On local Betti numbers

ON LOCAL BETTI NUMBERS

BY H. E. VAUGHAN, JR.

1. Introduction. Several types of local Betti numbers have been introducedrecently by Alexandroff and by ech. The local invariants introduced in thispaper were discovered during an attempt to define edge and kernel points of acompact metric space. Incidentally, they give a direct generalization of thenotion of the order, at a point, of a 1-dimensional set.

Section 2 consists of a list of theorems, a knowledge of which is necessary inthe later sections. In 3 the numbers #(a, ), i

_0, are defined for each point

a of a compact metric space , and this definition is illustrated in 4 by exam-ples. In 5 are given several definitions of edge and kernel points which leadto simple necessary conditions that a compact metric space be imbeddable inthe compact euclidean space of the same dimension. 6 is devoted to the deter-mination of the Borel class of the set of all points of M for which the numbers(a, M) satisfy certain inequalities.

In 7 the numbers (a, M) are related to the local connectedness of the setM, and also to that of its complement when M is considered as a subset of aeuclidean space. In order to extend these theorems, certain auxiliary theoremson the addition of irreducible membranes are required, and these are given "in 8.Their immediate consequences are then developed in 9.

There exist in the literature numerous characterizations of the plane, theclosed 2-cell and 2-manifold. The majority of these are purely set-theoretic,excepting certain definitions of Whitney and van Kampen which make use ofmixed methods. We give below, in 10, a characterization of the 2-manifoldin terms of the numbers (a, M). In 11 it is shown that a similar character-ization can be given for the closed 2-cell and, in fact, for any 2-dimensional setobtained from a 2-manifold by the omission of a finite number of open 2-cells.In 12 necessary and sufficient conditions are given that every point of a locallycompact metric space have a neighborhood homeomorphic with a 2-cell, andthese are applied to give characterizations of the open 2-cell (or euclidean plane)and of the class of cylinder-trees. The characterization mentioned in thisparagraph are of a purely combinatorial nature.

Received June 17, 1935; presented to the American Mathematical Society, April 19, 1935.On local properties of closed sets, Annals of Mathematics, vol. 36 (1935), pp. 1-35.Sur les hombres de Betti locaux, Annals of Mathematics, vol. 35 (1934), pp. 678-701.Menger, Kurventheorie, p. 96.van Kampen, On some characterizations of 2-dimensional manifolds, this journal, vol. 1

(1935), p. 87.Zippin, On continuous curves and the Jordan curve theorem, American Journal of Mathe-

matics, vol. 52 (1930), pp. 331-350.

117

118 H. E. VAUGHAN JR.

In 13 some properties of Alexandroff’s local Betti numbers are proved andan inequality is shown to exist between these and the local invariants of thepresent paper.

Section 14 contains a list of unsolved problems.I wish to take this opportunity to express my indebtedness to Professor R. L.

Wilder who has supervised this investigation and who has aided me constantlyby giving many valuable suggestions.

2. Theorems used in the succeeding sections. In this paper R will alwaysdenote the compact n-dimensional euclidean space.

DEFINITION. If A is any metric set and S is any system of open subsets of A,the S-regular part of A is the set of all points of A each of which is contained inarbitrarily small sets of the system S. The S-irregular part of A is the comple-ment with respect to A of the S-regular part.THEOREM A. For every metric set A and every system S of open subsets of

A, the S-regular part of A is a G in A, the S-irregular part of A an F in A.THEOREM B. If M is a compact metric space and p(M) is finite, there exists

an v > 0 such that every complete /-cycle of diameter < bounds on M. IfM is a compact metric space which is locally j-connected, 0 _-< j <_- i, then p(M)is finite.THEOREM C. Let C be the sum and C-1 the intersection of two closed sets

of points, A and B, in Rn. Then every k-cycle Lk,/ < n 1, of R" C whichr.k+l of R B must also bound inbounds a chain r.+ of R A and a chain .."A

R C provided the chains L+ and L+ may be so chosen that r+ + r+bounds in R C-. This is true even for k n 1, unless C-1 is vacuous,

THEOREM D. Let F’, F" be two closed subsets of R such that F’F" carriesa complete r-cycle which fails to bound on F’F" but which bounds on F’ andon F". There exists an (m r 2)-cycle in R" (F’ + F’) which boundsin R F’ and in R F’, but not in R (F + F").THEOREM E. Let F’, F" be two closed subsets of R, and --- a cycle in

R (F’ + F") which bounds in R F’ and in R F" but not in R(F’ + F’). Then FF" carries a complete r-cycle which bounds on F and onF" but not on F’F".THEOREM F. Let M be a compact metric space which is the irreducible car-

rier of an essential complete m-cycle and K a closed subset of M such thatp’-(K) k. Then M K has at most/ + 1 components.THEOREM G. A locally 0-connected compact metric space is homeomorphic

Menger, Kurventheorie, p. 103.Wilder, On locally connected spaces, this journal, vol. 1 (1935), pp. 543-555.Alexander, A proof and extension of the Jordan-Brouwer separation theorem, Trans-

actions of the American Mathematical Society, vol. 23 (1922), p. 342.Alexundroff, Untersuchungen iiber Gestalt und Lage abgeschlossener Mengen beliebiger

Dimension, Annals of Mathematics, vol. 30 (1928), p. 178.10 Wilder, Domains and their boundaries in En, Mathematische Annalen, vol. 109 (1933),

p. 281.

ON LOCAL BETTI NUMBERS 119

with a 2-dimensional manifold if it contains irreducibly a 2-cycle and is separated by each simple closed curve of diameter less than/t > 0.THEOREM H. A necessary and sufficient condition that a locally 0-connected,

locally compact metric continuum be a cylinder-tree is that it be cut by everysimple closed curve but by no arc.11

THEOREM I. A necessary and sufficient condition that a locally 0-connected,locally compact metric continuum S be a cylinder-tree is that it be cyclicallyconnected and, if K is any simple closed curve of S, every point of K is a limitpoint of S K and S K is the sum of precisely two components.1THEOREM J. A necessary and sufficient condition that a locally 0-connected,

locally compact cyclically connected metric continuum be homeomorphic witha subset of a spherical surface is that it do not contain a primitive skew curve.13

3. Definition of fl(a, M). Let M be a compact metric, space, a a point ofM, and/c dima M. There exist arbitrarily small neighborhoods of a whoseboundaries are (/ 1)-dimensional compact metric spaces. For every non-negative integer i and real number e > 0, let/ (a, M) be the smallest integer bsuch that there exists a neighborhood G of a such that ti(G) < e, dim(G G)k 1, and14p(G- G) b. Then fl (a, M) is defined for every e > 0and,as e approaches zero, is a monotone, non-decreasing function. Consequently itapproaches a limit, which is a non-negative integer or o. In case the limit isfinite it is denoted by t(a, M). If the limit is infinite, two cases arise" (1)t (a, M) is finite for all e > 0, in which case/(a, M) ; (2) for sufficientlysmall values of e, t (a, M) o, in which case (a, M) N0. in the definitionof (a, M) a 0-cycle is defined as an even number of points. The coefficientdomain is, of course, arbitrary, but in the present paper it will always be as-sumed finite, i.e., mod m __> 2, for reasons of convergence.

Remarks. I. From the fundamental properties of complete/-cycles it followsthat t(a, M) 0 for i > /c 1.

II. It is immediately evident from the definition that (a, M) is a localtopological invariant of M and, in particular, is independent of any space inwhich M may be considered to be imbedded.

III. Although the hypothesis that the t(a, M) have definite values makes itpossible to choose, for each value of i, an arbitrarily small neighborhood G satis-fying the conditions dim(G G) k 1, p(G G) (a, M), if this numberis finite, or p(G G) finite if f(a, M) o, it is not in general possible, as canbe shown by examples, to choose a single neighborhood G which satisfies these

n Zippin, loc. cit., p. 341.1 Zippin, loc. cit., p. 348.3 Claytor, Topological immersion of peanian continua in a spherical surface, Annals of

Mathematics, vol. 35 (1934), p. 832, and Zippin, On semi-compact spaces, American Journalof Mathematics, vol. 57 (1935),, p. 339.

4 Vietoris, )ber den hSheren Zusammenhang kompalcter Rdume und cine Klasse yon

zusammenhangstreuen Abbildungen, Mathematische Annalen, vol. 97 (1926), pp. 454-472.When we speak of complete cycles we refer to the Vietoris "Fundamentalfolgen".


conditions for all values of i. Of course, by a proper modification of the def-inition, this could be done, but as yet it has not appeared desirable to add thisextra complication.

IV. If the set M is considered as lying in a second space, R, for instance, thedefinition of (a, M) can be given in terms of neighborhoods of a in this space.It has not, however, been shown, and indeed seems unlikely, that similar topo-logical invariants can be defined in terms of any particular class of neighborhoodsin the imbedding space. This is due to the impossibility, in general, of extend-ing a topological transformation. It can be easily shown that definition interms of spherical neighborhoods would not give topological invariants. In thisconnection it is to be noted that while the local invariants introduced by Alexan-drof are defined in terms of spherical neighborhoods, a double limit process isrequired.

4. Examples. 1. For any point a of a compact metric space such thatdimaM 1, we have (a, M) ordaM 1, if the order is finite, (a, M)ordaM, if the order is o or N0, and (a, M) N0, if the order is N0 or c.

#(a, M)= O, if i > 0.2. For any point a of an n-dimensional (combinatorial) manifold, or interior

point of an n-cell, (a, M) 0, (0 _-< i < n 1), and n-l(a, M) 1. For anyboundary point a of a (closed) n-cell (a, M) 0, i _>_ 0.

3. Let M be a set consisting of n 2-cells with a common edge. If a is a pointof this edge, then ri(a, M) 0,/l(a, M) 0 or n 1 according as a is or is notan end point, and B(a, M) 0, if i > 1.

4. Let M be a set consisting of n 2-cells having only an interior point a incommon. Then (a, M) n- 1,l(a,M) nand(a,M) 0, if i> 1.

5. Definition of edge point and kernel point. The usual definitions ofboundary point and interior point of a set M imbedded in R are as follows.The point a is a boundary point of M if every sphere with center at a containsa point of M and a point not of M. The point a is an interior point of M if thereexists sphere with center at a which is entirely contained in M. These def-initions are taken to define edge points and kernel points respectively in the caseof an n-dimensional closed subset of R, and the purpose of this section is to givethem an invariant formulation in terms of the local Betti numbers.

First, if M is an n-dimensional closed subset of R, n 0, and -(a, M) 0,then a is a boundary point of M. For, since M is not 0-dimensional, a is a limitpoint of M, while inside any sphere with center at a there is a neighborhood (inR) whose boundary intersects M in a set whose (n 1)-th Betti number is zeroand hence is a proper subset of this boundary. Consequently, interior to thesphere there is a point of this boundary not belonging to M. This proves thestatement.

Conversely, if a is a boundary point of M, -(a, M) 0. For there exists an

arbitrarily small sphere having a as center whose boundary is not contained in M.These remarks lead to the following definitions, which will be further justified

later.


DEFINITION. The point a is called a k-edge point of the compact metric spaceM if dimaM =/ and k-l(a, M) 0.

DEFINITION. The point a is called a k-kernel point of the compact metricspace M if dimaM /c and/k-l(a, M) > 0.

DEFINITION. The point a is called an ordinary It-kernel point of the compactmetric space M if dimaM /c and fl-(a, M) 1.

DEFINITION. The point a is called a regular k-kernel point of the compactmetric space M if dimaM =/c and (a, M) 0, 0 _-< i < ]c 1, -i(a, M) 1.The preceding discussion shows that a necessary condition that it be possible

to imbed a compact metric space M in R is that n-I(a, M) _-< I for every pointa of M, while for every point such that tn-l(a, M) 1 it is necessary thatt(a, M) 0, 0 _-< i < n- 1. This may be restated in the followingTHEOREM 1. A necessary condition that a compact n-dimensional metric spaceM be imbeddable in R is that every point a satisfying dimaM n be either ann-edge point or a regular n-kernel point.The above condition is naturally not sufficient. In fact, it is not even suf-

ficient for "local imbeddability", as may be shown by the example of a spherewith infinitely many "handles". In this case a point a exists such that no neigh-borhood of a can be imbedded in R2. Moreover, it is possible to construct a2-dimensional set, every point of which is a regular 2-kernel point but whichcontains no open subset which can be imbedded in R2.

Several other definitions of kernel points and edge points have been given,including two by Alexandroff.15 By a result o his15 it follows that every n-dimensional set contains an n-kernel point as defined above.

6. Closure properties of certain sets. It is possible to apply Theorem Ato the solution of this problem. To do so, let S be the class of all open subsetsG of M such that dim(( G) =</c 1, let S be the class of all open subsets Gof M such that dim(G G) =< /c 1, p(( G) _-< p, let Sa be the class of allopen subsets G of M such that dim(( G) =< ]c 1, p(( G) finite. Usingthese for S in Theorem A and letting n dim M,/c =< n, the following resultsare obtained,is

1. The set of points a of M for which

dimaM _-< k is a G

__> Fk Gp, G, Fn

(These relations, due to Menger, ure well known.)15 See footnote 1, p. 27, and Dimensionstheorie, Mathematische Annalen, vol. 106 (1932),

pp. 161-238.1 If A represents a class of sets, the symbol Ap represents the class consisting of those

sets which may be obtained as the difference of two sets of the class A. See Menger,Kurventheorie, p. 105.


2. Those points a of the S.-regular part of M for which dimaM ]c haveti(a, M) =< p(< p + 1), and the S-regular part of M contains only points asuch that dimaM <= k. Those points a of the S-irregular part of M for whichdimaM /c have i(a, M) > p( >= p 1).

3. Those points a of the S3-regular part of M for which dimaM k havet(a, M) finite or w, and the S3-regular part of M contains only points a such thatdimaM =< k. Those points a of the S-irregular part of M for which dimaM khave t(a, M) 0.From these remarks several results may be deduced, of which the following

are the more important.The set of points a of M such that dimaM k and

(a, M)

THEOREM 2.THEOREM 3.THEOREM 4.

a Vpp.

The set of k-edge points of a compact metric space is a Gp.The set of k-kernel points of a compact metric space is a Gpp.The set of ordinary k-kernel points of a compact metric space is

7. Local connectedness. In the examples of 4 we have seen that thelocal Betti numbers give a measure of the ramification of the compact metricspace M. In the present section we give some theorems relating the numbersf(a, M) to the local i-connectedness properties of M, and, in case M is imbeddedin Rn, to the uniform local i-connectedness of R M.

DEFINITION. If a is a point of the compact metric space M such that toevery e > 0 there corresponds t > 0 such that every complete/-cycle carriedby S(a, ) bounds on S(a, ), then M is said to be locally i-connected at the point a.

If M is locally/-connected at each of its points it is said to be locally i-connected.The following theorem shows the relation between local i-connectedness and

the local Betti numbers.THEOREM 5. Let M be a compact metric space and a a point of M such that

(a, M) is finite or oo. Suppose further that one of the following three conditionsis satisfied.

1. pi(M) is finite.2. There exists a real number v 0 such that every complete i-cycle carried by

S(a, v) bounds on M.


3. pi(a, M) 0.17Then M is locally i-connected at the point a.

Proof. By theorem B, condition 1 implies condition 2. Consequently it issufficient to prove the theorem for each of the conditions 2 and 3. The proof ofthe first case follows the lines of the proof of Lemma 3 of the paper cited in foot-note 7 and will not be reproduced here. For the second case, suppose condition3 is satisfied. Let e > 0 be arbitrarily given. There exists an v > 0 such thatevery complete /-cycle on S(a, ) rood [M S(a, e)] bounds on M mod[M S(a, v)]. Let G be a neighborhood of a of diameter < and satisfyingthe conditions dim(G G) dimaM- 1, p(G G) m, finite. Theproof then proceeds as in the preceding case.

Remark. The preceding theorem is true even in the case fl(a, M) 0, ifthe neighborhood G may be chosen so that its diameter is < e and < andsuch that every complete/-cycle on G G bounds in S(a, ).As a corollary to the preceding theorem, we have the following well known

result.COROLLARY. If M is a compact metric continuum and a, is a point of M such

that ordaM is finite or oo, then M is locally O-connected at a.DEFINITION. i domain D of the compact euclidean space R is called uni-

formly locally i-connected (u.l.i-c.) if, for every e > 0, there exists a ti > 0 suchthat every/-cycle in D of diameter ti bounds a chain in D of diameter e.

THEOREM 6. Let K be a closed subset of R’*, D a domain of R K such thatthe following conditions are satisfied.

1. If a is a point of D D, then -(a, K) <__ 1;2. If K cuts R locally at a, one of the local domains is a subset of a domain

D1 of R K distinct from D and such that D Di i8 a subset of a domain com-plementary to some relative neighborhood of a in K.Then D is uniformly locally O-connected.

Proof. Suppose D is not u.l.0-c. There exists a point a of D D and ane > 0 such that, for every a > 0, S(a, a)D contains a 0-cycle which fails tebound in S(a, e)D. Let G be a relative neighborhood of a with respect to Kwhich satisfies 2, is contained in S(a, ) and is such that pn-2(G G) __< 1. Leta > 0 be chosen so that S(a, a)K is contained in G. Let18 x + x be a 0-cycleof S(a, ()D which fails to bound in S(a, e)D. Let y be a point of S(a, z)D1.Then there exist chains L and L in S(a, ) such that L - xi + y0, L --x -[-y in Rn- (K .- F(a, ) e). Let L L --L1. Then L--x -t-x. Since D is connected, there exists a chain Lo in D such thatL Xl +x inR- (. ThenLl-L isa 1-cycle inR- ((- G). If

17 See footnote 1, p. 2.18 For simplicity the proofs of the following theorems are stated in terms of mod 2 topol-

ogy, i.e., the coefficient domain consists of the integers mod 2. They may be modified tohold for any finite coefficient domain. Compare with proofs in R. L. Wilder’s paper, Aconverse of the Jordan-Brouwer separation theorem in three dimensions, Transactions ofthe American Mathematical Society, vol. 32 (1930), p. 635.


L + L 0 in R ( G), it follows from Theorem C that x -- x 0in R (K -- F(a, e)) and, consequently, in S(a, e)D, contradiction.From 2 there exists a chain L in R 0 such that L -- x - y0. ThenL -- L is l-cycle linking G G. For if not, by virtue of Theorem C,x -- y would bound in R- K.

Similarly, L -t- L - L links ( G. Hence, by 1, L -- L -L0 +L --L in R- ((- G) orL --L --L 0. This has been shown tolead to contradiction.THEOnE 7. Let K be a closed subset of Rn, D a domain of R K such that

the following conditions are satisfied, where r is any fixed integer, 1 <-- r <= n 2.1. If a is any point of D D, then -’-(a, K) O.2. If a is any point of D, there exists a relative neighborhood G’ of a

and a real number 0 such that any r-cycle in S(a, ()D bounds in the comple-ment of G’.Then D is uniformly locally r-connected.

Proof. Suppose D is not u.l.r-c. There exists a point a of D D and ane > 0 such that, for every > 0, S(a, a)D contains an r-cycle which fails tobound in S(a, e)D. Let G be a relative neighborhood of a contained in S(a, )and in the G’ of 2, and such that pn-’--(G G) 0. Let > 0 be chosen sothat S(a, a)K is contained in G, and less than the of 2. Let , be an r-cycleof S(a, (r)D. There exists chain L[ in S(a, ) such that L[+ -+ in

+ in R- ( such that L+ -- ,"R (K -- F(a, ) G) and a chain L.inRn- . ThenL[+ --L+ is an (r-l- 1)-cycleof R- (G- G) and, byhypothesis, bounds there. Consequently , bounds in R (K -- F(a, )),a contradiction.

8. Addition theorems. In this section we interpolate some general dditiontheorems which re necessary to the further development of our investigation.D,INTION. A compact metric spce K is sid to be n irreducible membrane

with respect to complete (n 1)-cycle /- if /- 0 on K but on no properclosed subset of K.

DEFINITION. An n-dimensional compact metric spce M is said to be nn-dimensional closed cantorian manifold if p’(M) > 0 while, if M’ is ny properclosed subset of M, pn(M’) O. It is sid to be regularly closed if p’(M) 12THEOREM 8. Let J be an (n 1)-dimensional regularly closed cantorian mani-

fold, K and K. two n-dimensional irreducible membranes with respect to the essen-tial complete (n 1)-cycle carried by J such that KK. J and p’(K) O,i 1, 2. Then K - K is an n-dimensional regularly closed cantorian manifold.

Proof. We may suppose K K imbedded in the compact euclidean spceR2n+. It is then sufficient to show that

1. K - K is linked by an n-cycle in R+ (K K),2. No proper subset of K K hs this property,3. The n-cycle in condition 1 is unique.Proof of 1. We apply Theorem D, setting K F’, K F", 2n 1 m,


n 1 r. Then J F’F", and carries an (n 1)-cycle which bounds onK1 and on K. but not on J, and hence pn(R2’+l (KI -- K2)) > O.

Proof of 2. We apply Theorem E. Let S be any proper closed subset ofK - K2 and set SK1 F’, SK F", 2n 1 m, n 1 r. Thenm r 2 n, and every n-cycle of R2n+ S bounds in R’+ SKI and inR’+ SK, since pn(SKi) pn(Ki) O, i 1, 2, and, consequently, everysuch cycle bounds in R’+ S unless SKK SJ carries an (n 1)-cyclewhich fails to bound on SJ but which bounds on SKi, i 1, 2. In order thatSJ carry a non-bounding (n 1)-cycle, it is necessary that SJ J. In orderthat such a cycle bound on SKi, it is necessary that SKi K. Since S is aproper subset of K -V K., these conditions cannot both be satisfied, and theproof that K nu Ks is a closed cantorian manifold is complete. ThatK nu K. is regularly closed follows from an addition theorem due to Mayer.9

In some cases in which the conditions pn(K) 0 are not known to be satis-fied, the following corollary is useful.COROLLARY. Let J, KI and K satisfy the hypotheses of the preceding theorem

except that pn(Ki) is not required to be zero. Then K 4- K. is the irreduciblecarrier of an essential complete n-cycle.

Proof. This proof is essentially the same as that of the theorem. It is onlynecessary to make use of the fact that the linking cycle given by Theorem Dbounds in the complement of K and hence, part 1 being as before, in part 2 thiscycle bounds in the complement of SK. This leads as before to a contradictionunless S K1 -t- K..The preceding theorem may be generalized as follows.THEOREM 9. Let J be the carrier of a complete (n 1)-cycle which fails to

bound on J, and let K and K. be two n-dimensional irreducible membranes withrespect to this cycle such that K1K. J, p’(K) O, i 1, 2. Furthermore,suppose that K and K. are irreducible membranes with respect to any complete(n 1)-cycle carried by J which fails to bound on J but which bounds on K andon K. Then K nu K. is an n-dimensional closed cantorian manifold. Also,pn(K1 - K2) is the number of (n 1)-cycles of the type described.

Proof. The proof follows the same lines as that of the preceding theorem.That of part 1 may be used as it stands. In the proof of part 2, there is thealternative that SJ may contain an (n 1)-cycle which fails to bound on SJbut which bounds on J, on SK and on SK.. In this case the argument ofpart 1 shows that J SK, which is contained in K, carries a non-boundingn-cycle. This contradicts the assumption that K is n-dimensional and thatpn(K) O. The last part of the theorem follows as before.TM

The corollary of Theorem 8 can be extended in the case of the precedingtheorem.

Monatshefte far Mathematik und Physik, vol. 36 (1929), p. 40. See also Whyburn,Cyclic elements of higher orders, American Journal of Mathematics, vol. 36 (1934), p. 136,footnote.


Combining the above theorem with one due to Alexandroff, we getTHEOREM 10. The necessary and sucient condition that the compact metric

space M be an n-dimensional closed cantorian manifold is as follows.1. M K - K with K, i 1, 2, n-dimensional compact metric spaces such

that p(K) O.2. Every complete (n 1)-cycle carried by KK which fails to bound on KK

but which bounds on K and on K has these sets as irreducible membranes.3. At least one such cycle as described in 2 exists.The lower dimensional connectivities of a closed cantorian manifold con-

sidered as the sum of two irreducible membranes may be found by pplying theMayer addition theorem.s The following special case may also be provedby the use of Theorem C:THEOREM 11. If, in addition to the hypotheses of Theorem 9, p--(KK) 0

and p’-(K) O, i 1, 2, then p-(K K.) O.Proof. To apply TheoremC, letK=A,K--B, 2n+ 1= m,n+r= k.

Thenp++(R’+- KK.) p’--(K1K) O, p+’(R+1 K) pn-’(K) O,and the statement follows.

9. Application of addition theorems. Using Theorem F it is possible toobtain the following extremely useful result.THEOREM 12. Let M be an n-dimensional compact metric space which is the

irreducible carrier of a complete n-cycle which fails to bound on M. If a is a pointof M such that n-l(a, M) is finite or o, then M is locally O-connected at a.

Proof. This follows from the fact that arbitrarily small neighborhoods of amay be chosen whose boundaries have finite (n 1)-dimensional Betti num-bers, and consequently, by Theorem F, separate M into a finite number ofcomponents. That component of such separation which contains the pointa has a diameter at most equal to that of the neighborhood whose boundarydetermines the separation and is itself a connected neighborhood of a. Conse-quently a has arbitrarily small connected neighborhoods and is a point of local0-connectedness of M.The question now arises as to whether or not the local condition in the hy-

pothesis of the preceding theorem is sufficient to insure the same conclusion forother classes of compact metric spaces. The following theorem answers this inthe affirmative, making use of the addition theorems already developed.THEOREM 13. Let M be an n-dimensional compact metric space with pn(M) O,

J a locally O-connected closed subset of M which carries a complete (n 1)-cyclewhich fails to bound on J but with respect to which M is an irreducible membrane,and such that every complete (n 1)-cycle carried by J which fails to bound on Jbut bounds on M has M for an irreducible membrane. Suppose further that if ais any point of M, ’-(a, M) is finite or o. Then M is locally O-connected at eachof its points.

Proof. M is locally 0-connected at all points of M J. Let M’ be a set

See footnote 9, p. 186.


homeomorphic to M nd so situated that MM’ J. Then by Theorem 9M -t- M’ is closed cntorin manifold, t every point a of which, exceptpossibly those of J, -(a, M -t- M’) is finite or . From the previous theoremit follows that M M’ is locally 0-connected at ech such point nd the sameis true of M itself.The space M is locally 0-connected t ech point of the set J. Suppose that

a is point of J t which M is not locally 0-connected. There exists n e > 0such that ny neighborhood of a of diameter less than e has n infinite numberof components. Let 5 > 0 be chosen corresponding to e with respect to thelocal 0-connectedness of J t the point a. Let G be neighborhood of a con-tained in S(a, ) such that dim(G G) n 1 nd p’-(G G) m, finite.There exists n infinite sequence, (g), of components of G. At least m -t- 1of these hve no limit points (nd hence no points) on GJ. For, if all but afinite number hd such points, they might be dded to the component of S(a, e)Jdetermined by a, and it would follow that any two points of M sufficiently nearto a would belong to a connected subset of S(a, ) nd M would be locally0-connected t a. Now let g be one of these m -t- I components. All of its limitpoints in G belong to it, and none of its limit points in G G belong to J. More-over, no point of g is limit point of M g, since such a point would lie inG J nd hence in M J and be a point of non-local 0-connectedness of Min M J. Consequently (( G) separates g from M. If the set M’ ofthe preceding paragraph is gain dded to M, it follows that g is separated bythe same set from M - M’ since, hving no limit points on J MM’, g cnhave none on M’. By Theorem F it follows that pn-(G G) >= m -- 1.The condition that J be itself locMly 0-connected is necessary, s is shown

by the following example. Let M be the compact plane set whose boundary,tken s J, consists of the following three prts" (1) the curve y sin l/x,0 x _-< 1/, (2) the segment x 0, -1 _<- y _-_ 1, (3) the rc (x 1/2) -(y- 1/2)= 1/4r, y _-__ 1/2.

10. Ctiaracterization of the 2-manifold.THEOREM 14. Let M be an n-dimensional compact metric space such that

p(M) m > O, while if M’ is any proper closed subset of M, p’(M’) < m. Leta be a point of M. There exists a positive integer k <= m such that, if G is anysujciently small neighborhood of a, pn(M G) m k, and n-l(a, M) >= k.

Proof. The existence of the number /c follows from the fact that, as thediameter of G decreases, pn(M G) increases, or remains constant, but neverexceeds the value m 1. That n-l(a, M) __> k follows from the Mayer addi-tion th.eorem,is since G G must carry at least k complete (n 1)-cycles whichfail to bound on G G.COROLLARY. Let M be an n-dimensional closed cantorian manifold such that,

for every point a of M,/-l(a, M) -<_ 1. Then M is regularly closed and locallyO-connected.



Proof. If pn(M) m, for every point a of M, t m. But this implies, bythe preceding theorem, that tn-l(a, M) >= m, and consequently, m 1. Thelocal 0-connectedness of M follows from Theorem 12.COROLLARY. Let M be an n-dimensional closed cantorian manifold imbedded

in Rn+l and such that, for every point a of M, ’-l(a, M) <- 1. Then M separatesR’+1 into exactly two uniformly locally O-connected complementary domains, ofwhich it is the common boundary.

Proof. It follows from the preceding corollary and Theorem 6.COROLLARY. Let M be a 2-dimensional locally 1-connected closed cantorian

manifold imbedded in R and such that, for each point a of M, #l(a, M) <- 1. ThenM is a 2-dimensional combinatorial manifold.

Proof. This follows from the preceding corollary and a theorem due toWilder.22

The following theorem shows, as might be expected, that the restriction inthe preceding corollary that M be imbedded in R is unnecessary.

PRINCIPAL THEOREM A. Let M be a 2-dimensional closed cantorian manifold,such that, if a is any point of M, (a, M) <-_ 1. Then M is a 2-dimensional com-binatorial manifold.

Proof. As immediate consequences of the hypotheses and of Theorem 12,it follows that M is locally 0-connected and that, for every point a of M,l(a, M) 1. By Theorem G it is sufficient to show the existence of a realnumber 5 > 0 such that every simple closed curve of M of diameter < 5 cuts M.Assuming that this is false, there exists a point a of M such that, for every realnumber ti > 0, S(a, 5) contains a simple closed curve which fails to cut M.

Let e > 0 be arbitrarily chosen. Then, since M is locally 1-connected,5 > 0 may be chosen in such a manner that every complete 1-cycle carried byS(a, 5) bounds on S(a, ). By hypothesis, S(a, ) contains a simple closed curveJ which fails to cut M. The essential complete 1-cycle carried by J bounds inS(a, ) and there exists a subset K of S(a, ) which is an irreducible membranewith respect to this cycle.

Let p’ be a point of K J, r a point of M K. Since M J is a connectedopen subset of the Peano continuum M, there is an arc a’, with end points p’and r, in M J. Let p be the first point of K, consequently a point of K J,on a’ in the direction from r to p’. Let a denote the subarc - of

Let > 0 be so chosen that < 1/2p(p, J - r). Let G be a neighborhood ofp such that (1) G C S(p, ), (2) dim(G G) 1, (3) p(G G) 1. Let, (,, ,, be an essential complete 2-cycle carried by M, , beingn e-cycle with lim e. 0. By an e-transformation of those vertices of ,which are within a distance e of ( G, we may insure that each ceil ofeither has all of its vertices on (, or none of its vertices on G. Let l denote thesubcomplex of , composed of all cells of the former class. The boundary of

"1l is then an e-cycle on ( G. By making a proper choice of a subsequenceolof the cycles ,’1 it is possible to obtain a complete 1-cycle i (i, .,



Since M is an irreducible carrier of the complete 2-cycle ,2, it follows that G isan irreducible membrane with respect to the complete cycle i1.

Let jl (j, j, be an essential complete 1-cycle on J, where jis an e-cycle, and let k be an e.-chain realizing the homology j 0 irre-

ducibly on K. As before, by an e,-transformation of the vertices of k2, it ispossible to insure that each cell of k2 either has all of its vertices on G or none ofits vertices on G. Let denote the subcomplex of k consisting of all cells ofthe former class. The boundary of k is then an e-cycle -1, onG- G. We

-1again suppose a proper subsequence of the cycles z to be chosen in such a way-1as to form a complete 1-cycle 1 (, z, ).

From (2) it follows that there exists an e-complex, m, on G G such that"I -I --2 -I -Im - on G G. Moreover, - - on M, l -l- m --. on .

Since p is on the carrier of ,2 but not on the carrier of 2 it is a part of the--2 --2carrier of .2 . Then , / l m 0, and this cycle is carried by

M (a p)G, a proper closed subset of M. Since this cycle differs from thenon-bounding complete 2-cycle ,2 only in a small neighborhood of a, it is alsoa non-bounding complete 2-cycle. This, however, contradicts the fact that M isa 2-dimensional closed cantorian manifold. This proves the theorem.COROLLARY. In the hypothesis of the preceding theorem the condition that M be

locally 1-connected may be replaced by any one of the conditions 1, 2 and 3 of Theo-rem 5 (for i 1).

Proof. Since l(a, M) is required to be not greater than 1 for every pointa of M, the hypothesis of Theorem 5 is satisfied and M is locally 1-connected ateach of its points.

Principal Theorem A can be stated in several ways. The following statementbrings out some points of interest.THEOREM 15. Let M be a compact metric space satisfying the following con-

ditions:1. dim M 2,2. p2(M) > O, but, if M’ is any proper closed subset of M, p2(M’) 0,3. pl(M) is finite,4. if a is any point of M, (a, M) <- 1.

Then M is a 2-dimensional combinatorial manifold.Thus we begin with the point set notion of a 2-dimensional compact metric

space, and by subjecting it to certain combinatorial conditions obtain the classof 2-dimensional combinatorial manifolds. Moreover, the only local restric-tion, except for the dimension, is that supplied by the number l(a, M). It isto be noted that if we wish to characterize any particular type of manifold, suchas the sphere, we need only require p(M) to have some particular value, in thiscase zero, and, in some cases, also require orientability or non-orientability.2

11. Characterization of the closed 2-cell.PRINCIPAL THEOREM B. Let M be a 2-dimensional compact metric space

with p2(M) p(M) O, J a simple closed curve contained in M and such that

Veblen, Analysis Situs, 2nd ed., p. 50.


M is an irreducible membrane with respect to some essential complete 1-cycle carriedby J. Suppose also that, if a is a point of M, (a, M) -<- 1, while, in particular,if a is a point of J, (a, M) O. Then M is a closed 2-cell.

Proof. The condition that M be locally l-connected, which ws necessaryin the hypothesis of Principal Theorem A, is here replaced by the stronger andcertainly necessary condition p(M) O.

Since M is n irreducible membrane with respect to n essential completel-cycle crried by J, it is n irreducible membrane with respect to ny suchcycle, since 11 of them re homologous on J.

Let C be 2-cell bounded by J nd such that MC J. From Theorem 8 itfollows that M + C is 2-dimensional closed cantorin mnifold, while fromTheorem 11 it follows that p(M - C) O. Moreover, M - C is evidentlylocally l-connected, nd, if a is ny point of M -- C J, t(a, M -t- C) 1.If this equality can be shown to hold for ech point of J, it will follow fromPrincipM Theorem A that M -t- C is 2-sphere nd, consequently, that M isa closed 2-cell.By Theorem 13, M is locally 0-connected t 11 points so that, if a is point

of J nd e > 0, there exists connected neighborhood G of a such that (G) < e,dim(G G) 1 nd p(G G) O. (All neighborhoods re with respectto M.)

It is first necessary to show that no point of J is local cut point of M. To dothis, suppose that a is point of J which is local cut point of M. Let G beconnected neighborhood of a such that dim(G G) 1, p(G G) 0 ndG a is not connected. Let G be neighborhood of a contained in G, hvingthe sme properties nd also being sufficiently smll so that JG is contained in thecomponent of JG determined by a. It follows that t most two components ofG a hve points in common with the set (G. a)J. Also, if component ofG a hs, s its only limit point on GJ, the point a, the portion of this compo-nent in G is separated from the closed cntorin mnifold M -t- C by a set con-sisting of the point a nd subset of the boundary of G which has no point on J.This subset must then hve positive first Betti number. This contradicts thehypothesis that dim(G G) 1 nd p(G G) 0. Since G is connected,every component of G a must either contain points of (G a)J or hve aas its only limit point. The preceding nMysis therefore shows that G ahs exactly two components determined by the two rcs of GJ a. Theclosures of these components will be denoted by A nd A.As the essential complete l-cycle on J, we my tke cycle / (,,., where / is n e.-cycle, e -+ 0, whose vertices re rrnged in definite

cyclic order on J nd include the point a. Let C be n e-chin on M boundedby ’n. Mke n e-deformtion of the vertices of C so that ny cell of Ceither hs M1 its vertices onA or none of its vertices on AG. This deformationmy be crried out so s not to ffect vertices of , near a. Let C be the sub-chain of C consisting of all cells of the latter whose vertices re on A. The

See footnote 21.


boundary of C consists of cycle which is the sum of two chains, one on G G,the other on J. The carrier of the latter must contain u subrc of J ending ut a,and a carries boundary vertex of this chuin. However, a cannot carryboundary vertex of the other chain since this chain is at positive distuncefrom a. Consequently the sum of the two chains cannot be a cycle and wereach contradiction. Therefore a is not u local cut point of M.

Since this is true, it follows that, if > 0, S(a, ()contains an arc a inS(a, o-) a joining two points p and q of the component of JS(a, ) determinedby a which are separated in this component by the point a. Let e > 0 bechosen arbitrarily and let the above a be so chosen that, if is the arc paq,any complete 1-cycle carried by -}- bounds in S(a, ). That this is possiblefollows from the local 1-connectedness of M. Now suppose a -}- does notseparate M. It is possible to repeat, word for word, the steps in the proof ofTheorem 14, replacing J in that theorem bySo far we have (M - C) (a -t- t) separated, (C being a 2-cell bounded by

J, MC J), one of the components, say M, being such that its closure issubset of M of diameter less than e, which is an irreducible membrane with re-spect to an (any) essential complete 1-cycle on a + ft. Since a - t separatesM from the remainder of M + C, which contains the arc J t, the only limitpoints of M on J are points of t. By Theorem F, (M -t- C) (J -t- a) has,from the connectivity of J, at most three components. One of these is C,another M, and the third, M, is the remainder of M.

Let (), (), (,) be essential complete 1-cycles on J,J, respectively, and such that ,. , -t- ’.-. bounds irreducibly on M, , bounds irreducibly on 2t). Consequently,

their difference, ,, bounds on M. Let K be an irreducible membrane withrespect to , contained in M. Since M is an irreducible membrane with respectto ,, it follows that K must contaia M, since K is closed and since the carrierof the homology ,

If a does not separate M, then M contains interior points of the arc f. Letr be one such point. Let C be a 2-cell bounded by J and such that MC’ J.By Theorem 8, K C’ is a closed cantorian manifold, the hypothesis p(K) 0of the theorem being satisfied, since M, which contains K, is 2-dimensionaland p(M) O.

Let G be any neighborhood of r in M - C’ so small that GC O, and suchthat dim(G G) 1. Then G is a neighborhood of r in M and KG is a neigh-borhood of r in K C’. Consequently, pI(KG KG) >. 0 and p(G G) O.But this contradicts the hypothesis that t(r, M)M-t- (t- p- q) +M.

It follows that if a is a point of J, there exist arbitrarily small neighborhoodsof a in M whose boundaries are arcs having both end points on J. Each ofthese may be extended tosimple closed curve. Consequently (a, M - C) -< 1, and the theorem isproved.


It is interesting, to note that Alexandroff25 raised the question as to whetheror not a locally 0- and 1-connected set M which is an irreducible membranewith respect to a simple closed curve and satisfies the conditions pt(M)p(M) 0 is necessarily a closed 2-cell. This is answered in the negative bythe example shown below.

The same method may be used to give characterizations of sets obtained by omittinga finite number of open 2-cells from 2-manifolds. In general, J will be replacedby a finite number of simple closed curves and pl(M) will be required to havesome non-zero finite value. The local conditions remaining as in Principal

Fro. 1. This figure represents a 2-cell, an interior portion of which has been stretched outinto a wedge-shaped surface and then bent down to make contact with the rest of the 2-cellalong a line, the sharp edge of the wedge coinciding with a portion of the boundary of the2-cell. The configuration evidently stisfies the conditions suggested by Alexandroff andfails to satisfy those of Principal Theorem B only along the T-shaped "locus of singulari-ties". At these points the 1-dimensional local Betti number has one of the values 2 and 3.

Theorem B will insure that when 2-cells or other simple elements bounded bythe simple closed curves are added the conditions of Principal Theorem A willbe satisfied. In some cases it may be necessary to make some hypothesis con-cerning orientability. As an example, we give the followingTHEOREM 16. Let M be a 2-dimensional compact metric space with p(M) O,

pl(M) 1, J a simple closed curve contained in M and such that M is an irreduci-ble membrane with respect to an essential complete 1-cycle carried by J. Supposealso that, if a is a point of M, (a, M) <= 1, and, if a is a point of J, l(a, M) O.Then M is homeomorphic to a Moebius strip.



Proof. Let C be a 2-cell bounded by J such that MC .J. Just as in theprevious theorem it can be shown that M + C is a combinatorial manifold, while,from the Mayer addition theorem, p2(M C) 1. Consequently, M + Cis a projective plane, and M is a Moebius strip.

12. Characterization of the open 2-cell. The following theorem is frequentlyuseful.THEOREM 17. Let M be an n-dimensional locally compact (or compact) metric

continuum such that, if a is any point of M, n-l(a, M) 1, and, infact, if e > O, aneighborhoodGof a exists such that (G) < , dim(G G) n 1, pn-1(G G) 1and G is an irreducible membrane with respect to a non-bounding complete (n 1)-cycle of G G. Then if J is any set which carries a complete (n 1)-cycle whichbounds on a set K such that K J and M (K J) are non-vacuous, M Jis not connected.Remark. It is sucient to assume that the local condition applies only to points

of K-J.Proof. Suppose M J is connected. Let r be a point of M K. Since

J does not cut M, the component of M K containing r has a limit point pon K J. Take e < p(p, J) and let G be a neighborhood of p satisfying thehypothesis of the theorem. Then G is an irreducible membrane with respectto the cycle on G G. However, GK is also an irreducible membrane withrespect to this cycle, since K is an irreducible membrane with respect to thecycle on J. But GK is a proper subset of G, since it contains no points of thecomponent of M K containing r, while G does. This contradiction provesthe theorem.Making use of this theorem we obtain a characterization of sets all points of

which have 2-cell neighborhoods, as follows.PRINCIPAL THEOREM C. Let M be a 2-dimensional locally compact metric

continuum such that, if a is any point of M, (a, M) 1 and, in fact, such that

for every > 0 there exists a (compact) neighborhood G of a of diameter < e suchthat dim(G G) 1, p(G G) 1, p(G) is finite and G is an irreducible mem-brane with respect to a complete cycle on G G. Then every point ofM has a 2-cellneighborhood.

Proof. Let a be a point of M, G1 a neighborhood of a of the type described.By Theorem 5, G1 is locally 1-connected.

Let G be a neighborhood of a of the type described and so small that everycomplete 1-cycle on G bounds on G1. Then Theorem 17 applies for every Jcontained in G, i.e., G1 J is not connected. We will show shortly that M islocally 0-connected. From this it follows that G J is not connected, and, inparticular, that any simple closed curve of G separates G.From the hypotheses on G G it follows that this set contains a regularly

closed 1-dimensional cantorian manifold C.26 Let G’ be an auxiliary set homeo-morphic with G and such that G--’ C. By the corollary to Theorem 8,

26 Wilder, Point sets in three and higher dimensions, Bulletin of the American Mathe-matical Society, vol. 38 (1932), pp. 649-692; see bottom p. 681.


G + G’ is the irreducible carrier of an essential complete 2-cycle. By Theorems12 and F it follows that G is locally 0-connected and is cut by .. arc of G.

Since G is locally 0-connected, locally compact, and cut by every simple closedcurve but by no arc, it follows from Theorem tI that G is a cylinder-tree, and,consequently, that every point of M has a 2-cell neighborhood.

It is evident that, if every point of M has a 2-cell neighborhood, then M satis-fies all the hypotheses of the theorem. These conditions are then necessary andsufficient. In place of the condition "pl(G) is finite" the equivalent hypothesis"M is locally l:connected" might be used.PRINCIPAL THEOREM D. Let M be a 2-dimensional locally compact, non-

compact, metric continuum such that every complete 1-cycle carried by M boundson a compact set on M and such that, if a is any point of M and 0 any realnumber, there exists a (compact) neighborhood G of a of diameter less than suchthat dim(G G) 1, p(G G) 1, and G is an irreducible membrane withrespect to a complete 1-cycle on G G. Then M is an open 2-cell.

Proof. By Theorems 5 and 18 any point of M has a 2-cell neighborhood and,by Theorem 17, every simple closed curve on M cuts M. By Theorem I itremains to show that, if J is a simple closed curve in M, M J has just twocomponents. To do this, we note that, since M is locally 0-connected, everycomponent of M J has every point of J as a limit point. For, if not, thereis a component C of M J and a point p of J which is a limit point of C and anend point of an arc of J, no interior point of which is a limit point of C. LetU be a 2-cell neighborhood of p. Since U is a 2-cell, exactly two componentsof U UJ have p as a limit point and each of these has as limit points all pointsof an arc of J to which p is interior. This contradiction proves the statement inquestion. Now suppose that M J has three components. Each has thepoint p of J as limit point, but we have just seen that in any 2-cell neighborhoodof p there are only two such components.

Principal Theorem C can also be used to give a characterization of cylinder-trees, i.e., subsets of the 2-sphere which are complementary to closed, totallydisconnected sets. All that is necessary is to add the condition that M is im-beddable in R. This is conveniently done by means of Theorem J.

PRINCIPAL THEOREM E. Let M be as described in Theorem 18 and, in addition,contain no primitive strew curve. Then M is a cylinder-tree.

13. Some properties of Alexandroff’s local Betti numbers.THEOREM 18. Let K be a closed subset of R’*, D a domain of R K such that,

if a is any point of D D, p’--(a, K) O. Then D is uniformly locallyi-connected.

Proof. If D is not u.l.i-c., there exists a number e > 0 and a point a of D Dsuch that, if a < e, S(a, ) K contains an/-cycle which does not bound in

S(a, ) K. However, sincep’--(a,K) q(a,R K) O, an ’ <and a ’ e’ and a exist such that every/-cycle in S(a, ’) K bounds in

S(a, ’) K, and, consequently, in S(a, ) K.


As a converse theorem we haveTHEOREM 19. Let K be a closed subset of R’, a a point of K such that only a

finite number of domains of R K have a as a boundary point and each suchdomain is uniformly locally i-connected, i O. Then p’--l(a, K) O.

Proof. Suppose pn-i-(a, K) O. Then qi(a, R K) 0 and for everye > 0 there exists a z > 0 such that, if ’ (, S(a, (’) K contains an/-cyclewhich fails to bound in S(a, ) K. This cycle may be taken as irreducible.Since i 0 we may suppose it is a connected set and therefore contained insome complementary domain of R K. Since there is, by hypothesis,only a finite number of domains having a as a boundary point one of these con-tains arbitrarily small/-cycles of the above type and hence is not locally i-con-nected.

For uniform local 0-connectedness we have the stronger result, supplementingTheorem 18,THEOREM 20. Let K be a closed subset of R’, D a domain of R K such that

if a is any point of D D, it is a boundary point of a finite number, exactly ka,of domains of R" K and p’-(a, K) ka 1. Then D is uniformly locallyO-connected.

Proof. If D is not u.l.0-c., there exists a point a of D D and a numbere > 0 such that, if a <: e, DIS(a, z) K] contains a 0-cycle which does notbound in DIS(a, ) K]. However, there exists an e’ <2 e and a z’ < e’ and

such that there are exactly/Ca 1 0-cycles’in S(a, ’) K which are inde-pendent in S(a, ’) K. But these must consist of pairs of points in differentdomains and, consequently, any 0-cycle of DIS(a, z’) K] bounds inS(a, ’) K and consequently in D[S(a, ) K].As a converse to this theorem we haveTHEOREM 21. Let K be a closed subset of R’, a a point of K which is a boundary

point of a finite number, ka, of domains of R K, each domain being uniformlylocally O-connected. Then p-(a, K) /Ca 1.

Proof. For every e > 0 there exists a z > 0 such that every 0-cycle inS(a, z) which is contained in a single domain of R K bounds in S(a, ).Hence pn-(a, K) <= /Ca 1, and the equality is an obvious conclusion.From these theorems we obtain a characterization of those of R. L. Wilder’s

generalized closed (n 1)-manifolds27 which can be imbedded in Rn.THEOREM 22. The necessary and sucient condition that a closed set M in R

be a generalized closed (n 1)-manifold is1. pn-i(M) 1, while, if M’ is any proper closed subset of M, p’-(M’) O.2. If a is any point of M, p’-i-i(a, M) O, 1 <= i <= n 2, p’-(a, M) 1.DEFINITION. Let M be a closed subset of Rn, N1 a neighborhood in R of

the point a of M. A cycle of NI M is said to be irreducibly linked with Mat a if does not bound in N1 M but bounds in N1 (M G), where Gis an arbitrarily small relative neighborhood of a.

Wilder, Generalized closed manifolds in n-space, Annals of Mathematics, vol. 35 (1934),pp. 876-903.


With respect to the fl’s we have the followingTHEOREM 23. Let M be a closed subset of R and a a point of M at which M is

irreducibly linked by m independent (in N1 M) r-cycles . / .. Then’-’-(a, M) >>- m.

Proof. Replace M by the boulldary of N1 together with the points of Minterior to N. From now on this set will be denoted by M. Let G be nyrelutive (to M) neighborhood of a interior to N and such that the distance ofany point of G from a is less than the distance of the /. from a. Let M’ be therelative boundary of G, G’ M (G + M’). Each of the ,:. bounds in R(G’ + M’) by hypothesis and also in R" (G + M’), since G + M’ is contained

r+l inin a sphere which excludes all the ’i. Therefore each bounds a chain Kr+inR (G’ + M’). ThenK+ +R" (G + M’) and a chain K

is a cycle in R" M’. If any linear combination of these cycles bounds in’s bounds in R" M,R’- M’, the corresponding linear combination of the

by Theorem C, a contradiction which proves the theorem.The following theorem gives a relation between the fl’s and Alexandroff’s

local Betti numbers?THEOnEM 24. If M has no (n r 1)-dimensional condensation at a and

p:--(M) is finite, then -’-(a, M) >= p:-’-(M). If p]--(M) is infinite,’-’-(a, M) oo or o, depending on whether or not the base determining pna-r-i(ican be so chosen that there exists a a > 0 such that all cycles of the base have points inM S(a,

Proof. Suppose that M has no (n r 1)-dimensional condensation at aand p:--i(M) m, finite. Then pa(R M) m. Let 7[" (/,,,...), (i 1, 2,...,m),beabaseata(inR- M). Let e > 0bennyreal number. The sequences ,. may be assumed to be such that every pair ofcycles of the sequence , are homologous in S(a, e) M. Take z so small thatsome cycle of each sequence lies in S(a, e) S(a, a). For simplicity of notationthis may be assumed to be i. Let G be a neighborhood of a in S(a,Let z’ > 0 be chosen so that S(a, r’)M is contained in G. Let ’i,

(i 1, 2, m), be a set of cycles contained in S(a, (r’). Then 0 inR’- (; "r /i 0 in R’- (M + F(a, e) G). These homologies deter-

+ If+ bounds in R (O G) then , boundsmine an (r + 1)-cycle ,in R (M + F(a, e)), or in S(a, e) M, a contradiction. Consequently

+1,.+ links O G, and similar reasoning shows that the m cycles are in-dependent in R (O G). Consequently p-’-(( G) >__ m and /--(a, M) >= m.

If pa-r-l(M) is infinite, -r-2(a, M) or 0, the first case occurring if itis impossible to choose a a > 0 so that some ’k lies outside S(a, ), for all i.

Since p’-r-l(a, M) finite or is a sufficient condition that M have no (n r

1)-dimensional condensation at a, we have theCOROLLARY. If p(a, M) is finite or oo, then -(a, M) >= p(a, M).

28 See footnote 1, pp. 16 and 25.


14. Unsolved problems. 1. It seems reasonable to suppose that many ofthe theorems in the theory of order of points of a 1-dimensional set could beextended to the n-dimensional case in terms of the ’s. However, in most cases,this seems to be very difficult.

2. Another problem is to give sufficient conditions, in terms of the ’s and,probably, local connectedness properties, that a point of an n-dimensional com-pact metric space have a neighborhood which can be imbedded in R.

3. A problem closely related to the preceding is that of extending the char-acterizations of sections 10, 11, and 12 to the corresponding n-dimensional sets.

4. Finally, under what conditions does the equality i-l(a, M) pi(a, M)hold? It seems probable that a partial answer is that it does whenever the latteris finite, but this has yet to be proved.

NIYERSITY OF MICHIGAIo

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