Homeomorphisms Between Topological Manifolds and Analytic Manifolds Author(S): Stewart S

Homeomorphisms Between Topological Manifolds and Analytic Manifolds Author(s): Stewart S. Cairns Source: The Annals of Mathematics, Second Series, Vol. 41, No. 4 (Oct., 1940), pp. 796-808 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1968860 Accessed: 15/12/2010 04:49

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http://www.jstor.org ANNA oFMv ATEMlATQCS Vol. 41, No. 4, October, 1940

HOMEOMORPHISMS BETWEEN TOPOLOGICAL MANIFOLDS AND ANALYTIC MANIFOLDS'

BY STEWARTS. CAIRNS (Received September 25, 1939) 1. Existence of the homeomorphisms. By a topologicalm-manifold, M, (m = 0, 1, ** * ) we mean a connectedtopological space which can be covered with a denumerableset of neighborhoods,each of whichis an m-cell. We will employ,on M, various coordinatesystems, each having an m-cellfor domain and each definedby a homeomorphismbetween its domain and a regionin a euclidean m-space,Em. Consider a set, X, of coordinatesystems whose domains cover M. We will say that M is analytic(in termsof thesystems X) if everytransformation (1.1) vs = vi(x) (i = 1, ... , m) betweentwo of the systems,(x) = (x1, * *, xm) and (v), whose domains over- lap, is analytic with a non-vanishingjacobian. Let (y) = (yi, ***, y.) denote a coordinate system in En. A topological m-manifoldin En will mean a set of points (1.2) M: y = y(P) (i=1,***,n) where (1) p is a variable point on a topological m-manifold,M', and (2) the correspondence(1.2) between M and M' is a homeomorphism. Let X be a set of coordinate systems whose domains cover M'. As p ranges over the domain of any system (x), the functionsyi(p) can be interpretedas functions of (x). If all such functionsare analyticand ifevery functional matrix (Oyj/Oxj) is of rank m on its domain, then M will be called an analyticmanifold in En (in termsof thesystems X and thecorrespondence (1.2)). Consider a point set, S, in En. A k-plane, 7rk (k > 1), througha point p of S will be called transversalto S at p if it makes angles bounded away from zero with the secant lines of some neighborhoodof p on S. Any plane, arks is called transversalto S (in thelarge) if it makes angles bounded away fromzero with all the secant lines of S. We will say that a topological m-manifold,M, in En is in normalposition if it is possible to define,through each point p of M, an (n - m)-plane,ir n-(p), in such a way that (1) 7r-m(p) varies continuouslywith p and (2) 7r`m(p) is transversalto M at p. Suppose the topologicalm-manifold, M, can be subdividedinto the cells of a

I Presented to the American Mathematical Society; March 27, 1937, February 25, 1939, and October 28, 1939. 796 HOMEOMORPHISMS 797 simplicial complex. It can then be mapped by a homeomorphisminto a polyhedralcomplex, Pm, in an En (n > 2m), wherethe faces of Pm are euclidean simplexescorresponding to the cells into whichM is subdivided. THEOREM I. Givena topologicalm-manifold, M, thereexists a setof coordinate systemsin termsof whichM is analytic.with an analyticRiemannian metric, if and onlyif M can be triangulatedso as to have a polyhedralrepresentation, Pm, in normalposition in someEV. This theorem,in so far as the sufficiencyof the conditionis concerned,is a consequenceof the following. THEOREM' IL. Arbitrarilynear any normalposition of Pm, thereexists an analyticmanifold in E", homeomorphicto Pm. Part of this paper is devoted to an investigationof conditionsunder which a polyhedralmanifold Pm can be put into normal position. By showingthis to be always possible when m = 3, we obtain the followingresult. The cases m < 3 can easily be dealt with by knownmethods. THEOREM III. If a topological 3-manifold, M, can be triangulated,then there exists a set of coordinatesystems in termsof whichM is analytic and has an analytic Riemannianmetric.

2. Normal positions and general positions. We now establish the necessity of the conditionsin Theorem I. Any analytic m-manifold,M, has a homeo- morph,M', which is analytic in some euclidean space En (DM, Theorem I). The writerhas shown' that M' can be so triangulatedinto cells (a) that (1) the vertices of each i-cell determinea non-degeneratem-simplex and (2) the totalityof the simplexesso determinedis a polyhedralmanifold, P"', homeomorphicto M' in such a way that correspondingrn-cells have identicalvertices and that the tangent i-plane to M' at any point of a cell, am' of (i) differs arbitrarilylittle in directionfrom the m-planeof the correspondingface of Pm. Now suppose (p, q) are correspondingpoints on (P', M') respectively. If r"tm(p) is the (n - m)-planethrough p parallel to the (n - m)-planenormal to M' at q, and if Pm is a sufficientlyclose approximationto M', then 7r`t(p) is transversal to Pm as required by the definition of normal position. For, since the faces of pm are approximatelytangent to M', the directionsof the secant lines of any neighborhoodon Pmare approximatelythe same as in the case of the correspondingneighborhood on M'. The sufficiencyproof for Theorem I will not be completeuntil ?8. Consider an arbitrarytriangulated topological m-manifold,M. We will assume first4that thereexists an upper bound to the numberof cells in a star

2 Our proof of Theorem II will involve methodsdue to Hassler Whitney. See Dif- ferentiablemanifolds, Annals of Mathematics,vol. 37 (1936),pp. 646-680. This paper will be referredto as DM. 3 Polyhedral approximations to regular loci, Annals of Mathematics, vol. 37 (1936), pp. 409-415. 4 In 99a methodis given which does not involve this hypothesis. 798 STEWAWVls. CAIRNS on M. The assumptionenables us to imbed Pm in an En, for n sufficiently large, so that the verticeson each star of simplexesare linearlyindependent. Pm is then said to be in generalposition. LEMMA. If it is possibleto put Pm into normalposition, then there exists a generalposition which is also a normalposition. PROOF. We commencewith an auxiliaryresult. (A) If a manifold,M, is in normalposition in EY,then it is in normalposition in any En whichcontains E' as a subspace. For, suppose that wr-`(p) in E' is transversalto M at p and that wrn-(p) in En is transversalto E' at p. Then it followsfrom our definitionsthat 7rn`'(p), determinedby ir`"'(p) and ir`-(p) is transversalto M at p. It remainsonly to requirethat 7r'-'(p) be continuousin p on M. We might,for example, use the (n - v)-planenormal to E' at p. Now let pm be in normalposition in E' C Et, n being so large that En can contain pm in general position. Then pm can be broughtinto a general position, *Pm, by arbitrarilysmall displacementsof its vertices. Let barycentric coordinatesbe introducedon the simplexesof pm and, in preciselythe same way, on the simplexesof *P`n. Two points on (Pm, *Pm), respectively,will correspondif theircoordinates are the same. Suppose ir -m(p) is transversal to a certainneighborhood, N(p), on Pm. The directionsof the secant lines of the correspondingneighborhood, N(p*), on *Pm can be made arbitrarilyclose to those of N(p) by suitable restrictionson the displacementscarrying Pm into *pm. Hence it can be arranged that rn-m(p*)11 f7r-m(p) shall be transversal to *Pm at p*, as requiredby the definitionof normal position.

3. Planes transversalto Brouwer stars. The triangulatedmanifold M, or its representationpm will be called a Brouwermanifold5 if the star of each vertexon pm can be mapped into an Em by a piecewiselinear homeomorphism; that is, a homeomorphismwhich is linear on each simplexof the star. THEOREM IV. No Pm can be put into normalposition unless it is a Brouwer manifold. This followsimmediately from the firstsentence in the lemma below. For m > 3, it is unknown6whether every triangulated m-manifold is a Brouwer manifold. We show, in ?8, that this is surelytrue for m = 3. It is obvious form < 3. Let trmsUrns) be planes, of the indicated dimensions,transversal to each otherin the euclidean space En. Consider the plane parallel to wn-mthrough any point p in En. This plane meets rmin a point,p', whichwill be called the T _nm-projectionof p on rm The locus of p' as p ranges over a point set S will

5 Brouwer, Uber Abbildungen von Mannigfaltigkeiten,Mathematische Annalen 71 (1912), pp. 97-115. 6 Since this was written, examples of non-Brouwer triangulated manifolds have been constructed. See Triangulated manifolds which are not Brouwer manifolds, immediately following the present article. HOMEOMORPHISMS 799 be referredto as the 7r -'projectionof S on irm. This same expressionwill be used forthe mappingof S onto irmin whichp and p correspond. The following result is then obvious. (A) The 7rn-m-projectionof S onto7rm is a homeomorphismif rn-m is transversal to S. As a partialconverse, if S is bounded and the 7r m-projection of the closure, 3, of S is a homeomorphism,then 7rn-m is transversalto 9 and henceto S. Let Sk = Sk(5s) be a set of simplexesof dimensions(j, *.. ,Ik) in EY, where (1) Sk is the star of a j-simplex,s, and (2) Sk, regardedas a point set, is a k-cell. We willrefer to Sk as a Brouwerc-star if it has a piecewiselinear homeo- morphin an Ek. LEMMA. If Sk(sj) can be put intonormal position, it is a Brouwerstar. Fur- thermore,every general position of a Brouwerstar is also a normalposition. As a firststep in the proof,we make the followingeasily verifiedstatement. (B) If N is an arbitraryneighborhood on Sk(sj) of a point on s', thenevery secantof Sk(8j) is parallelto a secantof N. It followsat once that, when Sk is in normal position, any (n - k)-plane rn-ktransversal to Sk at a point of s' is also transversalto Sk in the large. n-kofSknk The 7r -projectionof Sk onto any 7rktransversal to 7n-k affordsa piecewise linear homeomorphismas required by the definitionof Brouwer star. This establishesthe firststatement in the lemma. Consider,now, a Brouwerstar, sk, in generalposition in En. By definition, there exists a piecewise linear homeomorphism,A = A(Sk), mapping Sk onto an Ek. Let (PO, , Pi) denote the vertices of Sk, the notation being so assigned that (PO, ... , Pk) are the verticesof some k-cell of Sk. Let (y) = (Yl, . * *, yn) be a rectilinearcoordinate system in En, relative to which Po is the origin and Pi (i = 1, . . ., v) is unit point on the y,-axis. We restrictA so that it will map Sk onto the coordinate (Yi, . , yk)-plane with Pi self-corresponding(i = 0,. ,k). Let Q : (ail, ,ask, O.. ,O) denote the image of Pi (j =k + 1, .I , v) under A, and consider the transformationof coordinates + Xi= Yi asiiyj (i = 1, ...*, k) (3.1) x; = Ys .(.( j = k + 1, . , n).

In terms of the coordinate system (x), Po is still the origin, Pi (i = 1, . , k) is the unit point on the xi-axis, and the followingare the coordinatesof the remainingP's and of the Q's:

Xi O (3.2) Pi:(ail, , aik, 0,. .., 0, = 1, * , 0) (j = k + 1, ) Qi: (ail, **,ask, Oy..*** 0)

Hence, if 7rn-k denote the (Xk+l, .-. , xn)-plane,then A is the 7rn-k-projection onto the plane of (PO, * *, Pk). Therefore, by result (A), 7rn-k is transversal to Sk, and the proofis complete. 800 STEWART S. CAIRNS

4. Spaces of transversalplanes. Let Sk be a Brouwerk-star in generalposition in E'. We will denote withIH(Sk, E') the topological space each of whose pointsis a systemof parallel (v - k)-planes transversalto Sk in E', continuity being definedin terms of direction cosines.7 The followingstatements are easy to verify. (A) If

(4.1) Sk C E' C En then11(Sk, En) is theset of (n - k)-planes in En whichintersect E' in planes of theset H(Sk, E'). [Compare the proof of ?2(A)]. (B) If sko is a subsetof Sk, then

(4.2) ll(Sk, En) C II (Sk, En). We will referto two topologicalspaces, 11 and 22, as f3-equivalent8provided the followingstatement holds: For each j (j = 0, 1, ... ) every (singular or non-singular)j-sphere in :1 bounds a (j + 1)-cell if and only if every j-sphere in 12 also bounds a (j + 1)-cell. LEMMA. Any space II(Sm(sk), En) is ,8-equivalentto a certain space II( m-k (so)I En-k). PROOF. We take so as the barycenterof 5k, and Enk as the (n - k)-plane normal to 5k at so. The star Sm-k = S-k(s?) is defined as the projection of Stm= Sm(sk) onto En-k. We commencewith the followingauxiliary result. (C) Suppose rn-m belongsto ll(Sm, En), and let rk denotethe k-plane of sk. Then theplane 7n-m+k determinedby 7rn-r and 7rkbelongs to 11(Sm-k, En). In the firstplace, we note that the secants of S m-k are a subset of those of sm. With the aid of ?3(B), one can verifythat if 1 is a line in E n whose irk-projection is a secant of Sm-kk then 1 is parallel to a secant of St. Since every line on 7 n-m+k is parallel to irk or else has the same rk-projection as a line on 7rn-m it followsthat the only secants of Sm parallel to Tn-.+k are also parallel to 7rk. Hence no secant of Sm-k is parallel to 7rn-m+k. Since the secants of Smk are a closed set, result (C) followsat once, It also followsthat II(Sm kit En-k) con- sists of the intersectionsof En-k with planes such as ln-nm+k. An (n - m)-plane rn-, on 7rn-m+k belongs to ll(Sm, En) if it containsno line parallel to irk. Any subset of II(S', En) consistingof planes whose angles with k all exceed t9 > 0 can thereforebe homotopicallydeformed in ll(Sm, En) so that each rn-m remainsin a single plane such as rn-m+k and is carriedinto its 7rk-projection on E n-. Now suppose everysphere in ll(Sm, En) bounds a cell. Since 1(S mki En k) is a subset of H(Sm, En), any sphere, fl', in the former space bounds a cell, a,)+1 in the latter. Some deformationof the sort just

7 S. S. Cairns, The direction cosines of a p-space in euclidean n-space, American Mathe- matical Monthly, vol. 39 (1932), pp. 518-523. 8 The stronger condition of complete homology equivalence (cf. Alexandroff-Hopf, Topologie I, 1935) might be established for the spaces we treat. However, we need only ,8-equivalence. HOMEOMORPHISMS 801 describedwill leave 1i fixedand will carry au'l into a (j + 1)-cell in II(S"-m, Ef-k). Hence everysphere in nI(Smk E -k) bounds a cell in [I(S -k, En-k). Suppose converselythat, in fl(S,-k, En-k), every sphere bounds a cell: in otherwords, can be shrunkto a point. An arbitrarysphere in II(S', En) can be deformed,as above, into a spherein 11(S-k, En-k) and then furthershrunk to a point. This completesthe establishmentof the lemma.

5. 13-equivalentspaces of maps. Correspondingto a Brouwerk-star, Sk, we definea space of mappingsA(Sk) as follows. Each point of A(Sk) can be repre- sented by a piecewise linear homeomorphismof Sk into an Ek. Two such homeomorphismsrepresent the same point of A(Sk) if and only if one can be carriedinto the otherby a linear transformationof Ek. Let (P1,i , PO) be the verticesof Sk and (P, *.* , P) theirrespective images under a piecewise linear homeomorphismrepresenting a point, Xo, of A(Sk). A neighborhood, N(Xo), in A(Sk) will correspondas followsto any set of neighborhoods,Ni(P,) (i = i, o,v), in Ek: A point of A(Sk) belongs to N(Xo) if and only if it can be representedby a piecewise linear homeomorphismcarrying Pi into Ni(P') (i = 1, ** , v). Thus A(Sk) is definedas a topological space. LEMMA 5.1. There exists a homeomorphismbetween A(Sk) and II(Sk, Et) provided(1) Sk is in generalposition in E' and (2) no n-planewith n < v contains xk.

PROOF. Let Ek be determinedby the vertices (PO, * . , Pk) of a particular k-simplexof Sk. Since two piecewise linear homeomorphismsof Sk into EBk representthe same point of A(Sk) if they are related by a linear transformation ofEk, we can obtain unique representationsfor the pointsof A(Sk) by stipulating that (PO, ** , Pk) be self-corresponding.Comparing the proofof ?3, Lemma, we see that a homeomorphismbetween I(Ik EB) and A(Sk) is definedif each element irtk of the formerspace be associated with the 7r.k-proj ection of Sk onto Ek. LEMMA 5.2. If En D E', thenII(Sk, En) is 1-equivalent to 11(Sk, E') and hence to A(Sk). This can be proved,on the basis of Lemma 5.1, by reasoningas in the proof of ?4, Lemma. Given Sk = Sk(s), let po be the barycenter of s'. We will denote with AO(Sk) the subspace of A(Sk) consistingof those elementswhich map all the boundary verticesof Sk onto the unit (k - 1)-spherein Ek about the image of po . LEMMA 5.3. The spaces A(Sk) and AO(Sk) are A-equivalent. PRooF. We representall the points of A(Sk) by elementsmapping po into the origin,0, in Ek and mapping the vertices of some particular k-cellinto prescribedimages on the unit (k - 1)-sphereSk' about 0. Let Xodenote any elementof A(Sk), thus restricted,and let qo be the image of any vertex,p, of Sk under Xo. Let q1 denote the intersection of Sk-l with the ray OqO . We then denote with Xt that element of A(Sk) which carries each vertex p into the point qf on the segment qoql such that qoqt = tsqoql . As t increases from 0 to 1, Xt 802 STEWART S. CAIRNS definesa deformationof Xointo Xi. By applyingthis deformationsimultane- ouslyto all the elementsX, we deformthe wholespace A(Sk) intoAO(Sk). Using this deformation,one can completethe proofas in the case of ?4, Lemma.

6. The space of triangulationsT(Tki1). Given a Brouwerstar Sk = Sk( let the pointsof AO(Sk) be representedby homeomorphismscarrying the vertices (s8, P1, ... , Pk) of somek-cell into specifiedimages (0, Q1, ... , Qk) in Ek. Let Zkdenote the image of Sk undersome such homeomorphism,X. By definition of AO(S), the boundary,Bki, of 2k has all its verticeson the unit sphere, Sk-l, about 0. The centralprojection from 0 onto Sk-l maps Bk-i into a geodesic triangulation,-rk1, of sk ; that is, one in whicheach cell appears as a simplex relativeto some local coordinatesystem in whicharcs of great circlesare repre- sented as straightlines. This implies that the closure of each cell of trk is on an open hemisphereof Sk-i, in otherwords that each 1-cellis less than 1800. Let T(rki-) be the followingtopological space. Its points are geodesic tri- angulationsof sk-1 homeomorphicto t kl withQi (i = 1, * , k) self-correspond- - ing. If rT is any such triangulationand Qi (i = 1, io,v) are its vertices, then a neighborhood,N(r*-1), in T(rTk-) correspondsas followsto a givenset of neighborhoodsNj(Qj) (j = k + 1, ***, v) of the points Qj on Ski: N(rTo) consistsof all elementsof T(rk-i) forwhich the vertexcorresponding to Qj lies in N,(Q,). The followinglemma is a direct consequence of our definitions. LEMMA. The spaces T(rk-) and AO(Sk)are homeomorphicunder the correspond- enceinduced by thecentral projection from 0.

7. A sufficientcondition in the normal positionproblem. THEOREM V. A sufficientcondition that it be possibleto put a Brouwerm-manifold into normal po0?tion is that, in every space l(S i(80), E') [or A(Sk) or T(r-i)], every (m - k - 1)-spherebound an (m - k)-cell9 (k = 0,. , m - 1). The proofwill occupy this section and the next. (A) Let v be a positiveconstant less than 1/(m + 1). If s' be any j-simplex, j < m, then the n-core,A', of a' will mean the set of points whereall the barycentriccoordinates for 8' exceed q. In the case j = 0, we have y0 = . We now consideran m-simplex,sm, and defineon it certainneighborhoods, N(y'), where Y' is the n-coreof a typical bounding simplex, s', of Sm. The definitionwill be recurrentin j = 0, ** *, m. To defineN(y0), we choose barycentriccoordinates (uo, ... , ur) on s' so that us = 0, (i = 1, ... , m) at . We then make the definition

(7.1) N(y0): 0 _ ui ? We (i=1, *, m). To defineN(y'), 0 < j < m, choose the barycentriccoordinates so that u= 0

9 For k = 0, the space T(rk1) is vacuous, and the conditionbecomes trivial. HOMEOMORPHISMS 803

(i = j + 1, * , m) oniy'. Assume theN(ye) all defined(k = 0, .*. j-1). We definea regionN'(7y) as follows: (7.2) N'('y): O- ui (i = j + 1, ,m) and then make the definition

(7.3) N(y) = N'(ey') - N(ty),

the summationbeing over the n-coresof all boundingk-simplexes of s' (k < j). Then N(,y) is a closed,box-like, m-dimensional region with yj forone face. The simplexem is covered by ymplus the regionsN('y) (j < m). These regionsare distinct,save forcommon faces. The accompanyingfigure illustrates the definitions when m = 2. Any core y', 0 < j < m, is parallel to a certainbounding simplex, *Iyj of ym. In case j = 0, *by.will denote the vertexof ym nearest y0. Let Om"J-l denote the

off/ N(') \

boundingsimplex of y' opposite *,y. Consider any point, q, on y'. We will denotewith Bm-i(q) the intersectionof N(y') withthe (m - j)-plane determined by q and m-1-l- Then Bm-i(q) is a box-like (m - j)-dimensional point set. As q ranges over the closure of al, the sets Br-'(q) fillout the regionN(y') in continuousone-to-one fashion. (B) If q is on the boundaryof y', then Bmi(q) is commonto the boundaries of N(yj) and some N(Yk) wherek < j. Now let sm be any m-simplexof Pm and s' any boundingsimplex of Sm.with y' denotingits n-core. We will then use the notation

1(yi) = E N~y')

()q) = siB(q) q ony

where S(s') is the star of sj on Pm. Our proofwill consistin assumingthe conditionof the theoremand construct- ing suitable transversalplanes 7rn-(p). We take Pm in general position in E". The method will be recurrent,with the followingbasic hypothesis. 804 STEWART S. CAIRNS

HYPOTHESIS I. For somevalue j of theset (1, * *, m), a plane 7r'-m(p)has been definedso as to vary continuouslywith p on the sum of the neighborhoods 91(7k) (k < j) and so as to be transversalto Pm at p. The initialstep ofthe recurrencyfalls into two parts. We firstselect arbitrary transversalplanes, rn-m(p), at the vertices (s? , s? , ) = ('y 0, . ) [see (A) above] of Pm. This is possible,by ?3, Lemma, since Pm is a Brouwer manifold. We then define7rn-m(p) on the 9(y0) by the requirement

(7.5) Tn-m(p) j1 7ln-m(OY) p on %(y), togetherwith the requirementthat 7rn-m(p) pass throughp. The verification of hypothesisI forj = 1 depends on ?4 (B), to be applied wherek = m, Sk = S(s?), and Sk is the star of any simplexof Pm incidentwith s? .

8. The general step of the recurrency. Hypothesis I would be a sufficient basis forthe proofof TheoremV. The followinghypotheses are used to secure regularityrestrictions which will enable us to prove Theorem II. The initial step in ?7 satisfiesall these hypotheses. HYPOTHESIS II. If pi and P2 are commonto any ?3k(q), then

(8.1) 7n-m(pl) 117rn (p2)

HYPOTHESIS III. Thereexists a functiont -i(P) suchthat, if R(p) denotethe set of pointson 7rn-m(p)within distance t -i(p) of p, thenthe sets R(p) fill out,in one- to-onefashion, a closedneighborhood, R _1, in En of theinner points of the91(,yk) (k < j). HYPOTHESIS IV. If 7rn-m(p')denote the plane of theset 7r'-m(p)through any pointp' of Rj_1, then r7r-m(p'),regarded as a mappingof R3-1 into thespace of all (n - m)-planesin E , is differentiable.10 The generalstep of the recurrencyextends the definitionof 7rn-m(p) over the neighborhoods9(y'). We break this step into two parts. In the firstpart, we extend the definitionover a typical ti-core,yj. In the second part, omitted when" j = m, we extend it over the rest of 91('y'). The definitionof 7r'-m(p)maps s' - yj differentiablyl'-intoII(Sm(sj), En), which is p3-equivalentto some space II(Sm-j(so), En-j) [see ?4, Lemma]. Since the boundaryof yj is a (j - 1)-sphere,the conditionof Theorem V, read with k m - j, impliesthat the mappingof s' - can be extendedover yj. As a resultof Theorem7 in DM, the extensioncan be made so as to give a differenti- ablel0mapping of the whole of sj into II(Sm(Sj), En). By such a mapping,we extendthe definitionof 7r-m(p) over yj. If, now, q is any point of y', and p is any point of 13m(q), 7r'-m(p)will mean the (n - m)-planethrough p parallel to 7r`m(q). This completesthe definition of 7rn-,(p) on the neighborhoods %(,Y).

10 This term has meaning here, because nG(Sm(si), En) is a subspace of the space of all m-planes in En, and this latter space can be interpreted as an analytic manifold in some euclidean space (cf DM, ?24). 11The proof of the theorem is completed with the firstpart of the step j = m. HOMEOMORPHISMS 805

The preservationof Hypotheses II and IV is an immediateconsequence of the construction. In Hypothesis I, the transversalityrequirement is easy to verify,for the value j, with the aid of ?4 (B), and the continuityrequirement followswith the aid of ?7 (B) and HypothesisII. In establishingthe preserva- tion of HypothesisIII, it is convenientto considerthe two parts of our general step: (1) theextension of 7r'm(p) over y' and (2) the extensionover the rest of 91(y'). The preservationduring part (1) can, since HypothesisIV is preserved, be provedby the methodsof DM, Lemma 21. During part (2), HypothesisIII is preservedby virtue of the parallelismrequirement in Hypothesis II.

9. Completionof the proofsof TheoremsI and II. (A) If a Pm bein normal position,then transversal planes 7r'-m(p)can be constructedby therecurrency in ??7 and 8. For, if 7r'(p) denote any set of transversalplanes relative to which Pm is in normal position,then one can construct,using the methodsof the recurrency with -qsufficiently small, an arbitrarilyclose approximation,T"rm(p), to 7r'(p). We are now ready to establishTheorem II, thus incidentallycompleting the proofof TheoremI. We assume Pm in normalposition, with transversal planes ir"m(p) definedas in ??7 and 8. Hypotheses III and IV impose conditions whichpermit the applicationof Parts IV and V in DM, read withthe following substitutions. (1) The differentiablemanifold, M, in En is to be replaced by the polyhedral manifold,Pin in normal position in E'. (2) 7r-m(p) plays the role of the plane P(p), approximatelynormal to M at a point p. (3) For a given point p on Pm, let s'(p) denote the simplex containingp. Assuminga fixednumbering (se, sY , *** ) for the m-simplexesof Pt, let s~i denote the m-simplexwith the smallestsubscript belonging to the star of s'(p). The m-plane,r(p), of s'i replaces the tangent m-plane,T, to M at a point p. The relevant parts of Whitney's work can be outlined as follows,in their application to the constructionof an analytic manifoldM* homeomorphicto Pm. First, an analytic (n -l)-manifold S, surroundingPt, is defined. This is done with the aid of a function,('(p), continuousin R(Pm) = Rm (see Hy- pothesis III above), zero on Pt, and positive and analytic in R(Pm) - Pm. From V', thereis subtracteda small positive analytic function,W(p), such that the equation

(9.1) ?'(p) - @(p) = 0 determinesa suitably restricted'2analytic manifold,S. This manifoldis such that if 7rn-mpasses througha point p in R(Pm) and has directioncosines sufficientlyclose to those of r`tm(p) [cf. Hypothesis IV above], then (1) 7r'-' is transversalto r(p'), p' being the point where 7r'-m(p)intersects Pt, and (2) Tn-m meets S in an analytic (n - m - 1)-sphere S*(p, 7rn-m) contained in

12 The restrictionsare obtainedby conditionson A', w,and theirgradients. 806 STEWART S. CAIRNS

R(P"). Let Q*(p, i) be the part of mr" insideS*(p, ir"m). The following resultsare thenproved. (1) The vectorfunction g(p, wr)representing the center of mass of Q*(p, ir) is analytic. (2) If ir*(p) is a sufficientlyclose anlytic approximationto ir'"(p) throughfirst order derivatives,then the locus of the centersof mass g(p, ir*(p)) is an analyticmanifold, M*, homeomorphicto pm. This manifoldcan be made arbitrarilyclose to Pm. If it is desired merelyto make Pm, and hence M, differentiableto any given order r e (1, 2, ** *, X ), this can be done as follows. Construct7r'-(p) by the recurrencyof ??7 and 8 so that it is of class C7 in R(Pm). On each r(p), let therebe introduceda fixedrectangular cartesian coordinate system, with its domain restrictedto the part of r(p) inside R(Pm). If p' denote a point on such a domain,then the plane Tr -(p') [see HypothesisIV] meets Pmin a single point near p'. This affordsa mapping which carries the coordinate system fromr(p) onto Pm. In termsof coordinatesystems thus definedon Ptm,the latteris of class C' sincethe transformationbetween any two such systemsagrees with the correspondenceestablished by 7r'-m(p) in R(Pm) between rectangular cartesiansystems and two r-planes 'r(po) and r(pi). This method enables us to dispense with the hypothesisthat there be an upper bound to the number of cells in a star on pm [see footnote4]. For we can apply the above argument over a sequence of finitesubcomplexes each containingthe precedingand, in the limit,covering M. This makes the entiremanifold M of class Cr. It then remainsonly to apply DM, Theorem 1. The existenceof differentiableapproximations to polyhedralmanifolds was prematurelyasserted by the writer1,who is indebted to Hassler Whitneyfor calling his attention to the incompletenessof his work. The results of the present paper include the theoremof the abstract only for the case m = 3. This is the strongestsuch theoremwhich the writerhas thus far proved.

10. The Brouwernature of M (m _ 3). LEMMA. Let S be any star of an (m - k)-cellon a Pm. Then, if k ? 3, it is possibleto map S by a piecewise linear homeomorphisminto an Em. PROOF. In view of ?4, Lemma, and the work in ?6, it is sufficientto show that any triangulation(cr) of a (k - l)-sphere, Sk-', can be mapped homeo- morphicallyinto a geodesic triangulationof a (k - 1)-sphere. The proofis trivial for k < 3, so we restrictourselves to the case k = 3. We employ a recurrencywith the followingbasic hypothesis. HYPoTHEsIs. For some value j > 0, a subcomplex(o) j of (a), consistingof j 2-cells with theirboundaries, has been mapped topologicallyinto a geodesic complex (r,) on S2 so that the part of S2 not covered by (T) j is the sum of a finitenumber of convex regions,the closureof each of whichis a subset of an open hemisphereand none of which has three of its boundary vertices on a

Is Bulletin of the American Mathematical Society, vol. XL (1934), Abstract 67. HOMEOMORPHISMS 807

great circle. If ,3is the boundaryof any one of these regions,p, then the image, p3',of ,3bounds a subcomplex(a)* of (a) containingno 2-cellof (7) i . For the initialstep of the proof,consider any vertex,P, of (a). It is then a simplematter to map the closureof the star S(P), relativeto (a), into a complex (r) i so that the conditionsof the hypothesisare fulfilledfor j equal to the number of 2-cellson S(P). To definethe general step, using the notation of the hypothesis,let P be a vertexon #' and let S(P) denote its star relativeto the subcomplex(u)* of (o). There is no difficultyin mapping S(P) onto (,3 + p) so as to secure the conditions of the hypothesisfor a larger value of j. Aftera certain finitenumber of such steps, the mapping will be complete. COROLLARY. Every Pm (m < 3), and hence everytriangulated m-manifold (m < 3), is a Brouwermanifold. This is the special case of the lemma in whichm ? 3 and k = m.

11. Establishment of Theorem III. The case m = 4. LEMMA 11.1. It is possible,for any Pm in generalposition, to define7r'-m(p) on the regions9(y') (j > m - 3) so that7rn-m(p) will be continuousin p and will be transversalto Pm at p. PROOF. We will assume m ? 3, so that therewill exist stars S(sm3). The lower-dimensionalcases requireonly part of the followingargument. By ?10, Lemma, every S(sm-3) is a Brouwer star. Hence, given a point po on any m-32 we can definea plane rn-m(po) transversalto Pm at po. We can then extendthe definitionas p rangesover the y-3 containingpo by the requirement Tn-m(p) 11 7r-m(po). Since the ym-3 are bounded away fromone another,this can be done independentlyfor each of them. It is then possible to proceed with the recurrencyof ??7 and 8, startingwith the second part of the step j = m - 3. In order to apply the conditionof Theorem V, as in ?8, to the remainingsteps, we have to note that (1) in any T(r'), every (m - 3)-sphere bounds'4a cell, (2) in T(rO), every (m - 2)-spherebounds a cell, and (3) every (m - 1)-spherebounds a cell in T(r-') [see footnote9]. The case m < 3 gives the followingresult. COROLLARY. EveryPm (m ? 3) can be put in normalposition in some En. From this corollaryand Theorem II, we deduce Theorem III. LEMMA 11.2. If everyT(r2) is connected,then every Brouwer 4-manifold can be put in normalposition. PROOF. The connectednessof the T(r2) being assumed, we have only to note that the T(rl) are simply connected'4and that the higher-dimensional connectivitiesof T(r') and of T(Tr1) satisfythe conditionsof Theorem V. Thus the normal position problem for Brouwer 4-manifoldsreduces to the following.

14 For, T(r1) is a 1A-cell,where , + 2 is the number of vertices of r1. 808 STEWART S. CAIRNS

DEFORMATION PROBLEM. Given two geodesic triangulations,(a) and (r), of a 2-sphere,which correspondunder an orientation-preservinghomeomorphism, does thereexist a continuouslyvarying geodesic triangulation(U)t (O _ t _ 1), such that (U)o = (a), (a-)1 = (T), and (U)t is always homeomorphicto (U)o? Some of Tietze's work' has a bearing on this deformationproblem, but the writerhas not succeeded in obtaininga solution save in a few special cases. (A) It followsfrom TheoremII and Lemma 11.2 that every4-dimensional Brouwer manifoldcan be made into an analytic Riemannian manifoldif the deformationproblem has an affirmativesolution.

QUEENS COLLEGE, FLUSHING, N. Y.

16 Renditiconti del circolo matematico di Palermo, vol. 38 (1914), pp. 247-304, especially Satz IV on p. 280.