Baixe Elementary Concepts In Topology e outras Notas de estudo em PDF para Eletrônica, somente na Docsity! ELEMENTARY CONCEPTS OF TOPOLOGY By PAUL ALEXANDROFF with a Preface by DAVID HILBERT translated by ALAN E. PARLEY DOVER PUBLICATIONS, INC. New York TRANSLATOR'S PREFACE IN TRANSLATING this work, I have made no attempt to revise it, but have merely tried to preserve it in its proper historical perspective, and have made only a few minor modifications and corrections in the process. I have inserted a footnote and several parenthetical notes in an effort to clarify the material or to indicate the current terminology, and have actually changed the notation in several places (notably that for the homology groups) to avoid confusion and to bring it into consonance with modern usage. I wish to express my gratitude to Jon Beck, Basil Gordon and Robert Johnstone for their invaluable aid in the preparation of this translation. ALAN E. PARLEY Ann Arbor, Michigan January, 1960. CONTENTS Introduction . . . . . . . . 1 I. Polyhedra, Manifolds, Topological Spaces . 6 II. Algebraic Complexes . . . . . 11 III. Simplicial Mappings and Invariance Theo- rems . . . . . . . . 3 0 Index . . . . . . . . . 5 6 ' INTRODUCTION 1. “The specific attraction and in a large part the significance of topology lies in the fact that its most important questions and theorems have an immediate intuitive content and thus teach us in a direct way about space, which appears as the place in which continuous processes occur. As con- firmation of this view 1 would like to begin by adding a few examples! to the many known ones, 1. It is impossible to map an -dimensional cube onto a proper subset of itself by a continuous deformation in which the boundary remains point- wise fixed. That this seemingly obvious theorem is in reality a very deep one can be seen from the fact that from it follows the invariance of dimension (that is, the theorem that it is impossible to map two coordinate spaces of different dimensions onto one another in a one-to-one and bicontinuous fashion). The invariance of dimension may also be derived from the following theorem which is among the most beautiful and most intuitive of topo- logical results: 2. The tiling theorem. Tf one covers an a-dimensional cube with finitely many sufficiently small? (but otherwise entirely arbitrary) closed sets, then there are necessarily points which belong to at least a + 1 of these sets, (On the other hand, there exist arbitrarily fine coverings for which this number 2 + 1 is not excecded.) * One need only think of the simplest fixed-point theorems or of the well-known topological properties of closed surfaces such as are described, for instance, in Hilbert and Cohn-Vossen's Anschautiche Geametrie, chapter 6. [Published in English under the title Geometry and the Imagination by Chelsea, 1952—A.E.F.]. 1 “Sufficiently smalP” always means “with a sufficiently small diameter” 1 2 ALEXANDROFF For n =2, the theorem asserts that if a country is divided into suf- ficiently small provinces, there necessarily exist points at which at least three provinces come together, Here these provinces may have entirely Fic. 1 arbitrary shapes; in particular, they need not even be connected; each one may consist of several pieces. Recent topological investigations have shown that the whole nature of the concept of dimension is hidden in this covering or tiling property, and thus the tiling theorem has contributed in a significant way to the deepen- ing of our understanding of space (see 29 ff). 3, As the third example of an important and yet obvious-sounding theorem, we may choose the Jordan curve theorem.: À simple closed curve , the topological image of a circle) lying in the plane divides the plane into precisely two regions and forms their common boundary, 2. “The question which naturally arises now is: What can one say about a closed Jordan curve in three-dimensional space? The decomposition of the plane by this closed curve amounts to the fact that there are pairs of points which have the property that every polygonal path which connects them (or which is “bounded” by them) necessarily has points in common with the curve (Fig. 1). Such pairs of points are said to be separated by the curve or “linked” with it. In three-dimensional space there are certainly no paírs of points which are separated by our Jordan curve;? but there are closed polygons which * Even this fact requires a proof, which is by no means trivial. We can already see in wbat a complicated manner a simple closed curve or a simple Jordan arc can be situated in Rº from the fact that such curves can have points in common with all the says of a bundie of rays: it is sufficient to define a simple Jordan are in polar covrdinates ty the equations p=. d=/D vr=1+t ELEMENTARY CONCEPTS OF TOPOLOGY 5 4. All of these phenomena were wholly unsuspected at the beginning of the current century; the development of set-theoretic methods in topology first led to their discovery and, consequently, to a substantial extension of our idea of space. However, let me at once issue the emphatic warning that the most important problems of set-theoretic topology are in no way confined to the exhibition of, so to speak, “pathological” geo- metrical structures; on the contrary they are concerned with something quite positive. 1 would formulate the basic problem of set-theoretic topology as follows: To determine which set-theoretic structures have a comnection with the intuitively given material of elementary polyhedral topology and hence deserve to be considered as geometrical figures —even if very general ones. Obviously implicit in the formulation of this question is the problem of a systematic investigation of structures of the reguired type, particularly with reference to those of their properties which actually enable us to recognize the above mentioned connection and so bring about the geo- metrization of the most general set-theoretic-topological concepts. course of the first hour canals are to be dug, one from the sea, one from the warm lake, and one from the cold lake, in such a way that neither salt und fresh nor warm und cold water come into contact with one another, and so that at the end of the hour every point of land is at a distance of less than one kilometer from cach kind of water (i.e. salt, cald and warm). In the next halfhour, each of the canais is to be continued sa that the diflerent kinds of water remain separated, and at the end of'the work the distance of every point from each kind of water is less than one- half kilometer. In an analogous manner the work for the next 1/4, 1/8, 1/16, ..., hour is continued. At the end of the second hour, the dry land forms a closed set F nowhere dense in the plane, and arbitrarily near to each of its points there exists sea water as well as cold and warm fresh water. The set F is the common boundary of three regions: the sea, the cold lake and the warm lake (extended by their corresponding canals). [This example is due essentially to the Japanese ma- thematician Yoneyama, Tohoku Math. Journal, Vol. 12 (1917) p. 60.] (Fig. 4). The singular curves and surfaces in Rº which are also mentioned have been constructed by Antoine [J. Math. pures appl., Vol. (8) 4 (1921), pp. 221-325.] Also Alexander: Proc. Nat. Acad. U.S.4. Vol. t0 (1924), pp. 6-12. Concerning the in- variance of dimension, the tiling theorem and related questions see, in addition to Brouwer's classical work [Math. Ann. Vols. 70, 71, 72; J. reine angew. Math. Vol. 142 (1913), pp. 146-152; Amsterd. Proc. Vol. 26 (19231, pp. 795-800]; Sperner, Abh. Sem. Hamburg, Vol. 6 (1928), pp. 265-272; Alexandroff, Ann. of Math., Vol (2) 30 (1928), pp. 101-187, as well as “Dimensionstheorie” [Math. Anm., Vol. 106 (1932), pp. 161-238). Shortly a detailed work on topology by Professor H. Hopf and the author will be published in which all branches of topology will be taken into account. [Topologie, published by J. Springer, 1935—A.E.F.). 6 ALEXANDRORE “The program of investigation for set-theoretic topology thus formulated is to be considered—at least in basic outline—as completely capable of being carried out: it has turned out that the most important parts of set- theuretic topology are amenable to the methods which have been developed in polyhedral topology. Thus it is justified if in what follows we devote ourselves primarily to the topology of polyhedra. 1 Polyhedra, Manifolds, Topological Spaces. 5. We begin with the concept of a simplex. A zero-dimensional simplex is a point; a one-dimensional simplex is a straight line segment. A two- dimensional simplex is a triangle [including the plane region which it bounds--A,E.F |], a three-dimensional simplex is a tetrahedron. It is known and easy to show that if one considers all possible distributions of (non- negative) mass at the four vertices of a tetrahedron the point set consisting of the centers of mass of these distributions is precisely the tetrahedron itself; this definition extends easily to arbitrary dimension, We assume here that the r -- ] vertices of an r-dimensional simplex are not contained in an (r — 1)-dimensional hyperplane (of the R” we are considering). One could also define a simplex as the smallest closcd convex set which contains the given vertices, Anys + Lofther + 1 vertices of an 7-dimensional simplex (O < s < 7) define an s-dimensional simplex— an s-dimensional face of the given simplex (the zero-dimensional faces are the vertices). Then we mean by an r-dimensional polyhedron, a point-ser of Rº which can be decomposed into +-dimensional simplexes in such a way that two simplexes of this decompo- sition cither have no points in common or have a common face (of arbitrary dimension) as their intersection, The system of all of the simplexes (and their faces) which belong to a simplicial decomposition of a polyhedron is called a geometrical complex. The dimension of the polyhedron is not only independent of the choice of the simplicial decomposition, but indeed it expresses a topological 3 We defer these questions until sections 34 and 41. Conceraing the general standpoint appearing here and its execution, sec the works of the author mentioned in the preceding footnote. The basic work on general point set theory and at the same time the best introduction to set-theoretic topology is Mengenlehre by Hausdorft. See ulso Menger, Dimensionstheorie. ELEMENTARY CONCEPTS OF TOPOLOGY 7 invariant of the polyhedron; that is to say, two polyhcdra have the same dimension if they arc homeomorphic (if they can be mapped onto onc another in a one-to-one and bicontinuous fashion).* With the general viewpoint of topology in mind (according to which two figures—that is, two point sets—are considered equivalent if they can be mapped onto one another topologically), we shall understand a general or curved polyhedron to be any point set which is homeomorphic to a poly- hedron (defined in the above sense, that is, composed of ordinary “recti- linear” simplexes). Clearly, curved polyhedra admit decomposition into “curve” simplexes (that is, topological images of ordinary simplexes); the system of elements of such a decomposition is again called a geome- trical complex. 6. The most important of all polyhedra, indeed, even the most important structures of the whole of general topology, are the so-called closed n-dimensional manifolds M”. They are characterized by the following two properties. First, the polyhedron must be connected (that is, it must not be composed of several disjoint sub-polyhedra); second, it must be “homogencously n-dimensional” in the sense that every point p of M” possesses a neighborhood” which can be mapped onto the »-dimensional cube in a one-to-ane and bicontinuous fashion, such that the point p under this mapping corresponds to the center of the cube.* 7. In order to recognize the importance of the concept of manifold, it suflices to remark that most geometrical forms whose points may be One-to-one and bicontinuous mappings are called topological mappings or hameomorphisms. Properties of point sets which are preserved in such mappings are called topological invariants. "he theorem just mentioned is another form of the Brouwer theorem on the invariance of dimension. (Lt will be proved in sections 29.32.) ? The general concept of “neighborhood” will be further explained in section 8. The reader who wishes to avoid this concept may take “ncighborhood of a point of a polyhedron” to mean the set union of all simplexes of an arbitrary simplicial decomposition of a polyhedron which contain the given point in their interior or on their boundary. * On the subject of manifolds see, chiefiy, Veblen, Analysis Situs, second edition, 1931; Lefschetz, Topology, 1931 (both printed by the American Mathematical Society). Further references are Hopf, Math. Ann., Vol. 100 (1928), pp. 579-608; Vol. 102 (1929), pp. 562-623; Lefschetz, Trans. Amer. Math. Soc., Vol. 28 (1926), pp. 1-49; HopÍ, Jour. f. Math., Vol. 163 (1930), pp. 71-88, See also the literature given in footnote 41. 10 ALEXANDROFF A topological space is called a closed n-dimensional manifold if it is homeo- morphic to a connected polyhedron, and furthermore, if fts points possess neighborhoods which are homeomorphic to the interior of the n-dimensional sphere, 1t. We will now give some examples of closed manifolds. The only closed one-dimensional manifold is the circle. The “uniqueness” is of course understood here in the topological sense: every one-dimensional closed manifold is homeomorphic to the circle. The closed two-dimensional manifolds are the orientable (or two-sided) and non-orientable (or one-sided) surfaces. The problem of enumerating their topological types is completely solved.'º As examples of higher-dimensional manifolds—in addition to n-dimen- sional spherical or projective space-the following may be mentioned : 1. The three-dimensional manifold of line elements Íying on a closed surface F. (It can be easily proved that if the surface F is a sphere then the corresponding Mº is projective space.) 2. The four-dimensional manifold of lines of the three-dimensional projective space. 3. The three-dimensional torus-manifold: it arises if one identifies the diametrically opposite sides of a cube pairwise, The reader may confirm without difficulty that the same manifold may also be generated if one considers the space between two coaxial torus surfaces (of which one is inside the other) and identifies their corresponding points. The last example is also an example of the so-called topological product construction—a method by which infinitely many different manifolds can be generated, and which is, moreover, of great theoretical importance. “The product construction is a direct generalization of the familiar concept of coordinates. One constructs the product manifold MP+ == MP x Mº from the two manifolds M? and /Mº as follows: the points of M?+* are the pairs 2 = (x,y), where x is an arbitrary point of Mº? and y an arbitrary point of Mº. A neighborhood U(z,) of the point z, = (xy, X) is defined to be the collection of all points 2 = (x, y) such that x belongs to an arbitrarily 19 See for example Hilbert and Cohn-Vossen, Sec. 48, as well as Kerefijarto, Topologie, Chapter 5. ELEMENTARY CONCEPTS OF TOPOLOGY 1 chosen neighborhood of xy and y belongs to an arbitrarily chosen neigh- borhood of yo. It is natural to consider the two points and y (of M? and Mt respectively) as the two “coordinates” of the point (x, 7) of Meta, Obviously this definition can be generalized without difficulty to the case of the product of three or more manifolds. We can now say that the Euclidean plane [Not a closed manifold—A.E.F ] is the product of two straight lines, the torus the product of two circles, and the three-dimen- sional torus-manifold, the product of a torus surface with the circle (or the product of three circles). As further examples of manifolds one has, for example, the product S? x S1! of the surface of a sphere with the circle, or the product of two projective planes, and so on, The particular manifold S? x 8! may also be obtained if one considers the spherical shell Iying between two concentric spherical surfaces S? and sº and identifies the corresponding points (i.e., those lying on the same radius) of S? and s2. Only slightly more difhcult is the proof of the fact that, if one takes two congruent solid tori and (in accordance with the congruence mentioned) identifies the corresponding points of their surfaces with one another, one likewisc obtains the manifold S? x S1, Finally, one gets the product of the projective plane with the circle if in a solid torus one identifies each pair of diametrically opposite points of every meridian circle. These few examples will suffice. Let it be remarked here that, at present, in contrast to the two-dimensional case, the problem of enumerating the topological types of manifolds of three or more dimensions is in an apparently hopeless state. We are not only far removed from the solution, but even from the first step toward a solution, a plausible conjecture. IH. Algebraic Complexes. 12. There is something artificial about considering a manifold as a polyhedron: the general idea of the manifold as a homogencous structure ofn-fold extent, an idea which goes back to Riemann, has nothing intrinsic- ally to do with the simplicial decompositions which were used to introduce polyhedra. Poincaré, who undertook the first systematic topological study of manifolds, and thus changed topology from a collection of mathematical curiosities into an independent and significant branch of geometry, originally defined manifolds analytically with the aid of systems of equa- tions. However, within only four years after the appearance of bis first 12 ALEXANDROFF pionsering work!! he took the point of view which today is known as combinatorial topology, and essentially amounts to the consideration of manifolds as polyhedra.!º The advantage of this viewpoint lies in the fact that with its help the dificuht-—partially purely geometric, partially set- theoretic- considerations to which the study of manifolds leads are replaced by the investigation of a finite combinatorial model—-namely, the system of the simplexes of a simplicial decomposition of the polyhedron (ie. the geometrical complex)—which opéns the way to the application of algebraic methods. Thus, it turns out that the definition of a manifold which we use here is currently the most convenient, although it represents nothing more than a deliberate compromise between the set-theoretic concept of topological space and the methods of combinatorial topology, a compromise which, at present, can scarcely be called an organic blending of these two directions, "The most important of the difficult problems!s connected with the concept of manifold are by no means solved by the definition which we have adopted. 13. We shall now turn to the previously mentioned algebraic methods of the topology of manifolds (and general polyhedra). The basic concepts in algebraic topology are those of oriented simplex, algebraic complex and boundary of an slgebraic complex. An oriented one-dimensional simplex is a directed straight line segment (agay), that is, a line which is traversed from the vertex ag to the vertex aj. One can also say: an oriented one-dimensional simplex is one with a particular ordering of its endpoints. If we denote the oriented line (aça;) by x! (where the superscript | gives the dimension), the oppositely oriented 1 Analysis Situs, [J. Ec. Polyt. Vol. (2) 1 (1895) pp. 1-123]. 2 In the work; Complément à P Analysis Situs, [Palermo Rend., Vol. 13 (1899), pp. 285-343]. "This work is to be considered as the first systematic presentation of combinatorial topology. 13 These questions arise from the problem of the set-theorctic and the combina- torial characterizations of manifolds. The first problem is to establish necessary and sufficient set-theoretic conditions under which a topological space is homeomorphic toa polyhedron, that is, to establish necessary and sufficient conditions under which its points possess neighborhovds homesmorphic ta Rr. T'he second secks a charac- terization of those complexes which appear as simplicial decompositions of poly- hedra and which possess the manifold property, (in other words, each of whose points possesses a neighborhood homeomorphic to Rº). Both problems remain unsalved and, doubtless, are among the most difficult questions in topology. ELEMENTARY CONCEPTS OF TOPOLOGY 15 The above expression a 1 Dust is called the doundary of the oriented complex C? and is denoted by e, Examples. 1. Let K be the system composed of the four triangular faces of a tetrahedron; let the orientation of each of the faces be as indicated by the directions of the arrows in Fig. 5. The boundary of the resulting oriented complex e spisra Cosparara equals zero, because each edge of the tetrahedron appears in the two triangles of which it is a side with different signs. In formulas: a = (aga;), 5 — (araras), as = (ajaçãs), 4 = (aaa), and nam), =), (ai) (ma) m=(ma) a=(ma): thus 1d a + —% + “a 1 A a + *s ra dido = ara 1 10d +al-al +a 2. If one orients the ten triangles of the triangulation of the projective plane shown in Fig. 6 as indicated by the arrows, and puts 16 ALEXANDROFF then 3) C = 22 +20 + 284. The boundary of the oriented complex consists, therefore, of the pro- jective line 44º (composed of the three segments x), x), x) counted twice. With another choice of orientations x?, a7, x2, ..., Xtg Of the ten triangles of this trianguiation one would obtam another oriented complex Fic. 6 and its boundary would be different from (3). Hence, it is meaningless to speak of the “boundary of the projective plane”; one must speak only of the boundaries of the various oriented complexes arising from different triangulations of the projective plane. One can easily prove that no matter how one orients the ten triangles of Fig. 6, the boundary of the resulting complex Coat = ELEMENTARY CONCEPTS OF TOPOLOGY 17 is never zero. In fact, the following general result (which can be taken as the definition of orientability cf a closed surface) holds: A closed surface is orientable if and only if one can orient the triangles of any cf its triangulations in such a way that the oriented complex thus arising has boundary zero. z g a J , J; Hg A b 2 27 Fic. 7 3, In the triangulation and orientation for the Móbius band given in Fig. 7, we have: C>dtatararal 15. Oriented complexes and their boundaries serve also as examples of so-called algebraic complexes. A (two-dimensional) oriented complex, that is, a system of oriented simplexes taken from a simplicial decomposition of a polyhedron, was written by us as a linear form, E x?; furthermore, as the boundary of the oriented complex C! = E 43, there appeared a linear form E us! whose coefficients are, in general, taken as arbitrary integers. Such linear forms are called algebraie complexes. The same considerations bold in the n-dimensional case if we make the general definition: Definition 1. An oriented r-dimensional simplex x" is an r-dimensional simplex with an arbitrarily chosen ordering of its vertices, =" = (apar ... às), where orderings which arise from one another by even permutations of the vertices determine the same orientation (the same oriented simplex), so that each simplex | x” | possesses two orientations, x" and — ar. l Remark. Let! be an oriented simplex. Through the r 4 1 vertices of x” passes a unique r-dimensional hyperplane Rº (the R' in which x” lies), and to each r-dimensional simplex |3" | of R” there exists a unique 15 A zero-dimensional simplex has only one orientation, and thus it makes no sense to distinguish between xo and | xº |. 20 ALEXANDROPF complex is a (finite) set whose elements are simplexes, and, indeed, sim- plexes in the naive geometrical sense, that is, without orientation. An algebraic complex is not a set at all: it would be false to say that an algebraic complex is a set of oriented simplexes, since the essential thing about an algebraic complex is that the simplexes which appear in it are provided with coefficients and, therefore, in general, are to be counted with a certain multiplicity. This distinction between the three concepts, which often appear side by side, reflects the essential difference between the set- theoretic and the algebraic approaches to topology. 17. The boundary C” of the algebraic complex C” = E t'xf is defined to be the algebraic sum of the boundaries of the oriented simplexes x!, ie., E f'x, where the boundary of the oriented simplex x” = (aa ... à,) is the (r — 1)-dimensional algebraic complex! (4) 4 =3- 1) (ao À, a), where á, means that the vertex a, is to be omitted, In case the boundary of €r is zero (for instance, in the case of example | of Sec. 11) C! is called a cycle Thus in the group LR”), and analogously in L(K) and LUG), the subgroup of all r-dimensional cycles Z(R”), or, respectively, Z(K) and Z(G), is defined. We can now say (sec Sec. 14): a closed surface is orientable if and only ifone can arrange, by a suitably chosen orientation of any simplicial (i.e., in this case, triangular) decomposition of the surface, that the oriented com- plex given by this orientation is a cycle, Without change, this definition holds for the case of a closed manifold of arbitrary dimension, Let us remark immediately: orientability, which we have just defined as a property of a definite simpli jal decomposition of a manifold, actually expresses a 15 A zero-dimensional simplex has boundary zero; the boundary of a one- dimensional oriented simplex, 1.€., the directed Ime segment (asãi), 15 found by formula (4) to be a — ay, one endpornt having coeflicient + 1, one having coef- ficent — 1, In the symbolic notation of footnote 13, one can write formula (4) in the form: a-Bonã. 1º In particular, every zero-dimensional algebrarc complex 18 obvinusly a cycle, ELEMENTARY CONCEPTS OF TOPOLOGY 2 property of the manifold itself, since it can be shown that if one simplicial decomposition of a manifold satisfes the condition of orientability, the same holds true for every simplicial decomposition of this manifold, Remark. WE x" and y* are two equivalently oriented simplexes of Rr which have the common face | x |, then the face x"? (with some orienta- tion) appears in *º and 5” with the same or different signs according to whether the simplexes | x? ! and | 3º | lie on the same side or on different sides of the hyperplane Rº"1 containing | 4º”! |. The proof of this assertion we leave to the reader as an exercise, 18. As is casily verified, the boundary of a simplex is a cycle. But from this it follows that the boundary of an arbitrary algebraic complex is also a cycle. On the other hand, it is easy to show that for each cycle Z",r > 0, in R” there is an algebraic complex Jying in this R? which is bounded by 2":% indeed, it suffices to choose a point O of the space different from 2 Fic. 10 all the vertices of the cycle Z* and to consider the pyramid erected above the given cycle (with the apex at 0) (Pig. 10). In other words, if E =x, and (aja a), o 2º On the other hand, a zero-dimensional cycle in R* bounds if and onty if the sum of its coefficients equals zero (the proof is by induction on the number of sides of the bounded polygon). 22 ALEXANDROFF then one defines the (+ + 1)-dimensional oriented simplex 7” as AH (0, ap sa) and considers the algebraic complex sta art c Xe , The boundary of Cr is Z”, since everything else cancels out. Tf we consider, however, instead of the whole of R”, some-region G in R" (or more generally, an arbitrary open set in Rº), then the situation is no longer so simple: a cycle of R? lying in G need not bound in G. Indeed, Fic. 11 if the region G is a plane annulus, then it is casy to convince oneself that there are cycles which do not bound in G (in this case closed polygons which encircle the center hole) (Fig. 11). Similarly, in a geometrical com- plex, there are generally some cycles which do not bound in the complex. For example, consider the geometrical complex of Fig. 12: the cycle ABC as well as the cycle abr obviously does not bound, Consequently, one distinguishes the subgroups BG) of Z4G), and BAK) of Z(K), of bounding cycles: the elements of B,(G), or B(K), are cycles which bound some [(r + 1)-dimensional—A.E.F.] algebraic com- plexin G, or respectively, K. In the example of the triangulation given in Fig. 6 of the projective Plane, we see that it can happen that a cycle z does not bound in K, while a certain fixed integral multiple of it (i.e., a cycle of the form tz where t isan integer different from zero) does bound some algebraic subcomplex of K. We have, in fact, seen that the cycle 2x! + 2x1 4 25] = 23! (the ELEMENTARY CONCEPTS OF TOPOLOGY 25 every one-dimensional cycle satisfies a hornology of the form 2 = tz,, and, in Fig. 13, a homology of the form z — uz, + vz, where £, u, and q are s Z Fig. 15. 2,025 (in G) integers; furthermore, the strong homology classes coincide with the weak in both complexes (for there are no boundary divisors which are not at the same time boundaries). Fio. 16. 230073 + 2 (in 0). IF(, and L, are two homology classes and 2, and 2, are arbitrarily chosen eycles in £, and £, respectively, then one denotes by é, + £, the ho- mology class to which 2, + 2, belongs, This definition for the sum of two homology classes is valid because, as one may easily convince onesclf, the hamology class designated by £, -- £, does not depend on the parti- cular choice of the cycles z, and x, in £y and lp. 26 ALEXANDROFF The r-dimensional homology classes of K therefore form a group—the so-called factor group of ZLK) modulo B(K), or modulo B/(K); it is called the r-dimensional! Betti group of K. Moreover, one differentiates between the full and the free (or reduced) Betti groups-—the first corres- ponds to the strong homology concept [it is, therefore, the factor group Z(K) modulo B(K), denoted H (A), while the second is the group of the weak homology classes [the factor group Z,(K) modulo B/(K), denoted F(K))?! For examples sec Sec. 44. From the above discussion it follows that the full onc-dimensional Beiti group of the triangulation of the projective plane given in Fig. Gisa finite group of order two; on the other hand, the free (one-dimensional) Betti group of the same complex is the zero group. The one-dimensional Betti group of the complex K (Fig. 12) is the infinite cyclic group, while in Fig. 13 the group of all linear forms ul, + va (with integral « and 0) is the one-dimensional Betti group. In the latter two cases the fult and reduced Betti groups coincide, From simple group-theoretical theorems it follows that the full and the reduced Betti groups (of any given dimension 7) have the same rank; that is, the maximal number of linearly independent elements which can be chosen from eách group is the same. This common rank is called the r-dimensional Betti number? of the complex K. The one-dimensional Betti number for the projective plane is zero; for Figs. 12 and 13 it is, respectively, 1 and 2. 21. “The same definitions are valid for arbitrary regions G contained in Rr, It is especially important to remember that, while in the case of a geometrical complex all of the groups considered had a finite number of generators, this is by no means necessarily the case for regions of Rº, Indeed, the region complementary to that consisting of infinitely many circles converging to a point (Fig. 18) has, as one may casily sec, an infinite one-dimensional Betti number (consequently, the one-dimensionat Betti 2 [In fact, one may write HAK) = F4K) OD THK) where T4K) is the subgroup of HAK) consisting of the elements of finite order; the so-called torsion subgroup of HAK). In current usage, the group HK) is more often referred to as the r-dimensional homology group rather than the r-dimensional Betti group — AE) *» The reader will be able to prove easily that the zero-dimensional Betti number of a complex K equals the number of its components (i.e., the number of disjaint pieces of which the corresponding polyhedron is composed). ELEMENTARY CONCEPTS OF TOPOLOGY 2 Fig. 17. 34 mo 229 (in O). group does not have finite rank, therefore, certainly not a finite number of generators). Fic. 18 22. The presentation of the basic concepts of the so-called algebraic topology?? which we have just given is based on the concept of the oriented simplex, In many questions, however, one does not need to consider the orientation of the simplex at all —and can still use the algebraic methods extensively, In such cases, moreover, all considerations are much simpler, because the problem of sign (which often leads to rather tedious calcula- tions) disappears. The elimination of orientation throughout, wherever it is actually possible, leads to the so-called “modulo 2” theory in which all coefhicients of the linear forms that we have previously considered are replaced by their residue classes modulo 2. Thus, one puts the digit O 23 We prefer this expression to the otherwise customary term “combinatorial” topology, since we consider much broader applications of algebraic methods and concepts than the word “combinatorial” would include. 30 ALEXANDROFF subdivision of K*, and ! y” | some simplex of C"; then |? | lies on some particular simplex | x? | of 0%, We now orient the simplex | y* | the same as x7 (see Sec, 15), and give it the coefficient +. In this way, we obtain an algebraic complex which is called a subdivision of the algebraic complex €". One can easily see that the boundary of the subdivision Cj of Cris a subdivision of the boundary of C". (Considered modulo 2, the process gives nothing beyond the subdivision of a geometrical complex.) HI. Simplicial Mappings and Invariance Theorems, 24. Ifwe review what has been said up till now, we see that the discussion bas turned around two main concepts: complex on the one hand and topolo- gical space on the other. The two concepts correspond to the two inter- pretations of the concept basic to all of geometry—the concept of géo- metrical figure. According to the first interpretation, which has been inherent in synthetic geometry since the time of Euclid, a figure is a finite system of (generally) heterogeneous elements (such as points, lines, planes, ete,, or simplexcs of different dimensions) which are combined with one another according to definite rules—hence, a configuration in the most general sense of the word, According to the second interpretation, a figure is a point set, usually an infinite collection of homogencous elements. Such a collection must be organized in one way or another to form a geometrical structure—a figure or space. This is accomplished, for example, by introducing a coordinate system, a concept of distance, or the idea of neighborhoods.*s As we mentioned before, in the work of Poincaré both interpretations appear simultancously. With Poincaré the combinatorial scheme never becomes an end in itself; it is always a tool, an apparatus for the investiga- tion of the “manifold itself,” hence, ultimately a point set. Set-theorctic methods sufficed, however, in Poincaré's earliest work becausc his investi- gations touched only manifolds and slightly more general geometrical structures.?” For this reason, and also in view of the great difficulties 2º "Yhe set-theoretic interpretation of a figure also goes back to the oldest times— think, for example, of the concept of geometrical locus. “his interpretation became prominent in modern geometry after the discovery of analytic geometry. = Of course, in the felds of differential equations and celestial mechanics, the works of Poincaré already lead us very close to the modern formulation of questions in set-theoretic topology. ELEMENTARY CONCEPTS OF TOPOLOGY 3 connected with the general formulation of the concept of manifold, one can hardty speak of an intermingling or merging of the two methods in Poincaré's time, The further development of topology is marked at first by a sharp separation of set-thcorctic and combinatorial methods: combinatorial topology had been at the point of believing in no geometrical reality other than the combinatorial scheme itself (and its consequences), while the set- theoretic direction was running into the same danger of complete isolation from the rest of mathematics by an accumulation of more and more specialized questions and complicated examples. In the face of these extreme positions, the monumental structure of Brouwer's topology was erected which contained—at least in essence — the basis for the rapid fusion of the two basic topological methods which is presently taking place. In modern topological investigations there are hardiy any questions of great importance which are not related to the work of Brouwer and for which a tool cannot be found--often readily applicable --in Brouwer's collection of topological methods and concepts. In the twenty years since Brouwer's work, topology has gone through a period of stormy development, and we have been led—mainly through the great discoveries of the American topologists!—to the present “Aowering” of topology, in which analysis situs—still far removed from any danger of being exhausted—lies before us as a great domain developing in close harmony with the most varied ideas and questions of mathematics, At the center of Brouwer's work stand the tbpological invariance theorems. We collect under this name primarily theorems which maintain that if a certain property belonging to geometrical complexes holds for one simpli- cial decomposition of a polyhedron, then it holds for all simplicial decom- positions of homeomorphic polyhedra. “The classícal example of such an invariance theorem is Brouwer's theorem on the invariance of dimensioi if an n-dimensional complex K” appears as a simplicial decomposition of a polyhedron P, then every simplicial decomposition of P, as well as every simplicial decomposition of a polphedron P, which is homeomorphic to P, is tikewise an n-dimensional complex, Along with the theotem on invariance of dimension we mention as à 2º Alexander, Lefschetz and Veblen in topology itself, and Birkhoff and his successors in the topological methods of analysis. 32 ALEXANDROFF second example the iheorem on the invariance of the Betti groups proved by Alexander: if K and K, are simplícia] decompositions of two homeo- morphic polyhedra P and P,, then every Betti group of K is isomorphic to the corresponding Betti group of K,.º 25. In the proof of the invariance theorems one uses an important new device—the simplicial mappings and simplicial approximations of continuous mappings introduced by Brouwer. Simplicial mappings are the higher- dimensional analogues of piecewise lincar functions, while the simplicial approximations of a continuous mapping are analogous to linear inter- polations of continuous functions. Before we give a precise formulation of these concepts, we remark that their significance extends far beyond the proof of topological invariance: namely, they form the basis of the whole general theory of continuous mappings of manifolds and are— together with the concepts of topological space and complex—among the most important concepts of topology. 26. To each vertexa of the geometrical complex K let there be associated a vertex b = f(a) of the geometrical complex K” subject to the following restrictions: if «,, ..., a, are vertices of a simplex of K, then there exists in K' a simplex which has as its vertices precisely f(a), -.., f(a,) (which, however, need not be distinct). From this condition it follows that to each simplex of K there corresponds an (equal- or lower-dimensional) simplex of K'.3º One obtains in this way a mapping / of the complex K into the 3º 'The scope of these two theorems is not lessened if one assumes that K and Ky are two curved simplicial decompositions of one and the same polyhedron, for under a topological mapping an (arbitrary, also curved) simplicial decomposition of P, goes over into an (in general, curved) simplicial decomposition of P. One could, on the other hand, timit oneself to rectilinear simplicial decompositions of ordinary (“rectilinear”) polyhedra, but then Soth polyhedra P and P, must be considered. 15, indeed, P is an arbitrary polyhedron in a (curved) simplicial decomposition X, then there is a topological mapping of 2 into à sufficienty high dimensional Euelidean space in which P goes over into a rectilincar polyhedron P, and K, into its rectílinear simplicial decomposition K. 30 TF onc thinks of K as an algebraic complex modulo 2, then it is found to be convenient in the case where a simplex | 4º , of K is mapped onto a lower-dimen- sional simplex oÉ K' to say that the image of | xº | is mero (i.e., as an 1-dimensional simplex, it vanishes). ELEMENTARY CONCEPTS OF TOPOLOGY 35 29. We apply the third conservation theorem to the proof of the tiling theorem, already mentioned in Sec. 1; however, we shall now formulate it not for a cube but for a simplex: For sufficiently small e > 0, every e-covering"” of an n-dimensional simples has order = n 4 1. First, we choose e so small that therc is no set with a diameter less than € which has common points with all (n — 1)-dimensional faces of | 2” |. In particular, it follows thar no set with a diameter less than e can simul- taneously contain a vertex a; of |xº | and a point of the face |xj!| opposite to the vertex q. Now let €) Fo Fo Pop ces Fr be an e-covering of !x” |, We assume that the vertex a, 10, 1, ..,n lies in F,38 If there are more than n -+ 1 sets F,, then we consider some set F,j > m, and proceed as follows: we look for a face | x! | of ja” | which is disjoint from F, (such a face exists, as we have seen), strike out the set F, from (1) and replace F,by FU Fg renaming this last set Ps. By this procedure, the number of sets in (1) is diminished by | without increasing the order of the system of sets in (1). At the same time, the condition that none of the sets F, contains simultaneously a vertex and a point of the face opposite to the vertex will nor be violated. By finite repetition of this process, we finally obtain a system of sets (2) Fo Fc Fa containing the sequence of vertices, ag, 44, ..., G, Of [4º | a;eF, with the property that no set contains both a vertex a, and a point of |x71|]. Furthermore, the order of (2) is at most equal to the order of (1). It there- fore suffices to prove that the order of (2) is equal ton | 1, ie., to show that there is a point of |” | which belongs to al! sets of (2). As quite elementary convergence considerations show, the later goal will be reached if we show that in each subdivision | X” | of |x"“ |, no matter 2º By an e-covering of a closed set F one means a finite system Fi, Fa, uy Fe of closed subsets of F, which have as their union the set P and which are less than e in diameter. The order of a covering (or more generally, of an arbitrary Ênite system of point sets) is the largest number À with the property that there are & sets oÉ the system which have at least one common point. 3º According to our assumption, two different vertices cannot belong to the same set K; a vertex a; can, however, be contained in sets of our covering other than Fo 36 ALEXANDROFF how fine, there is necessarily a simplex [3º | which possesses points in common with all the sets Fy, Fi, Fe Let b be an arbitrary vertex of the subdivision | X” |. Now »b belongs 10 at least one of the sets F;; if it belongs to several, then we choose a particular one, for example, the one with the smallest subscript. Let this be F,, then we define f(b) — a,. In this way, we get a simplicial mapping / of |X"|into x” | which, 1 assert, satisfies the conditions of the third conservation theorem.º? Indeed, if b is interior to the face , x" [of |xº |, then (6) must be a vertex of x” .; because otherwise, if the whole sim- plex |x” |, and in particular the point é, were to lie on the face | xº! | of [xº | opposite 10 a; = f(b), then the point 5 could not belong to PF, Since, according to the third conservation theorem (understood modulo 2), [at [=X |), there must be among the simplexes of | X"| at least one which will be mapped by f onto | x” | (and not onto zero);'? the vertices of this simplex must lie successively in Fy, Fy u., Fy qe dt 30. If Fis a closed sex, then the smallest number » with the property that F possesses for each e >> Q an e-covering of order r + 1, is called the general or Brouwer dimension of the set H, It will be denoted by dim F, J€ F'isa subsetof F, then clearly, dim 4º < dim F. It is easy to con- vince oneself that two homeomorphic sets F, and F, have the same Brouwer dimension, In order to justify this definition of general dimension, however, one must prove that for an r-dimensional (in the elementary sense) polyhedron P, dim P —r; one would, thereby, also prove Brouwer's theorem on the invariance of dimension. Now, it follows at once from the tiling theorem that for an 7-dimensional simplex, and consequently for every r-dimen- sional polyhedron P, that dim P >> 7. For the proof of the reverse inequal- ity, we have only to construct, for cach e > 0, an e-covering of P of order + + L Such coverings are provided by the so-called barycentric coverings of the polyhedron. 3º We consider | x! | as an algebraic complex mod 2, so that footnotes 30 and 32 are valid. 49 "The above proof of the tiling theorem is due in essence to Sperner; the arrangement given here was communicated to mc by Herr Hopf, We have carried through the considerations modulo 2, since the theorem assumes no requirement of orientation. 'The same proof is also valid verbatim for the oriented theory (and, in fact, with respect to any domain of coefficients). ELEMENTARY CONCEPTS OF TOPOLOGY 37 31. First, we shall introduce the barycentric subdivisions of an n-dimen- sional complex K*, If n = 1, the barycentric subdivision of K! is obtained by inserting the midpoints of the one-dimensional simplexes of which K! consists [i.e., if the midpoint of each I-simplex of K! is called the bary- center of that simplex, then the barycentric subdivision of K1is the complex consisting of all the vertices of K? and all barycenters of K! together with the line segments whose endpoints are these points —A.E.F.]. H'n —2, the barycentric subdivision consists in dividing each triangle of Kº into six triangles by drawing its three medians (Fig. 19). Suppose that the barycentric subdivision is already defined for all r-dimensional complexes, then define it for an (+ + 1)-dimensional complex K by barycentrically subdividing the complex K, consisting of all +-dimensional simplexes (and their faces), and projecting the resulting subdivision of the boundary of each (+ + 1)-dimensional simplex of K from the center of gravity (barycenter) of this simplex. It is easy to see by induction that: 1. Each n-dimensional simplex is subdivided barycentrically into (n + 1) simplexes. 2. Among the n + 1 vertices of an n-dimensional simplex | yº | of the barycentric subdivision K, of K, (0) one vertex is also a vertex of K (this vertex is called the “Icading” vertex of | 9º |), (1) one vertex is the center of mass of an edge |x! | of K (which possesses the leading vertex of | y” | as a vertex), 40 ALEXANDROFF system of all vertices of the complex decomposes into the vertex-systems of the individual simplexes.“ Therefore, if one wants to define an abstract geometrical complex, it is most convenient to begin with a ser & cf (arbitrary) objects, which are called vertices; the set E we call a vertex domain. In E we then pick out certain finite subsets, which are called the frames; here the following two conditions must be satisfied: 1. Each individual vertex is a frame. 2. Every subset of a frame is a frame. The number of vertices of a frame diminished by one will be called its dimension. Finally, we suppose that to cach frame is associated a new object—the simplex spanned by the frame; here we make no assumptions about the nature of this object; we are concerned only with the rule which associates to every frame a unique simples. The dimension of the frame is called the dimension of the simplex; the simplexes spanned by the sub-frames of a given simplex a" are called the faces of x". A finite system of simplexes is called an abstract geometrical complex of the given vertex domain. Furthermore, one introduces the concept of orientation exactly as we have done previously, KH this is done then the concept of an abstract alge- braic complex ecith respect to a definite domain of coefficients*? necessarily results. From the fact that we formulate the concept of a complex abstractly, s one adheres to the elementary geometrical conception of a complex as a simplicial decom- position of a polyhedron, one cannot free oneself from the impression that there is something arbitrary which is connected with the choice of this concept as ihe basic concept of topology: why should this particular notion, simplicial decomposition of polyhedra, constitute the central point of all topology? The abstract conception of a complex as a finite scheme its range of application is substantially enlarged. As long 41 This general standpoint was formulated with complete clarity for the first time in the works of the author: “Zur Begrundung der n-dimensionalen Topologie,” Math. Ann. Vol. 94 (1925), pp. 296-308; “Simpliziale Approximationen in der allgemeinen Topologie,” Math. Ann. Vol. 96 (1926), pp. 489-51t; see also the correction in Vol. JOI (1929), pp. 452-456. 42 The general concept of algebraic complex thus arises by combining two different concepts: those of vertes and coefficient domains. An algobraic complex is finally nothing but a prescription which associates to each simplex of a given vertex domain a definite element of the chosen coefficient domain, ELEMENTARY CONCEPTS OF TOPOLOGY 4 which is, a priori, suitable for describing different processes (for example, the structure of a finite system of sets) helps to overcome this skepticism. Here, precisely those abstract complexes which arc defined as nerves of finite systems of sets play a decisive role: that is, it can be shown that the topological investigation of an arbitrary closed set therefore, the most general geometrical figure conceivable—can be completely reduced to the investigation of a sequence of complexes Ie) KE KO o KB (n is the dimension of the set) related to one another by certain simplicial mappings, Expressed more exactly: for every closed set one can construct a sequence of complexes (1) and of simplicial mappings f, of Kpsa into Ky th=1,2, ..) (which also satisfics certain secondary conditions which, for the moment, need not be considered). Such a sequence of complexes and simplicial mappings is called a projection spectrum. Conversely, every projection spectrum defines in a certain way, which we cannot describe here, a uniquely determined class of mutually homeomorphie closed sets; moreover, one can formuiate exact necessary and suflicient conditions under which two different projection spectra definc homcomorphie sets. In other words: the totality of all projection spectra falls into classes whose definition requires onty the concepts “complex” and “simplicial mapping”, and which correspond in a one-to-one way to the classes of mutualty homeo- morphic closed sets, It turns out that the clements of a projection spectrum are nonc other than the nerves of increasingly finer coverings of the given closed sets. These nerves can be considered as approximating complexes for the closed set. 35. We now go over to a brief survey of the proof of the invariance of the Betti numbers of a complex promised at the close of Sec. 25. Since we are only going to emphasize the principal ideas of this proof, we shall forgo a proof of the fact that a gcometrical complex! has the same Betti numbers as any one of its subdivisions. We begin the proof with the following fundamental lemma: “ Concerning this, see P. Alexandroff, “Gestaltu. Lage abgeschlossener Men- gen,” Ann. of Math. Vol. 30 (1928), pp. 101-187. 43 Until further notice, we are again dealing only with geometrical complexes, i.e. simplicial decompositions of (perhaps curved) polyhedra of a coordinate space. 4 Concerning this, see for example Alexander, “Combinatorial Analysis Situs,” Trans, Amer. Math. Soc., Vol. 28 (1926), pp. 301-329. 42 ALEXANDROFF Lebesgue's lemma. For every covering (1) S=(Fy Fy us Fo) of the closed set P, there is a number o = o(S)-—the Lebesgue number of the covering S—with the following property: if there is a point a whose distance from certain members of the covering S—say F,, Fi, Fi is less than o, then the sets Fi Fr ue Fi have a non-empty intersection. Proof: Let us suppose that the assertion is false, 'V'hen, there is a sequence of points 2 das dg cos Gm and of sub-systems fo) Soo Ses ca Say of the system of sets .$ such that a, has a distance less than 1/m from all sets of the system S, while the intersection of the sets of the system S, is empty. Since there are only finitely many different sub-systems of the finite system of sets S, there are, in particular among the S,, only finitely many different systems of sets, so that at least one of them—say S,— appears in the sequence (3) infinitely often. Consequentiy, after repla- cing (2) by a subsequence if necessary, we have the following situation: there is a fixed sub-system Sp = (Fio Fig es Fi) of and a convergent sequence of points (4 dy Oyo cos Amy um with the property that the sets F,, A = 1, 2,..., k, have an empty inter- section, while, on the other hand, the distance from «,, to each Fr, is less than 1/m; however, this is impossible, because, under these circum- stances the limit point a, of the convergent sequence (4) must belong to all sets of the system S,. q.e.d. 36. For the second lemma we make the following simple observation. Let P be a polyhedron, K a simplicial decomposition of P, and K, a subdivision of K. If we let each vertex 6 of K, correspond to the center of a barycentric star containing , then (by the remark made at the begin- ning of Sec, 32) the vertex » is mapped onto a vertex of the simplex of K ELEMENTARY CONCEPTS OF TOPOLOGY 4 their subdivisions in K,. The cycles gl), g(zo), e glzo) are independent in (2, since if U is a subcomplex of (2 bounded by a linear combination estao, then f(U) will be bounded by Sete, ie. E c'Z,, which, according to the assumed independence of the Z, in K, implies the vanishing of the coeficients ct. Under the topological mapping 1, the linearly independent cycles g(z,) of the complex Q go over into linearly independent cycles of the complex KJ (indeed, both complexes have the same combinatorial structure), so that there are at least p linearly independent r-dimensional cycles in Kj. Since we have assumed the equality of the Betti numbersof K and K, it follows, therefore, that p' >> p. q.ed. With the same methods (and only slightly more complicated considera- tions) one could also prove the isomorphism of the Betti groups of K and K'. 40. The proof of the theorem of the invariance of Betti numbers which we have just given, following Alexander and Hopf, is an application of the general method of approximation of continuous mappings of polyhedra by simplicial mappings. We wish to say here a few more words about this method, Let f be a continuous mapping of a polybedron P” into a poly- bedron P”, and let the complexes K' and K” be simplicial decompositions of the polyhedra P' and P”, respectively. Let us consider a subdivision Ky of K” so fine that the simplexes and the barycentric stars of Ky are smaller than a prescribed number e; then, we choose the number 3 so small that two arbitrary points of P' which are less than ô apart go over by means of f into points of P” whose separation is less than the Lebesgue number o of the barycentric covering of Ky'. Now consider a subdivision Kj of K' whose simplexes are smaller than 5. The images of the vertex frames of Kj have a diameter < o, and their totality can be considered as an abstract complex 2; because of the smallness of the simplexes of Q, one 46 ALEXANDROFF can apply to this complex the procedure of Sec. 37, i.e., one can map it by mcans of a canonical displacement g into the complex Ky. The transi- tion of Kj to Q and the map g from Q to g(9) together produce a simplicial mapping f, of K/ into K/. This mapping (considered as a mapping from P' into P”) differs from f by less than e (i.e., for every point a of P' the distance between the points (a) and fi(a) is less than e). The mapping f is called a simplicial approximation of the continuous mapping f (and, indeed, one of fineness e). By means of the mapping f, there corresponds to each cycle z of K' (where x is to be regarded as belonging to the subdivision K/of K? a cycle fiz) of K/. Morcover, one can casily convince oneself that if mm zo in K' then it follows that (21) » fi(z)) in KY, so that to a class of homologous cycles of K' there corresponds a class of homologous cycles of K;. In other words, there is a mapping of the Betti groups of K' into the corresponding Betti groups of K!'; since this mapping preserves the group operation (addition), it is, in the language of algebra, a homo- morphism. But there also exists a uniquely determined isomorphism *º between the Betti groups of K(' and K”, so that as a result, we obtain a homomorphic mapping of the Betti groups of K' into the corresponding groups of K”. Consequently, we have the following fundamental theorem (first formulated by Hopf): A continuous mapping f of a polyhedron P' into a polyhedron P" induces a uniquely determined homomorphic mapping of all the Betti groups of the simplicial decomposition K of P" into the corresponding groups of the simplicial decomposition K” of P'3 If the continuous mapping f is one-to-one (therefore, topological) it induces an isomorphic mapping of the Betti groups of P' onto the corresponding Betti groups of P'50 By this theorem a good part of the topological theory of continuous “s Resulting from the canonical displacement of X' with respect to K”, *º Because of the isomorphism between Betti groups of the same dimension of different simplicial decompositions of a polyhedron, one can speak simply of the Betti groups of P' or P”, 29 The proof of this last assertion must be omitted here. Our considerations up to now contain all the elements of thc prof; its exccution is, therefore, left to the reader. 'The reader should observe, however, that an arbitrarily fine simplicial approximation to a tupological mapping need by no means be à one-to-one mapping. ELEMENTARY CONCEPTS OF TOPOLOGY 4“ mappings of polybedra (in particular of manifolds) is reduced to the in- vestigation of the homomorphisms induced by thesc mappings, and thus to considerations of a purely algebraic nature. In particular, one arrives at far reaching results concerning the fixed points of a continuous mapping of a polyhedron onto itself. 8! 41. We shall close our treatment of topological invariançe theorems with a few remarks about the gencral concept of dimension which are closely related to the ideas involved in the above invariance proofs. Our previous considerations have paved the way for the following definition. A continuous mapping f of a closed set Fof Rº onto a set H' Iyingin the same Rº is called an e-transformation of the set F (into the set E) if every point a of F is at a distance less than « from its image point (4). We shall now prove the following theorem, which to a large extent justifies the general concept of dimension from the intuitive geometrical standpoint, and allows the connection between sct-theoretic concepts and the methods of polyhedrai topology to be more casily and simply under- stood than do the brief and, for many tastes, too abstract remarks con- cerning projection spectra (Sec. 34). Transformation theorem, For each e > O, every r-dimensional set F can be mapped continuonsly onto an r-dimensional polyhedron by means of an e-transformation ; on the other hand, for sufficiently small e there is no e-transformation of E into a polyhedron whose dimension is at most r — 1. The proof is based on the folowing remark. If €) FP, Foco Es is an e-covering of F, then the nerve of the system of sets (1) 1s defined first as an abstract complex: to each set F;(1 <2i < s) let there correspond a “vertex” a, and consider a system of vertices dig Cp o Fis 52 [ mean here principally the Lefscheiz-Hopf fixed-point formula which completely determines (and, indeed, expresses by algebraic invariants of the above homomerphism) the so-called algebraic number of fixed points of the given continuous mapping (in which every fixed point is to be counted with a definite 'y which can be positive, negative, or zero). Concerning this, see Hopf, Wiss. Cottingen (1928), pp. 127-136, and Math. Z., Vol. 29 (1929), pp. 493-525. 50 ALEXANDROFF oforder < r, and denote by F, the set of all points of F which are mapped into F$ by our transformation. Clearly, the sets F, form a 3e-covering of F of the same order as (2), therefore of order <, 7. Since this holds for alle, we must have dim 7 <7 — |, which contradicts our assumption. With this, the transformation thcorem is completely proved. 43, Remark. If the closed set F of R” has no interior points, then for every e it may be e-transformed into a polyhedron of dimension at most n — |: it suffices to decompose the Rº into e-simplexes and to “sweep out” each n-dimensional simplex of this decomposition, A set without interior points is thus at most (4 — 1)-dimensional. Since, on the other hand, a closed set of R? which possesses interior points is necessarily n-dimen- sional (indeed, it contains n-dimensional simplexes!), we have proved: A closed subset of Rº is n-dimensional if and only if it contains interior points. With this we close our sketchy remarks on the topological invariance thcorems and the general concept of dimension—the reader will find a detailed presentation of the theories dealing with these concepts in the literature given in footnote 4 and above all in the books of Herr Hopf and the author which have been mentioned previously. 44. Examples of Betti groups. 1. The one-dimensional Betti group of the circle as well as of the plane annulus is the infinite cyclic group; that of the lemniscate is the group of all linear forms ul, + vb a (with integral u and 0). 2. The one-dimensional Betti number of a (p -+ I)-fold connected plane region equals p (see Fig. 13, p = 2). 3. A closed orientable surface of genus p has for its one-dimensional Betti group the group of all linear forms 2 » ug + D, cima (with integral uº and 09); & a here one takes as gencrators £, and m, the homology classes of the 29 canonical closed curves. 35 See for example: Hilbert and Cohn-Vossen, p. 264, p. 265, and p. 284 [German edition—A.E.F). ELEMENTARY CONCEPTS OF TOPOLOGY 51 The non-orientable closed surfaces are distinguished by the presence of a non-vanishing one-dimensional torsion group, where by torsion group (of any dimension) we mean the subgroup of the full Betti group con- Fis. 21 sisting of all elements of finite order. The one-dimensional Betti number of a non-otientable surface of genus pisp — 1. 'The two-dimensional Betti numberof a closed surfaceequais lor Oaccord- ing as the surface is orientable or not, The analogous assertion also holds for the n-dimensional Betti number of an 2-dimensional closed manifold. 4. Let P be a spherical shell (Fig. 21), and Q be the region enclosed between two coaxial torus surfaces (Fig. 22). The one-dimensional Betti number of P is 0, the one-dimensional Betti number of 2 is 2, while the two-dimensional Betti numbers of P and Q have the value 1, 5. One can choose as generators of the one-dimensional Betti group of the three-dimensional torus (Sec. 11) the homology classes of the three cycles xt, z!, 2! which are obtained from the three axes of the cube by identifying the opposite sides (Fig. 23). As generatora of the two-dimen- sional Betti group, we can use the homology classes of the three tori into which the three squares through the center and parallel to the sides are transformed under identification. Therefore, the two Betti groups are isomorphic to one another: each has three independent generators, hence, three is both the one- and two-dimensionai Betti number of the manifold, 6. For the one- as well as the two-dimensional Betti group of the manifold S? x S! (see Sec. 11) we have the infinite cyclic group (the corresponding Betti numbers are therefore equal to 1). As 23 choose the cycle (Fig. 21) which arises from the line segment aa” under the identifica- 52 ALEXANDROFF tion of the two spherical surfaces, and as 23, any sphere wbich is con- centric with the two spheres S2 and s? and lies between them. Fig. 23 It is no accident that in the last two examples the one- and two-dimen- sional Betti numbers of the three-dimensional manifolds in question are equal to one another; indeed, we have the more general theorem, known as the Poincaré duality theorem, which says that in an a-dimensional closed orientable manifold, the r- and the (x — r)-dimensional Betti numbers are equal, for 0 < 7 < n. The basic idea of the proof can be discerned in the above examples: it is the fact that one can choose for every cycle 2” which is not = Q in JM a cycle 2” such that the so-called “intersection number” of these cycles is different from zero. 7, The product of the projective plane with the circle (Sec. LI) is a non-orientable three-dimensional manifold Mº. It can be represented as a solid torus in which one identifies, on each meridian circle, diametrically opposite pairs of points. The one-dimensional Betti number of Mº is | (every one-dimensional cycle is homologous to a multiple of the circle which goes around through the center of the sotid torus); the two- dimensional Betti group is finite and has order 2; therefore, it coincides with the torsion group** (the torus with the aforementioned identification indeed does not bound, but is a boundary divisor of order 2). Here again he (n — 1)-dimensional torsion group of a closed there is a general law “4 The +-dimensional torsion group TAK) of a complex K is the finite group which consists of all elements of nite order of the Betti group H(K). The factor group HAK)/TAK) is isomorphic to F(K). [See footnote 21—A. ELEMENTARY CONCEPTS OF TOPOLOGY 55 almost exclusively; the first applications to differential equations, mechan- ics, and algebraic geometry lead back to Poincaré himself. In the last few years these applications have been increasing almost daily. It suffices here to mention, for example, the reduction of numerous analytical existence proofs to topological fixed point theorems, the founding of enu- merative geometry by Van der Waerden, the pioneering work of Lefschetz in the field of algebraic geometry, the investigations of Birkhoff, Morse and others in the calculus of variations in the large, and numerous dif- ferential geometrical investigations of various authors, etc. One may say, without exaggeration: anyone cho wishes to learn topology with an interest in its applications must begin with the Betti groups, because today, just as in the time of Poincaré, most of the threads which lead from topology to the rest of mathematics and bind topological theories together into a recogniz- able whole lead through this point. SE A rather complete bibliography will be found at the end of the book by Lefschetz, already mentioned many times before. abstract geometrical complex, 40 Alexander duality theorem, 53 Alexander's theorem on invariance of Betti groups, 32 algebraic complex, 12, 17, 18 subdivision of, 29 with arbitrary coefficient domain, 29 (1.25) algebraic subcomplex, 18 algebraic topology, 12, 17 approximating complex, 41 approximation, simplicial, 46 barycenter, 37 barycentric covering, 36 barycentric star, 38 barycentric subdi Betti groups, 26 free, 26 full, 26 invarianee theorem for, 32 reduced, 26 Betti number, 26 boundary, 12 boundary divisor, 23 bounding cyele, 22 Brouwer dimension, 36 Brouwer's theorem on invariance of dimension, 31 on, 37 canonical displacement, 43 modified, 43 center (of barycentric star), 38 class, homology, 24 closed manifold, 7, 10 coefficient domain, 29 (n.25) combinatorial topology, 12 complex, abstract geometrical, 40 algebraic, 12, 17, 18 geometrical, 6 n-dimensional, 28 oriented, 14 NDEX component, 26 (n.22) connected, 7 consistently oriented, 18 continuous mapping, 9 covering, barycenteie, 36 e, 35 curve, Jordan, 2 simple closed, 2 curved polyhedron, 7 curved simplex, 7 eycle, 20 bounding, 22 dimension, Brouwer, 36 Brouwer's theorem on invariance of, 31 general, 36 of frame, 40 displacement, canonical, 43 modified canonical, 43 divisors, boundary, 23 domain, coefficient, 29 (n.25) vertex, 40 duality theorem, Alexander, 53 Poincaré's, 52 ecovering, 35 e-transformation, 47 equivalently oriented, 18 face (of simples), 6 abstract, 40 fineness, 46 frame, 40 dimension of, 40 simplex sparmed by, 40 free Betti group, 26 full Betti group, 26 geometrical complex, 6 abstract, 40 INDEX 57 geometrical complex algebraic subtomplex of, 18 subdivison of, 29 group, Beth, 26 homology, ses Beta groups torsion, 26 (n 21), 51 Hausdort” axioms, 9 Hausdorf! space, 9 homeomorphic, 21 homeomorphusm, 21 (n 6) homologous, 23 homology, strong, 23 weak, 23 bomology class, 24 homology group, see Betti groups homomorphism, induced, 46 induced homomorphism, 46 into, 33 (n 31) invariance of dimension, Brouwer's theorem on, 31 invariance of Betti groups, Alexander's theorem on, 32 invariance theurem, topological, 31 invaniant, topological, 6-7 (n 6) Jordan arc, sample, 4 Jordan curve, 2 Jordan curve theorem, 2 generalized, 2, 3 leading vertex, 37 Lebesgue number, 42 Lebesgue's lemma, 42 linked, 2 mamiíold, closed, 7, 10 produet, 10 mapping, conunuous, 9 simpheral, 32-33 topological, 7 modified canonical chisplacement, 43 modulo 2 theory, 27 modulo theory, 29 (n 25) a-dimensional complex, 28 neighborhood, 7, 9 nerve, 39 onto, 33 (n31) oreler (of à covering), 35 (n 37) onentable, 17, 20 onented, consstently, 18 equivalentiy, 18 oriented complex, 14 oriented smplex, 12, 17 Poincaré's dualty theorem, 52 polyhedron, 6 curved, 7 projection spectrum, 41 projective plane, 8 reduced Bett group, 26 simple closed curve, 2 smple Jordan arc, 4 simplex, 6 curved, 7 face of, 6 oriented, 12, 17 simplex spanned by frame, 30 smpleial approximanion, 46 ampheial mapping, 32-33 space, Hausdorff, 9 product, 10 topological, 8 spectrum, projecuon, 41 star, barycentrie, 38 strong homology, 23 subcomplex, algebraic, 18 of algebraie complex, 29 of geometric complex, 29 tiling theorem, 1 prof of, 35-36 topological invariance theorem, 31 topological mvariant, 6-7 (n 6) topological mapping, 7 (n 6) topological space, 8 topology, algebraic, 12, 27 combinatorial, 12 torsion group, 26 (n 21), 51 triangulaton, 13 transformauon, e, 47 vertex, abstract, 40 leading, 37 vertex domain, 40 weak homology, 13