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Homeomorphisms Between Topological Manifolds and Analytic Manifolds Author(S): Stewart S Annals of Mathematics 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 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=annals. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics. 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"', homeo- morphicto 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 posi- tion, *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.
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