Topology bare facts: part 6

NB: same reference: Elementary Concepts of Topology: Paul Alexandroff

A:

The proof of the theorem of the invariance of Betti numbers which was presented last, 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 polyhedra P^{'} into a polyhedron P^{''}, and let the complexes K^{'} and K^{''} be simplicial decomposition of the polyhedra P^{'} and P^{''} respectively. Let us consider a subdivision K_{1}^{''} of K^{''} so fine that the simplexes and the barycentric stars of K_{1}^{''} are smaller than a presribed number \epsilon; then, we choose the number \delta so small that the two arbitrary points of P^{'} which are less than \delta apart go over by means of f into points of P^{''} whose separation is less than the Lebesgue number \sigma of the barycentric covering of K_{1}^{''}. Now consider a subdivision K_{1}^{'} of K^{'} whose simplexes are smaller than \delta. The images of the vertex frames of K_{1}^{''} have a diameter less than \sigma, and their totality can be considered as an abstract complex Q; because of the smallness of the simplexes of Q, one can apply to this complex the following procedure: one can map it by means of a canonical displacement g in to the complex K_{1}^{"}. The transition of K_{1}^{'} to Q and the map g from Q to g(Q) together produce a simplicial mapping f_{1} of K_{1}^{'} into K_{1}^{''}. This mapping (considered as a mapping from P^{'} into P^{''} ) differs from f by less than \epsilon (that is, for every point a of P^{'} the distance between the points f(a) and f_{1}(a) is less than \epsilon). The mapping f_{1} is called a simplicial approximation of the continuous mapping f (and, indeed, one of fineness \epsilon).

By means of the mapping f_{1} there corresponds to each cycle z of K^{'} (where z is to be regarded as belonging to the subdivision K_{1}^{'} of K^{'}) a cycle f_{1}(z) pf K_{1}^{''}. Moreover, one can easily convince oneself that if z_{1} \sim z_{2} in K^{'} then it follows that f_{1}(x_{1}) \sim f_{2}(x_{2}) in K_{1}^{''} so that to a class of homologous cycles of K^{'} there corresponds a class of homologous cycles of K_{1}^{''}. In other words, there is a mapping of the Betti groups of K^{'} into the corresponding Betti groups of K_{1}^{''}; since the mapping preserves the group operation (additIon), it is, in the language of algebra, a homomorphism. But there also exists a uniquely determined isomorphism between the Betti groups of K_{1}^{''} 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 homeomorphic mapping of all the Betti groups of the simplicial decomposition K^{'} of P^{'} into the corresponding groups of the simplicial decomposition K^{''} of P^{''}.

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^{''}.

By this theorem a good part of the topological theory of continuous mappings of polyhedra (in particular of manifolds) is reduced to the investigation of the homomorphisms induced by these mappings, and thus to considerations of a purely algebraic notion. In particular, one arrives at far reaching results concerning the fixed points of a continuous mapping of a polyhedron onto itself.

(Note: We mean here principally the Lefschetz-Hopf fixed point formula which completely determines (and indeed expresses by algebraic invariants of the above homomorphism) 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 multiplicity which can be positive, negative or zero.))

B.

We close our topic of topological invariance theorems with a few remarks about the general concept of dimension which are closely related to the ideas involved in the previous invariance proofs. Our previous considerations have paved the way for the following definition:

A continuous mapping f of a closed set F of \mathcal{R}^{n} onto a set of F^{'} lying on the same \mathcal{R}^{n} is called an \epsilon-transformation of the set F (into the set F^{'}) if every point a of F is at a distance less than \epsilon from its image point f(a).

We now present the proof of the following theorem, which to a large extent justifies the general concept of dimension from the intuitive geometrical standpoint, and allows the connection between set-theoretic concepts and the methods of polyhedral topology to be more easily and simply understood than do the brief and, for many tastes, too abstract remarks concerning projection spectra:

Transformation theorem:

For each \epsilon>0, every r-dimensional set F can be mapped continuously onto an r-dimensional polyhedron by means of an \epsilon-transformation; on the other hand, for sufficiently small \epsilon, there is no \epsilon-transforation of F into a polyhedron whose dimension is at most r-1.

The proof is based on the following remark. If

I: F_{1}, F_{2}, \ldots, F_{s}

is an \epsilon-covering of F, then the nerve of the system of sets I is defined first as an abstract complex: to each set F_{i}, where 1 \leq i \leq s let there correspond a “vertex” a_{i} and consider a system of vertices

a_{i_{0}}, a_{i_{1}}, \ldots, a_{i_{s}}

as the vertex frame of a simplex (of the nerve Kof I) if and only if the sets F_{i_{0}}, F_{i_{1}}, \ldots, F_{i_{r}} have a non empty intersection. However, one can realize this abstract complex geometrically if one chooses for a_{i} a point of F_{i} itself, or a point from an arbitrarily prescribed neighbourhood of F_{i}, and then allows the vertex frame of the nerve to be spanned by the ordinary geometrical simplexes.. This construction is always possible, and yields as the nerve of the system of sets I an ordinary geometrical polyhedral complex provided the coordinate space \mathcal{R}^{n} in which F lies is of high enough dimension, (note:) but this condition can always be satisfied because one can, if need be, imbed the \mathcal{R}^{n} in which F lies in a coordinate space of higher dimension.

C:

In any case, we now assume that a_{i} is at a distance less than \epsilon for each F_{i} and prove the following two lemmas:

C1:

If K is a geometrically realized nerve of the \epsilon-covering I of F, then every complex Q whose vertices belong to F, and whose simplexes are smaller than the Lebesgue number \sigma of the covering I, goes over into a subcomplex of K by means of a 2\epsilon-displacement of its vertex.

Indeed, associate to each vertex b of Q as the point f(b) one of those vertices a_{i} of K which correspond to the sets F_{i} containing the point b. Thereby, a simplicial mapping f_{i} of Q into K is determined; since, the distance between a and f(a) is clearly less than 2\epsilon our lemma is proved. QED.

C2:

The conclusion of lemma C! also holds (with 3\epsilon in place of 2\epsilon) if hte vertices of Q do not necessarily belong to F but if one knows that they lie at a distance of less than 1/3 \sigma from F, and that the diameters of the simplexes Q do not exceed the number 1/3 \sigma.

In order to reduce this lemma to the preceding one, it is only necessary to transform the vertices of Q into points of P by means of a 1/3 \sigma displacement.

We now decompose the \mathcal{R}^{n} into simplexes which are smaller than 1/3 \sigma, and denote by Q the complex which consists of all those simplexes which contain points of F in their interiors or on their boundaries; then apply to this complex the lemma just proved. This gives us the following:

A sufficiently small polyhedral neighbourhood Q of F is transformed by means of a 2\epsilon transformation into a polyhedron P, consisting of simplexes of K.

Since F was r-dimensional and the dimension of the nerve of a system of sets is always 1 less than the order of the system of sets, we may assume that P is at most r-dimensional. From the fact that a certain neighbourhood of F is transformed onto the polyhedron P by the 2\epsilon-transformation in question, it follows that F itself will be mapped onto a proper or improper subset of P (that is, in P_{r}).

Thus, we have proved: For every \epsilon>0 F can be mapped onto a subset \Phi of an r-dimensional polyhedron by an \epsilon-transformation.

We now consider a simplicial decomposition K of P whose elements are smaller than \epsilon. Since \Phi is closed, there exists — if \Phi \neq P — an r-dimensional simplex x^{r} of K which contains a homothetic simplex x_{0}^{r} free of points of \Phi. If one now allows the domain x^{r} - x_{0}^{r} which lies between the boundaries of x^{r} and x_{0}^{r} to contract to the boundary of x^{r}, then all the points of \Phi contained in x^{r}, and the points of the set \Phi will be “swept out” of the interior of the simplex x^{r}. By a finite number of repetitions which do not belong to \Phi will be freed of points of the set. One continues the process with (r-1)-dimensional simplexes, and so on. The procedure ends with a polyhedron composed of simplexes, and so on. The procedure ends with a polyhedron composed of simplexes (of different dimensions) of K. \Phi is mapped onto this polyhedron by means of a continuous deformation in which no point of \Phi leaves that simplex of K to which it originally belonged; consequently, every point of \Phi is displaced by less than \epsilon. Hence, the whole passage from F to P is a 2\epsilon-transformation of the set F so that the first half of our theorem is proved.

QED.

In order to prove the second half, prove the following more general statement: there exists a fixed number \epsilon(F)>0 such that the r-dimensional set F can be mapped by an \epsilon(F)-transformation into no set whose dimension is at most (r-1).

We assume that there is no such \epsilon(F). Then, for every \epsilon>0 there exists a set F_{\epsilon} of dimension at most (r-1) into which F can be mapped by means of an <\epsilon-transformation. Consider an \epsilon-covering of the set F_{\epsilon}

II \ldots F_{1}^{\epsilon}, F_{2}^{\epsilon}, \ldots, F_{s}^{\epsilon} of order less than or equal to r, and denote by F_{s} the set of all points of F which are mapped into F_{i}^{\epsilon} by our transformation. Clearly, the sets F_{i} form a 3\epsilon-covering of F of the same order as (II), therefore of order less than or equal to r. Since this holds for all \epsilon we must have dim F less than or equal to (r-1), which contradicts our assumption. With this, the transformation theorem is completely proved.

QED.

D:

Remark:

If the closed set F of R^{n} has no interior points, then for every \epsilon it may be \epsilon transformed into a polyhedron of dimension at most (n-1); it suffices to decompose the \mathcal{R}^{n} into \epsilon-simplexes and to “sweep out” each n-dimensional simplex of this decomposition. A set without interior points is thus at most (n-1)-dimensional. Since, on the other hand, a closed set of \mathcal{R}^{n} which possesses interior points is necessarily n-dimensional (indeed, it contains n-dimensional simplexes!), we have proved the following:

A closed subset of \mathcal{R}^{n} is n-dimensional if and only if it contains interior points.

With this we close our sketchy remarks on the topological invariance theorems and the general concept of dimension — the reader will find a detailed presentation of the theories dealing with these concepts in literature and above all in the book of Paul Alexandroff and Herr Hopf.

E:

Examples of Betti groups:

E1:

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 u\zeta_{1} + v \zeta_{2} (with integral u and v).

E2:

The one dimensional Betti number of a (p+1)-fold connected plane region equals p.

E3:

A closed orientable surface of genus p has for its one-dimensional Betti group the group of all linear forms:

\sum_{i=1}^{p}u_{i}\xi_{i} + \sum_{i=1}^{p}v_{i}\eta_{i}….with integral u^{i}, v^{i}

Here one takes as generators \xi_{i} and \eta_{i} the homology classes of the 2p canonical closed curves. (For example, in the text Geometry and Imagination by David Hilbert and Cohn-Vossen)

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 consisting of all elements of finite order. The one-dimensional Betti number of a non-orientable surface of genus p is (p-1).

The two dimensional Betti numbers of a closed surface equals 1 or 0 according as the surface is orientable or not. The analogous assertion also holds for the n-dimensional Betti number of an n-dimensional closed manifold.

E4:

Let P be spherical shell and Q be the region enclosed between two coaxial surfaces. The one dimensional Betti number of P is 0, the one dimensioinal Betti number of Q is 2, while the two-dimensional Betti numbers of P and Q have the value 1.

E5:

One can choose as generators of the one-dimensional Betti group of the three dimensional torus the homology classes of the three cycles z_{1}^{1}, z_{2}^{1}, z_{3}^{1} which are obtained from the three axes of the cube by identifying the opposite sides. As generators of the two dimensional Betti group, we can use the homology classes of the three tori into which the three squares through the centre 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-dimensional Betti number of the manifold.

E6:

For the one- as well as the two-dimensional Betti group of the manifold S^{2} \times S^{3} we have the infinite cyclic group (the corresponding Betti numbers are therefore equal to 1). As z_{0}^{1} choose the cycle which arises from the line segment aa^{'} under the identification of the two spherical surfaces, and as z_{0}^{2}, any sphere which is concentric with the two spheres S^{2} and s^{2} and lies between them.

It is no accident that in the last two examples the one- and two-dimensional Betti numbers of the three-dimensional manifolds in question are equal to one another; indeed, we have the more general theorem, known as the Poincare duality theorem, which says that in an n-dimensional closed orientable manifold, the r- and the (n-r)-dimensional Betti numbers are equal, for 0 \leq r \leq 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 z^{r} which is not \approx 0 in M^{n} a cycle z^{n-r} such that the so-called “intersection number” of these cycles is different from zero.

E7:

The product of the projective plane with the circle is a non-orientable three-dimensional manifold M^{3}. 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 M^{3} is 1 (every one dimensional cycle is homologous to a multiple of the circle which goes around through the centre of the solid torus); the two-dimensional Betti group (the torus with the aforementioned identification indeed does not bound, but is a boundary divisor of order 2) (Note: the r-dimensional torsion group T_{r}(K) of a complex K is the finite group which consists of all elements of finite order of the Betti group H_{r}(K). The factor group H_{r}(K)/T_{r}(K) is isomorphic to F_{r}(K)). Here again there is a general law; the (n-1)-dimensional torsion group of a closed non-orientable n-dimensional manifold is always a finite group of order 2, while an orientable M^{n} has no (n-1)-dimensional torsion. One can also see from our example that for non-orientable closed manifolds Poincare’s duality theorem does not hold in general.

F:

If we consider the polyhedra mentioned in examples 1, 2, and 3 as polyhedra of three-dimensional space, we notice immediately that both the polyhedron and the region complementary to it in \mathcal{R}^{3} have the same one-dimensional Betti numbers. This can be seen most easily if one chooses as generators of the group H_{1}(P) the homology classes of the cycles x_{1}^{1} and x_{2}^{1} respectively, x^{1} and y^{1}, and as the generators of the group H_{1}(R^{3}-P) the homology classes of the cycles Z_{1}^{1} and Z_{2}^{1} respectively, X^{1} and Y^{1}. This remarkable fact is a special case of one of the most important theorems of all topology, the Alexander duality theorem: the r-dimensional Betti number of an arbitrary polyhedron lying in \mathcal{R}^{n} is equal to the (n-r-1)-dimensional Betti number of its complementary region R^{n}-P for 0 < r < n-1.

The proof of Alexander’s duality theorem is based on the fact that for every x^{r} not \approx0 in P, there exists a x^{n-r-1} in R^{n}-P which is linked with it —- an assertion whose intuitive sense is made sufficiently clear by drawing diagrams. This fact also hols for r=n-1 (since pairs of points which are separated by the (n-1)-cycle concerned appear as zero-dimensional linked cycles. From those considerations the theorem easily follows that the number of regions into which a polyhedron decomposes R^{n} is1 larger thanthe (n-1)- dimensional Betti number of the polyhedron — a theorem which contains the n-dimensional Jordan theorem as a special case. Both this decomposition theorem and the Alexander duality theoremhold for curved polyhedra.

G:

I have intentionally placed in the centre of the presentation those topological theorems and questions which are based upon the concepts of teh algebraic complex and its boundary first, because today this branch pf topology — as no other — lies before us in such clarity that is is worthy of the attention of the widest mathematical circles; second, because since the work of Poincare it is assuming an incresingly more prominent position within topology. Indeed, it has turned out that a larger and larger part of topology is generated by the concept of homology. This holds true especially for the theory of continuous mappings of manifolds, which in recent years — principally through the work of Lefschetz and Hopf — has shown a significant advance to a large extent, this advance has been made possible by the reduction of a series of important questions to the algebraic investigations of the homomorphisms of the Betti group induced by continuous mappings. Recently, the development of set theoretic topology, especially that of dimension theory, has taken a similar turn; it is now known that the concepts of cycle, boundary, Betti groups, etc. hold not onlyfor polyhedra, but also can be generalized to include the case of arbitrary closed sets. Naturally, the circumstances here are much more complicated, but even in these general investigations we have now advanced so far that we are at the beginning of a systematic and entirely geometrical oriented theory of the most general structures of space, a theory which has its own significant problems and its own difficulties. This theory is also based principally on the concept of homology.

Finally, the part of topology which is concerned with the concept of cycle and homology as the part on which the applications of topology depend almost exclusively; the first applications to differential equations, mechanics, and algebraic geometry lead back to Poincare himself. In the last few years, 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 enumerative geometry by Van der Waerden, the pioneering work of Lefschetz in the field of algebraic geometry, the investigation of Birkhoff, Morse and others in the calculus of variations in the large, and numerous differential geometrical investigations of others, etc. One may say, without exaggeration, ANYONE WHO WISHES TO LEARN TOPOLOGY WITH AN INTEREST IN ITS APPLICATIONS MUST START WITH BETTI GROUPS, because today, just as in the time of Poincare, most of the threads which lead from topology to the rest of mathematics and bind topological theorems together into a recognizable whole lead through this point.

Cheers, cheers, cheers, 🙂

Nalin Pithwa

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