conditi.on (22) would imply an inconsistency (more precisely, the orientable case Jis defined icn (1) without imposition of acyclicity on neighborhoods, while (2') would impose it). This brings out the fact that the difference noted between (2) and (2!) is quite essential. We shall begin with an investigation of conditions under whicvh local, and related medial, properties of a locally compact space extend to compactifications thereofo In particular, irn Jl we find corditions on a space which enslre that local connectedness properties extend to certain types of compactifications, such as the one-point and Freudenthal "end" compactifications.1 In ~2, analogous problems concerning medial properties are treated; such medial properties have been systematically discussed in [10]o And, of importance for the study of manifolds, conditios obtained which ensure that the local Betti numbers pr(x) shall be - (D in the compactifications. In 53 some applications are i:dicated for continuous mappings and in h4 applications are made to the matters discussed above We shall take p = 0 in &ech s definition of "n-manifold of rank p'y [2], since it is in this form that the resulting manifolds become a subclass of the generalized manifolds currently employed under a number of equivalent definitions (see for instance [6; ViII]) including Cech's earlier defin.itions (see references in [2]). For purposes of the present paper only, we designate the former by the term "Cech manifolds"3and as for the latter, we use the symbol "n-gm" to denote "n-dimensional generalized manifold" and "n-gem" to denote "n-dimensional generalized closed manifold. " For the classical type of local connectedness. this was treated by L. Zippin [11|]. 2

1 EXTEINSION OF LOCAL CONMECTEDNESS PROPERTIES TO COMPACTIFICATIONS Since the case of the common one-point compactification is so simple, we dispose of it separately, saving generalization for subsequent treatment. We employ Cech homology and cohomology with coefficients in an arbitrary algebraic field; the ordinary homology and cohomology groups are indicated by use of the capital H-thus "Hn(X)" denotes homology group of X. Since we make such frequent use of the "compact" groups (based on compact carriers of chains and cycles), we indicate these by the lower case "h"-as in "hn(X)."T By "Pr(X)" and "pr(X)" we denote the dimensions of hn(X) and hn(X), respectively. Lemma. If X is compact and M a closed subset of X such that both p (X) and p n(M) are finite, n > 0, then pn(X;M) is finite. Proof. Immediate consequence of the exactness of the homology sequence of the compact pair X, Mo Corollary 1.1o If X is compact and lc, n > 0, and M is a closed subset of X such that pnl(M) is finite, then pn(X,M) is finite. Proof. Immediate consequence of the complex-like character of X (see [6; 180]) and the lemma. Corollary 1.2. If X can be imbedded as an open subset of an lcn compact space S, n > O. so that S - X is complex-like in dimensions 0 to n - 1, then X is complex-like in its compact cohomology in dimensions 1 to n. Proof. We recall p(X,M) = pr(X - M) for a compact pair X, M. Remark. That p (X) is not necessarily finite under the hypothesis of Corollary 1.2 is shown by the example in E2 of X = ) Xi where Xi = ((x,y) x = 1/i, 0 _ y < oo}, and S is the one-point compactification of X. A Theorem 1.1. In order that the one-point compactification X of a connected and lcn space X should be lcn, it is necessary and sufficient that X be complex-like in compact cohomology in dimensions 1 to n...............,._..3

Proof of necessityo A consequence of Corollary 1lo2 Proof of sufficiency. Since Hr(s) = 0 for all r(where p = X ~HrX) AA A Hr(X p) hr(X) for r =,.., n; and pr(X) is finite since hr(X) is of finite 2 A dimension. Then by Theorem 4 of [5 ] if X were not lcn, it would fail to be le at a nondegenerate set of pointsO If by "IlcO" we denote possession of the r - Ic property for all r, then we can state a similar theorem for lcm spaces. A Theorem 1.2. In order that the one-point compactification X of the con00 X nected and lc space X should be ic, it iJs necessary and sufficient that the compact cohomology groups of X 'be finitely generated in all dimensions greater than 0. Proof of necessity. As aboveo Proof of sufficiency. The proof of Theorem 4 of [5] can be applied to show that the property of being lc0 is expansive (opo cit.) relative to the class of compact spaces that are complex-like in all dimensions. The proof of Theorem 1ol is then adaptable to the present theorem. 2 Although the results of [5] were stated only for metric spaces, their extension to the nonmetric cases presents no difficultyo 4

2. RELATIONS OF MDIAL PROPERTIES OF A SPACE TO ITS COMACTIFICATIONS AND OTHER TYPES OF EXTENSION We recall (see [8]) that a subset M of a space X is said to haVe property (P, Q)r if for every canonical pair of open sets P, Q (Io.e, Q is compact and P Q), the group hr(M^Q4IMP) is finitely generatedo (By hr(UIV) we denote the image of hrU) under the inclusion mapping U — V). Property (P, )r is similarly defined in terms of cohomologyo Of equal interest are medial properties defined in terms of bounding (or cobounding): Thus a subset M of X Ias property (P., QPr if for every canonical pair P, Q of open subsets of X, the image of i. in the sequence of homomorphisms hr+l(M, MnQ) -- hr(M Q) -- hr(MgP) where i is induced by inclusion and a by the boundary operator, is finitely generated. The corresponding cohomology property is denoted by (Pg Qpr) It is clear from their definitions that these medial properties, as applied to subsets of a space, are positional or relative in character, inasmuch as the sets P and Q are taken as open in the "parent space." However, as applied to a space X and its topological images, they are topological invariants, since here the sets P and Q are open relative to X (or its images)o Consequently in discussions where the medial properties of a space X and those of its compactifications are concerned, it becomes necessary to distinguish between those which are relative to X itself (and hence topological) and those properties of X which are relative to the compactifications (and hence positional); we shall call the former intrinsic and the latter extrinsic. The following example will make this distinction clearer~ 5

Example o In E2, let A = (xy) 0 < x 1/jr, y = sin (l/x)}, B = (x,y) x = 0, -1 < y - 1), and let C be an arc joining (1/7, O) and (o,-i) in the fourth quadrant of E2 but not meeting A UB otherwise. Let X be the bounded domain having AuB uC as bounldary, and = X (each with the subspace topology induced by the topology of E2)o Then X, as a subspace of X, does not have property (P, Q)o extrinsically; however, it does have property (P, Q)o intrinsically, since X is homeomorphic with the open circular disk bounded by x2 + 2 y =1. A Remark. Clearly if X, as a subset of a space X, has one of the medial properties defined above extrinsically, then it has it intrinsically. To indicate these medial properties over a range of dimensions k to n inclusive, k < n, we use pairs of indices; thus k(P Q)n" indicates property (P, Q)r for r = k, k + 1,...,n. Since many of our conclusions hinge upon certain groups being finitely generated, we shall use the abbreviation "f.g." to denote "finitely generatedo" "Image of f" will be abbreviated to "Im f" and "Kernel of f" to "Kern f." Theorem 2.1. Let X be a locally compact space, T a closed, totally disA A connected subset of X, and X = X - T. If X has property (P, Q,^)n intrinA A sically, then X has property (P, Q,'J)no Conversely, if X has property (P, Q, v)n), then X has property (P, Q?')n both extrinsically and intrinsically. A Proofo Let P, Q be a canonical pair of open subsets of X; we may assume A A A that P is comt P Q U en T is compact.closed., and. X - P and Q are disjoint closed subsets of. We assert that there exists an open subset R of X such that (1) P)R)Q and (2) TFF(R) = 0~ A/\~ A To see this, we note that since T is a locally compact subspace of X and A / A\ A Q is a compact subset of T there exists a decomposition T = Ti /T2 separate, Awhe- P e A where Tl:Q and T2X - P. (See [6; 100, Th. 1.3]). For each xT_1 there 6

A - - A A exists an open subset Ux of X such that UxCP and Uxn T2 =. As T, is compact, A a finite number of such sets Ux covers T1 and their union, R, is a set of the type desired. Since T is closed, there exist open sets P1 and Qi such that (1) P P1i R DQ1 XQ and (2) Tn (P1 - Q1) = 0; and open sets U1 and V1 such that (1) U1 F(P1), Vi=F(Q1), (2) T(NUUUV1) = 0 (3) Ui1P - R VlCR - Q. Now suppose Q contains an infinite collection (Zn} of compact n-cycles A that are lirh in P and bound on X. Then there exists, for each i, a cycle i - -ri i i i Zn+l mod QUF(P1) on P such that Zn+ = Zn - 7n, where 7n is a cycle on F(Pi)o ~~~~~i -17 i The portion of Zn+l on P1 - Q1 is a relative cycle Zn such that Znl = in * p wn, where wn is on F(Q1); and since X has property (P, QY')n intrinsically, there exists a homology Za 7n - Zaiwn-'O in UlU Vlo But the sets U1, VI are i i disjoint, so that this implies a homology Zaiyn JO in U1. But the 7n must be i i lirh in P since the Zn are, so the existence of the cycles Zn must be impossible. To prove the converse, let P, Q be a canonical pair as before, and select R as above. Since X is open, there exists an open set U such that P - Q3U 3 F(R) and UCX. This time we suppose the Zn lie in XOQ and bound in Xo They are therefore homologous in PtnX to cycles 7i on F(R), and using the (P, Q, ^)n property of X, we find that the y are not lirh in U. But as UCX, this implies i the Zn not lirh in P \X. We conclude that X has property (P, Q, ~ extrinsically and hence intrinsically. Corollary 2.1. With X and X as in Theorem 2.1, if X has property (P, Q n intrinsically, then it has property (P, Q"^)n extrinsically. Remark. That Theorem 2.1 fails if "(P, Q)1n" is substituted for "(P, " Q,)n" is shown by the following examples Let X be the space of [6; 541, 5.19], consisting of a denumerable set of circles Cn successively tangent and converging to a point p. Let T consist of p together with a point xn of each Cn, which 7

may as "well be distinct from the poinrts of tangeney with C j1 and C Hno Eere A A the set X = X- T has property (P, Q) ~ even extrinrslcally, yet X does noto This example also shows, incidentally, that property T (P, Q) " car aot be substituted for "(P, Q,) n. l Corollary 2.2. Tf X is a locally compact space having property (P, Q,t), A A and X is a compactification of X such that X - X is a closed, totally disconA A nected subset of X, then X has property (P, Q& )ro Corollary 2o2ao If X is a locally connected, locally compact space having property (P, Q^)n, then the Freudenthal compaetification [5] of X has property (P, Q, -)n A Theorem 2.2 Let X be a compact space and T a closed, totally disconnected A A subset of X such that X = X - T has property (P, QVc)- intrinsically, is (n + 1) - lc, and has finite pn l (X). Then X is (n + 1) - lc and, moreover, has property (P, Q)n+l~ Proof Since Hn+l(X, 'T) hn+1(X) and pn+l(X) is finite, the group Hn+l(X, T) is f.g. And since T is closed and totally disconnected, Hn+l(T) 0. It follows, from the homology sequence of the pair X, T, that Hn+l(X) is fogo And since X is compact, X is semi-(n + 1) - connected. Since X has property (P, Q,-')p and is (n + 1) - lc, X has property (P, Q,')n+l [10; Th III 1]o By Theorem 2ol, X has property (P, Q^)n+l' Hence by [10; Lemma II 1], X has property (P, Q)l and is a fortiori (n + 1) - c. Corollary 2.5. If X is a locally compact, (nT + 1) - lo space having property ( Q ) and X is a compatification of X such that the set T = - X is a elosed, totally disconnected subset of X, then a su~ficient conditionr that X be (n + 1) - Ic is that pn'-(X) be finiteo And if either (1) n > 0 or (2) T is finite, then this condition is necessaryo 8

A Proofo For the recessity, X has property (P, Q~)n by Theorem 2.1 and Pof.FrteneQ n A.,,A together with the fact that X is (n + 1) - lc this implies that pn,(X) is finite [10; Cor. III 2]. Hence if n > 0 or T is finite it will follow from A A \ n+lA the homology sequence of the pair X, T that pn+l(X T) = p (X) is finite. That the necessity fails when n = 0 and T is not.finite is shown by the familiar examples of dendrites having a closed. infinite set of endpoints; denoting such a dendrite by X and the set of endpoints by T, pl X - T) is infiniteo m Corollary 2.4. If X is a locally compact, lC+1 space, n < m s o0, having property (P, Q,')n and X is a compactification of X such that the set T = X - X is a closed3 totally discoriected subset of X, then a sufficient condition that 2X be lcm is that the numbers pr(X), r = n + 1,..., m all be finiteo And if - _,n+leither (1) n > 0 or (2) T is finite, then this condititon is necessary. Proof of sufficiencyo By [10; Th. III 2], X has property n+l(P' Q)m and a fortiori property n+l(P, Q, )m-' Hence by Corollary 2.3, X is lcn+l. m Corollary 2.o5 In order that the one point compactification of a locally compact, lcn+ space X having property (P, Q,^) should be e +l' n < m oo, it is necessary and sufficient that the numbers pr(X) r = n + 1,..., m all be finite. In particular, if X is lem then for X to be lcm it is sufficient that the numbers pr(X), r = 0,, ooo, n, all be finite. Proof For the lcm case, we recall that the 0-lc and (P, Q)0 condition are equivalent. And if p~(X) is finite, X has only finitely many components so that the proof of sufficiency for 0 - reduces to an. appeal to the fact m that no continuum can fail to be 0-lc at one point. That X is also 1c1 follows from Corollary 2.3. That p~(X) is not of necessity finite, in general, is shown. by such an example as that in the Remark following Corollary 1.20 9

A Corollary 2060 In order that the Freudenthal compactification X of a connected, locally connected, lcncl, n < m oo, locally compact space X having property (P, Q')n should be lcn+l it iS necessary and sufficient that the n shounld be l __ numbers pr(X), r = n + 1, o o o m all be finite In particular if X is lcm, then for X to be c1m it is sufficient that the numbers pr(X), n = 1 o o, m all be finite. Let us turn now to the cohomology caseo Here we can expect substantial differences, inasmuch as r-colc at a point x is equivalent to pr(x) = 0, while the range of possible values of pr(x) is infinite. (See [6; 190, 6.6]) On the other hand, in the ease of. homology the corresponding numbers gr(x) have only two possible values, 0 and oo the former corresponding to r-le at x (see [6; 192]). However, corresponding to Theorem 2ol we have~ A Theorem 2.3. Let X be a compact space, T a closed totally disconnected A A/n subset of X, and X = X - To If X has property (P, Q,)n intrinsically, then /\.~~~~~ r X has property (P,,)n Conversely, if X has property (P, Q,)n, then X has property (P, Qn)n both extrinsically and intrinsically. Proof. By the fundamental duality between homology and. cohomology of "(P, Q&,')" properties [10; Th. II 1], if n > 0O X has property (P, Q')n-l intrinsically, so that by Theorem 2.1, X has property (P, Q^)n 1 and, by duality, property (P, Q,/)n When n = 0, and P, Q form a canonical pair, every cobounding 0-cocycle of Q OX is in the same cohomology class of QAX as a 0-cocycle of QnX, so that the (P, Q/s) property of X yields the desired result immediately. Conversely, if X has property (P Q ) intrinsically since X is openo However, to show the property is extrinsic we proceed as in the converse of Theorem 2.1, this time letting U1 and U2 be opensets such that P - ~ U1:DU2aDF(R), XDUi and U2is compact. Then U1, U2 form a canonical pair in t a c i Xn X; and if zi are cobounding cocycles of X in Q. they are cohomologous in XnP 10

to cocycles yr' in U2 which are related, (because of the properties of X) by cohomologies in.UiL, hence n P/IXo A Theorem 204o Let X be a compact space, T a closed, totally disconnected A A n+1 subset of X, and X = X- T If X has property (P, Q,, ) and p+(x)! 03 for all x ~X, n 1, then in order that pn(x) _ c) for x E, it is sufficient that pn(X) be finite. Anrd if n > 1 or T is finite, this condition is also necessary. Proof of sufficier.cy By [10,; Tho VI 2], X has property (P, Q.,) and hence by Theorem 2.53 X has property (P, Q, )n And if pn(X) is finite, it follows from the exactness of the sequence h, (X ) h.3 hn - h)(X) h"ll-x) and the fact that hn(X, X) = 0 for n _ 1, that p"'(X) is finite. Hence by [10; Lemma II 2] X has property (P, Q) and a fortiori that p (x) s o for all x~. Proof of necessity for case n > 1 or T finite~ We are given that X has property (P, Q, )nl, p (x) C for x all x and must show that pn(X) is A n+l finite. Since, by Theorem 2.3, X has property (P, Q,), it follows that has property (P, Q) [10; Tho VI 2 jo Hence pn() is finite. It then follows from the above exact sequence that if n > 1 (in which case hn-l( X) = hn" (T) =0 = hn(1, X) is finite) or T is finite then pn () is finite. Remark. In the sufficiency proof of Theorem 2.4, we also proved~ Theorem 2o4ao If., X and T are as in the hypothesis of Theorem 2.4, then A X has property (P, Q)no The extensions of the preceding two theorems by induction are obvious. In particular, we have~ A Theorem 2o5. Let X be a compact space, T a closed, totally disconnected A A subset of X and X = X - To If k and n are positive integers such that k ' n and X has property (P Q )1 p and X has property (P, P,,), xr(X) (i) f or all x X acd Pr(X) finite for 11

r = k k + 1 o... n then X has property (Pp Q) o Moreover, X is lo That is lac follows from [10; Tho VI 10 o Theorem 2.6. If X ts the compactification of a locally compact space X by the addition of a point set T which is a closed and totally disco.:iected subset of X and X has property (P, Qy) and pr() s t at all x~X, r = k, k + 1,,o., n where 1 - k s n, then for pr(x) to be _-< at all points of X (for the same range of r), it is sufficient that pr(X) be finLiteo And if k > 1 or T is finite, this condition is also necessaryo 12

3 APPLICATIONS TO CONTISJOUS MAPPINGS Generally, the riage of an ic'"n space, n > 0, unx.der a contirnuous mapping is not Ic 9 For i.stancel if on the cirsle C = i(xy)!x y2 = 1) the pointS pcn qn obtained by intersection of C with the line x = (n - 1)/n are identified for n = 1, 2? 35,.., the resu.lting configgrat.ioi~s. C not lc although iS It is therefore of importance to know under what condi..tions a mapping preserves n the lc property (see [8. VIII], for instance). From the theorems of ~2 we can J obtain conditions of this natureo Theorem 3olo Let X be a loally compat spa I sce (n 0), and T a closed, totally disco:mr.ected subset of X such that the groups hr(X- T), r s n, are al finitely generatedo Then the space Y formed by identifying all points n of T is lIce Proof. Denoting by y the point of Y formed by identfication of the points of T, we have hr(Y - y) - h(X - T), r < n, so that the groups hr(Y - y) are finitely generatedo But Y can be considered as the one-point compactification of Y - y, so that Y is lcn by Corollary 2o5. m Theorem 3.2. Let U be an lcn+l open subset, n < m, of a compact space X such that U has property (Pg Q. )n intrinsically and prlU) iis finite for r = n + 1,..., m. Then if f X - Y is a continuous mapping of X onto a locally compact space Y such that fJU is a homeomorphism, f(U)nf(X j L) = 0 and f(X - U) is a closed totally discon nected subset of Y then Y is ley+, Proof The set V = f(U) has property (P.S QY)n inTtrizisically, is Ics 1 and the numbers pr(V), r = n + 1,,.m are all finiteo Accordingly, by Corollary 2 A, Y is Icmo

4. APPLICATIONS TO THE CECH MANIFOLDS We return now to the discussion of the Introduction concerning Cech manifolds Theorem 4o1o Let M be an n-gm and x a point of M having arbitrarily small neighborhoods LU whose one-point compactifications are orientable n-gcmso Then x has arbitrarily small neighborhoods V which are orientable n-gms for which x n-r < < r (V) h (vx) = 1 =r n - 1o Proof Since an n-gm is r-lc for all r, we may litit our attention to Ux's such that hr(UxlM) = 00 Let Ux denote a one-point compactifieation of such a Ux, forming an orientable n-gem by the addition of an ideal point p. We assert that Ux is a Vx of the desired typeo Let Zr be any compact r-cycle of Ux, r n - 1; it is carried by a compact A A.A ^.,/ subset K of Ux - po Let N be a neighborhood of p in Ux such that NtK = Since Ux is an orientable n-gem, it is completely r-avoidable [6; 229] at p for r < n- 2, and locally (n - 1)-avoidable [6; 218] at Po Hence there exist neighborhoods P and Q of p in U such that NaPnPdQ~Q, and such that every r-cycle of F(P) bounds on Ux - Q, r = n - 1: Since hr(UxII) = O, Z^O on M. Hence ZrO0 mod M- Ux, and accordingly -- A ZrO mod P on Ux (we continue to use the same symbols for subsets and eycles of Ux, whether considered as a subset of M or of Ux)) It follows that there exists a cycle Cr on F(P) such that ZrCr on Ux - Po And since CftO on - Q = U - so must ZrO on a compact subset of UO Hence hr(U) = 0; and since U is an orientable n-gm, h (j = 0 by duality [6; 260, Lemma 5ol6]o It is interesting to note that Theorem 4,1 has a converseo Theorem 4,2,~ Let M be an n-gm and x a point of M having a neighborhood U which is an r-acyelic, orientable n-gm, r < n - 1, Then the one-point compacti 15

fication of U is a spherelike n-gcn. A ^A Proof -Let U denote the compactification of U by an ideal point po Consider the exact sequence (1)... hr (U - - h( - h) Since hr(T) [ hn-r(U) [6; 260, Lemma 5.16], and hr, U) hr(p = C, 1 <- r n - 1, it follows from (L) that h r() = 0 Hence by duality, hr() so that hr(U) = 0 for all r < n - 1 r Let P be any neighborhood of p in., and let Zr 1 < r < n - 1, be a cycle Ar <A a. A of U mod U - Po Since hr(U) = 0, there is a cycle Cr of U such that, for some neighborhood Q of, Cr zr Zr mod U - Q And since hr(U) = 0, we have Cre 0 on A A rA 4 U and accordingly Zr, 0 mod U - Q It follows that pa(U) = 00 By Theorem 2.6, p r() co for r = 1,.., n - 1, and by [7; Th. 4], p (P) = ps() Oo It remains to show that pn(p) 1 Since U is orientable, it carries a nonbounding infinite cycle Cn Let P be a neighborhood of p such that U - P I 0, Since by Theorem 1,1, T is lc there is a neighborhood Q of ~ such that QCP and hn- n(QP) = Oo As Cn is a cycle mod Q, its boundary aCn is a cycle of Q, and since aCn# 0 in P, there is an absolute cycle Zn of U such that Cn Zn mod P. Now Zn^ 0 mod U - Q else (since U is n-dimensional) Z = Cn o Q p, impying Cn s carried by the closed proper subset U - Q of U and hence is O.on U, We conclude ^,A >2 that Zn 0 mod U - Q and that pn(p) = 1. Finally, suppose Zn, Zn are cycles A A A of U mod U - P for some neighborhood P of po Since hn-l(U) = 0, they are extendible (as was Z) to cycles n, C2n respectively, of U in such a way that When this paper was completed, we noticed that what is really proved here is that the one point compactifieation of an r-acyclic, orientable n-gmn, r < n - 1, is a spherelike n-gem; and that the latter result has recently been established by F, Raymond in a paper to appear in the Pacific Journ0 Math, Since our proof is evidently quite different from Raymond s, as well as for reasons of completeness, we include it hereo This is the dimension of the Alexandroff group H(U); [7 2]1

i i.Cn~ Zn mod U Q for some neighborhood QCP, i = 1, 2o But U is an orientable '2 mA 2 n-gm, so there must be a relation aC o bC2 mod p, implying that aC= - bC mod n U = Qo W e onvlude that p,(A) < 1 and.9 with the above relation, that pn(p) = l That U is orientable follows from the sequence (i), which gives hr(U) ) hr h (.U) Example of a 3-gem having a point p which does not have arbitrarily small 1-acyclic neighborhoodso Let A denote the solid Alexander horned sphere in S3; ie,., the "wild" sphere of [1] together with its (tame) interioro Let S denote the quotient space resulting from identifying all points of A, and p the point of S corresponding to Ao Then S is an orientable 3-glm of the same homology type as s3 (see [8]). We shall show that p does not have arbitrarily small 1-acyelic neighborhoods, or, which is equivalent, that A does not have arbi-, trarily small 1-acyclic neighborhoods in S3"' Referring to the Alexander construction [1], let E denote the totally disconnected, closed set of "endpoints" needed to complete the "horns", and suppose U is a 1-acyclic neighborhood of A, Define stages of construction of the horned sphere such that. (1) At stage 1, there are just two "interlocked" horns; (2) at stage 2 there are just four new "interlocked" horns, emanating in pairs from the horns of stage 1; w; (n) at stage n there are 2n new "interlocked" horns, etc' Clearly there exists n such that all 2n horns of the nth stage lie in U; moreover, we may assume (see the Figure) that the connecting cylinders On, C n 2n noo. Cn (which do not form part of the horned sphere, of course) all lie in U. 1 1 2 Let C be the connecting cylinder Cn-i of stage n - 1, containing Cn and Cn (see the Figure), The curves J1 and J2, lying as shown on the n'th stage horns 1 2 and running through Cn and Cn lie in U, Consider J1; let Z1 denote its fundamental 1-cyele0 Since U is 1-acyclic, Zl 0O in U, Then Z1 is homologous in CATU to a cycle Z. on F(C)-see [6; 203, 17

1<1513 We may assume that there exists a chain C2n O. U such that Zi<v Zi on IC211I irredueibly, so that K = |2 Zi Z.ll is conrected (see [4; 299, Lemma 5]) However, since Zi 0 on F(G) it follows that K minst meet J2, inasmuch as J alnd J2 are linked~ This implies that the arc A, on the parent horn (see the Figure) can be extended through C over to a simple closed curve Al in U; and similar situations prevail in regard to each of the 2 parent horns of the other horn-pairs of stage nr Now consider the pair A. A4 of closed curves &btained by extensions of Al and A2 as described aboveo As these lie in U, the fundamental I-cyele on AO bounds in U, and we can. proceed as before to show that corresponding to the associated parent horn of the 2 th stage there exists a simple closed curve analogous to J1 and Ai. And this process can be continued back to the first stage. But U, when taken permit bounding of the as a sufficiently close approximation to A, will not curve indicated at the first stage. If D is a chain, then by I|D|i we denote a carrier of Do 18

2 A A, A2 Cn-i CIn Jl C J2 Cn l 7 A \10r~_/ \ H I~nI \ \ \ \ \ \ / I / / I I

REFERENCES [1] Jo W. Alexarnder, An example of a simply connected surface bounding a region which is not simply connected, Proco Nat. Acado Sci., volo 10 [2] E. Tech, On general manifolds, Proc. Nat. Acado Sci,, volo 22 (1936), 110-111. [3 ] Ho Freudenthal, TUber die Enden topologische Raume und Gruppen, Math. Zeit., vol. 33 (19351), 692-7153 [4] R. Lo Wilder, On the properties of domains and their boundaries in E., Math. Ann., vol. 109 (1933), 273-305 3 [5 ], Decompositions of compact metric spaces, Amer. Jour. Math., vol. 6 (1941), 691-697 - [6], "Topology of Manifolds," Amer. Math. Soc. Coll. Pi'b., vol. 32, [7], Some consequences of a method of proof of J. Ho C. Whitehead, Micho th. Jour. volo 4 (19577, 27-31o [8] Monotone mappings of manifolds, Pacific Jouro of Math., vol. 7 (1957), 1519-1528. [9], Monotone mappings of manifolds, II, Mich. Math Jouro, vol. 5 (1958), 19-25. [10], A certain class of topological properties, Bullo Amero Math. Soc., vol. 6E (1960), 205-239. [11] L. Zippin, On semicompact spaces, Amero Jour. Matho, vol, 57 (1935), 327 -341. 21

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