Topology…bare facts…part 4

Ref: Elementary Concepts of Topology by Paul Alexandroff, Available Amazon India.

Simplicial Mappings and Invariance Theorems:

A.

If we review what has been said uptil now, we see that the discussion has turned around two main concepts: complex on the one hand and topological space on the other. The two concepts correspond to the two interpretations of the concept basic to all of geometry — the concept of geomtrical 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, etc. or simplexes 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 homogeneous elements. Such a collection must be recognized 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 neighbourhoods. (Note: the set theoretic interpretation of a figure also goes back to the oldest times — think, for example, of the concept of geometrical locus. This interpretation became prominent in modern geometry after the discovery of analytic geometric.)

As we mentioned before, in the work of Poincare both interpretations appear simultaneously. With Poincare the combinatorial scheme never becomes an end in itself; it is always a tool, an apparatus for the investigation of the “manifold itself,” hence, ultimately a point set. Set-theoretic methods sufficed, however, in Poincare’s earliest work because his investigations touched only manifolds and slightly more general geometrical structures. (Note: Of course, in the fields of differential equations and celestial mechanics, the works of Poincare lead us very close to the modern formulation of questions in set-theoretic topology) For this reason, and also in view of the great difficulties connected with the general formulation of the concept of manifold, one can hardly speak of an intermingling or merging of the two methods in Poincare’s time.

The further development of topology is marked at first by a sharp separation of set-theoretic and combinatorial methods: combinatorial topology had been at the point of believing in no geometric 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 hardly 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 flowering of topology, (Alexander, Lefshetz and Veblen in topology itself and Birkhoff and his successors in the topological methods of analysis), 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 centre of Brouwer’s work stand the topological invariance theorems. We collect under this name primarily theorems which maintain that if a certain property belonging to geometrical complexes holds for one simplicial decomposition of a polyhedron, then it holds for all simplicial decompositions of homeomorphic polyhedra. The classical examples of such an invariance theorem is Brouwer’s theorem on the invariance of dimension: If an n-dimensional complex K^{n} appears as a simplicial decomposition of a polyhedron P, then every simplicial decomposition of P, as well as every simplicial decomposition of a polyhedron P_{1} which is homeomorphic to P, is likewise an n-dimensional complex.

Along with the theorem on invariance of dimension we mention as a second example the theorem on the invariance of the Betti groups proved by Alexander: if K and K_{1} are simplicial decomposition of two homeomorphic polyhedra P and P_{1}, then every Betti group of K is isomorphic to the corresponding Betti group of K_{1}. (Note: The scope of these two theorems is not if one assumes that K and K_{1} are two curved simplicial decompositions of one and the same polyhedron, for under a topological mapping an (arbitrary, also curved) simplicial decomposition of P_{1} goes over into an (in general, curved) simplicial decomposition of P . One could, on the other hand, limit oneself to rectilinear simplicial decomposition of ordinary (“rectilinear”) polyhedra, but then both polyhedra P and P_{1} must be considered. If, indeed, P is an arbitrary polyhedron in a (curved) simplicial decomposition K_{1} then there is a topological mapping of P into a sufficiently high dimensional Euclidean space in which P goes over into a rectilinear polyhedron P^{'}, and K, into its rectilinear simplicial decomposition K^{'}.)

B.

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 linear functions, while the simplicial approximations of a continuous mapping are analogous to linear interpolations 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.

C.

To each vertex a 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_{1}, a_{2}, \ldots, a_{s} are vertices of a simplex K, then there exists in K^{'} a simplex which has as its vertices precisely f(a_{1}), \ldots, f(a_{x}) (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^{'}. (Note: If one thinks of K as an algebraic complex modulo 2, then it is found to be convenient in the case where a simplex |x^{r}| of K is mapped onto a lower dimensional simplex of K^{'} to say that the image of |x^{r}| is zero (that is, as an r-dimensional simplex, it vanishes))One obtains in this way a mapping f of the complex K into the complex K^{'}. (If to each element of the set M there corresponds an element of the set N, then one speaks of a mapping of the set M into the set N. The mapping is a mapping of M onto N if every element of N is the image of at least one element of M) A mapping of this kind from K to K^{'} is called a simplicial mapping of one geometrical complex into the other.

D.

If now, x^{r}=(a_{0}a_{1}\ldots a_{r}) is an oriented simplex of K, then two cases are to be distinguished: either the image points b_{0}=f(a_{0}), \ldots, b_{r}=f(a_{r}) are distinct vertices of K^{'}, in which case, we set f(x^{r})=(b_{0}b_{1}\ldots b_{r}); or, else at least two of the image points b_{i}, b_{j} coincide, in which case, we define f(x^{r})=0. Thus, the simplicial image of an oriented simplex will be either an oriented simplex of the same dimension or zero. (the geometrical meaning of the occurence of zero is clear: if two vertex images coincide, then the image simplex is degenerate; that is, it vanishes if one considers it as an r-dimensional simplex. The same mapping convention also holds when a non-oriented simplex is interpreted as an element of an algebraic complex modulo 2).

Now, let an algebraic subcomplex C^{r} = \sum t^{i}x_{i}^{r} of the complex K be given. According to what was just said, the simplicial mapping f of K into K^{'} yields a well-defined image f(x^{r}) for each oriented simplex x^{r}, where f(x^{r}) is either an oriented r-dimensional simplex of K^{'}, or zero. Consequently, f(C^{r}) = \sum t^{i}f(x_{i}^{r}) is a uniquely determined (perhaps vanishing) r-dimensional algebraic subcomplex of K^{'} which is called the image of C^{r} under the simplicial mapping of K into K^{'}. (One can also speak directly of the simplicial mapping f of the algebraic complex C^{r} into the geometrical complex K^{'})

E.

From these definitions follows easily the intuitive and extremely important:

(E1) Conservation Theorem: If the oriented simplex x^{r} is simplicially mapped into K^{'}, then f(\dot{x}^{r})= [f(x^{r})]^{'}. From this, by simple addition: f(\dot{C}^{r}) = [f(C^{r})]'

In words, the image of the boundary (of an arbitrary algebraic complex) is (for every simplicial mapping) equal to the boundary of the image.

From the first conservation theorem follows without difficulty the extraordinarily important:

(E2) Conservation Theorem: If the algebraic complex C^{n} is simplicially mapped into the complex consisting of a single simplex |x^{n}|, and if f(C^{n})=\dot{x}^{n} (where x^{n} is some particular orientation of the simplex |x^{n}|), then

f(C^{n}) = x^{n}.

For, on the one hand, it is necessary that f(C^{n})=tx^{n} (where t is an integer which, a priori, could be zero), while, on the other hand, according to the assumption, f(C^{n})=\dot{x}^{n}, and by the first conservation theorem, f(\dot{C}^{n})=t \dot{x}^{n}. Therefore, it must be that t=1. QED.

As an immediate application of the second conservation theorem, we prove the following remarkable fact:

(E3) Conservation Theorem: Let C^{n} be an arbitrary algebraic complex, and C_{1}^{n} a subdivision of C^{n}. To each vertex a of C_{1}^{n} we let correspond a completely arbitrary vertex f(a) of that simplex of f(C^{n}) which contains the point a in its interior; then, for such a simplicial mapping f of the complex C_{1}^{n} it follows that :

f(C_{1}^{m})=C^{m}.

(Note: In particular, if a is not only a vertex of C^{n} but also a vertex of C^{n} then our condition means that f(a)=a).

Proof:

For n=0, the theorem is trivially true.

We assume that it is proved for all (n-1)- dimensional complexes, and consider an n-dimensional complex C^{n}. Let x_{i}^{n} be a simplex of C^{n} = \sum t^{i}x_{i}^{n}, and let X_{i}^{n} be the subdivision of x_{i}^{n} which is given by C_{1}^{n}. The mapping f of the boundary of X_{i}^{n} obviously fulfills the assumptions of our theorem, so that (because of the assumption of its validity for (n-1)) f(\dot{X_{i}}^{n}) = \dot{x_{i}}^{n}; therefore, by the second conservation theorem, f(X_{i}^{n})=x_{i}^{n}. Summing this over all simplexes x_{i}^{n} one has

f(C_{1}^{n}) = f(\sum_{i} t^{i}X_{i}^{n})= \sum_{i} t^{i}x_{i}^{n}=C^{n}.

QED.

Remark: Clearly, all three conservation theorems together with the proofs given here are valid modulo 2; in this case, they may be considered as statements concerning geometrical complexes. (Note : they are valid quite generally for an arbitrary coefficient domain). It is recommended that you verify this by examples — it suffices to choose a triangle for C^{n}, and an arbitrary subdivision of it for C_{1}^{n}.

F.

We apply the third conservation theorem to the proof of the tiling theorem, already mentioned in blog article 1; however, we shall now formulate it not for a cube but for a simplex:

For sufficiently small \epsilon>0, every \epsilon-covering of an n-dimensional simplex has order greater than or equal to (n+1). (Note: By an \epsilon-covering of a closed set F one means a finite system F_{1}, F_{2}, \ldots, F_{s} of closed subsets of F, which have as their union the set F and which are less than \epsilon in its diameter. The order of a covering (or more generally, of an arbitrary finite system of point sets) is the largest number k with the property that there are k sets of the system which have at least one common point)

First, we choose \epsilon so small that there is no set with a diameter less than \epsilon which has common points with all (n-1)- dimensional faces of |x^{n}|. In particular, it follows that no set with a diameter less than \epsilon can simultaneously contain a vertex a_{i} of |x^{n}| and a point of the face |x_{i}^{n+1}| opposite to the vertex a_{i}. Now let

(Equation 1) F_{0}, F_{1}, F_{2}, \ldots, F_{s}

be an \epsilon-covering of |x^{n}|. We assume that the vertex a_{i} i=0,1,2, \ldots, n lies in F_{i} (According to our assumption two different vertices cannot belong to the same set F_{i}; a vertex a_{i} can, however, be contained in sets of our covering other than F_{i}). If there are more than (n+1) sets F_{i}, then we consider some set F_{j}, j>n, and proceed as follows: we look for a face |x_{i}^{n+1}| of |x^{n}| which is disjoint from F_{j} (such a face exists, as we have seen), strike out the set F_{j} from (1) and replace F_{i} by F_{i} \bigcup F_{j}, renaming this last set F_{i}. By this proceddure, the number of sets in (1) is diminished by 1 without increasing the order of the system of sets in (1). At the same time, the condition that none of the sets contains simultaneously a vertex and a point of the face opposite to the vertex will not be violated. By finite repetition of this process, we finally obtain a system of sets:

(Equation 2) F_{0}, F_{1}, \ldots, F_{n}

containing the sequence of vertices, a_{0}, a_{1}, a_{2}, \ldots, a_{n} of |x^{n}|, a_{i} \in F_{i}, with the property that no set contains both a vertex a_{i} and a point of |x_{i}^{n+1}|. Furthermore, the order of (2) is at most equal to the order of (1). It therefore suffices to prove that the order of (2) is equal to (n+1), that is, to show that there is a point of |x^{n}| which belongs to all sets of (2). As quite elementary convergence considerations show, the latter goal will be reached if we show that in each subdivision |X^{n}| of |x^{n}|, no matter how fine, there, is necessarily a simplex |y^{n}| which possesses points in common with all the sets F_{0}, F_{1}, F_{2}, \ldots, F_{n}.

Let b be an arbitrary vertex of the subdivision |X^{n}|. Now b belongs to at least one of the sets F_{i}; if it belongs to several, then we choose a particular one, for example, the one, with the smallest subscript. Let this be F_{i}, then we define f(b)=a_{i}. In this way, we get a simplicial mapping f of |X^{n}| into |x^{n}| ; because otherwise, if the whole simplex |x^{r}|, and in particular the point b, were to lie on the face |x^{n+1}| of |x^{n}| opposite to a_{i}=f(b), then the point b could not belong to F_{i}. Since, according to the third conservation theorem (understood modulo 2), |x^{n}| = f(|X^{n}|), there must be among the simplexes of |X^{n}| at least one which will be mapped by f onto |x^{n}| (and not onto zero); (note: we consider |x^{n}| as an algebraic complex modulo 2) the vertices of this simplex must lie successively in F_{0}, F_{1, F_{2, \ldots, F_{n}}} Q.E.D. (The above proof of the tiling theorem is due in essence to Sperner: 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 is in fact, with respect to any domain of coefficients))

G.

If F is a closed set, then the smallest number r with the property that F possesses for each \epsilon>0 an \epsilon-covering of order r+1 is called the general or Brouwer dimension of the set F. It will be denoted by dim F. If F^{'} is a subset of F, then clearly, dim \hspace{0.1in} F \leq dim \hspace{0.1in} F^{'}. It is easy to convince oneself that two homeomorphic sets F_{1} and F_{2} 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 r-dimensional simplex, and consequently for every r-dimensional polyhedron P, that dim P is greater than or equal to r. For the proof of the reverse inequality, we have only to construct, for \epsilon>0, an \epsilon-covering of P of order (r+1). Such coverings are provided by the so-called barycentric coverings of the polyhedron.

H.

First, we shall introduce the barycentric subdivision of an n-dimensional complex K^{n}. If n=1, the barycentric subdivision of K^{1} is obtained by inserting the midpoints of the one-dimensional simplexes of which K^{1} consists (that is, if the midpoint of each 1-simplex of K^{1} is called the barycentre of that simplex, then the barycentric subdivision of K^{1} together with the line segments whose endpoints are these points). If n=2, the barycentric subdivision consists in dividing each triangle of K^{2} into six triangles by drawing its three medians. Suppose that the barycentric division is already defined for all r-dimensional simplexes (and their faces) and projecting the resulting subdivision of the boundary of each (r+1) dimensional simplex of K from the centre of gravity (barycentre) of this simplex. It is easy to see by induction that :

(H1) Each n-dimensional simplex is subdivided barycentrically into (n+1)! simplexes.

(H2) Among the n+1 vertices of an n-dimensional simplex |y^{n}| of the barycentric subdivision K_{1} of K.

(H2_a) one vertex is also a vertex of K (this vertex is called the leading vertex of |y^{n}|)

(H2_b) one vertex is the center of mass of an edge |x^{1}| of K (which possesses the leading vertex of |y^{n}| as a vertex).

(H2_c) one vertex is the centre of mass of a triangle |x^{2}| of K (which is incident with the edge |x^{1}|.

\vdots

(H2_n) one vertex is the centre of mass of an n-dimensional simplex |x^{n}| of K (which contains the previously constructed |x^{1}|, latex |x^{2}|, \ldots, |x^{n-1}|$ among its faces).

I.

One means by a barycentric star of K, the union of all simplexes of the barycentric subdivision K_{1} of K which possess a fixed vertex a of K as their common (leading) vertex. The vertex a is called the centre of the star.

It is easily shown that a point of a simplex |x| of K can belong only to those barycentric stars whose centres are vertices of the simplex |x|. From this we conclude that:

(I_a) If certain barycentric stars B_{1}, B_{2}, \ldots, B_{s} have a common point p, then their centres are vertices of one and the same simplex of K (namely, that simplex which contains the point p in its interior).

(I_b) There is a positive number \epsilon = \epsilon(K) with the property that all points of the polyhedron P (whose simplicial decomposition is K) which are at a distance of less than \epsilon from a simplex x of K can belong only to those barycentric stars which have their centres at the vertices of s. (This follows simply from the fact that all other stars are disjoint from s, and consequently have a positive distance from this simplex.)

The second of these two properties we will use later. We remark in passing that the converse of proposition (I_a) is also true: if the centers of the barycentric stars B_{1}, B_{2}, \ldots, B_{s} lie at the vertices of one and the same simplex s of K, then they have a common point (namely, the centre of mass of the simplex x). As a consequence, we have the following theorem:

Arbitrarily chosen barycentric stars of the complex K have a non-empty intersection if and only if their centres are vertices of a simplex of K.

In particular, the last statement includes the following corollary:

The system of all barycentric stars of an n-dimensional complex has order n+1.

If one chooses a sufficiently fine simplicial decomposition K of an n-dimensional polyhedron P, then one can arrange it so that the barycentric stars of K are all of diameter less than \epsilon, which therefore gives an \epsilon covering of P of order (n+1). QED.

The agreement of Brouwer’s general dimension with the elementary geometrical dimension of a polyhedron, as well as the invariance of dimension are, hereby, completely proved.

J.

Remark I. If a finite system of sets

(Equation 3)F_{1}, F_{2}, \ldots, F_{s}

and the system of vertices

a_{1}, a_{2}, \ldots, a_{s}

of a complex K are related to one another in such a way that the sets F_{i_{0}, F_{i_{1}}}, \ldots, F_{i_{r}} have a non-empty intersection if and only if the vertices a_{i_{0}}, a_{i_{1}}, \ldots, a_{i_{r}} belong to a simplex of K, then the complex K is called a nerve of the system of sets given by (Equation 3). Then, one can formulate the theorem of the preceding section in the following way:

Every complex K is a nerve of the system of its barycentric stars.

K.

Remark II:

The previous remark leads us to the point at which the concept of complex attains its complete logical rigour and generality: it is exactly this example of the nerve of a system of sets which shows that the conceptual content of the word “complex” is, frequently, to a great extent independent of the “geometrical stuff” with which our concept operates. A complex, considered as the nerve of a system of sets, (for example, the system of its own barycentric stars), is above all an abstract scheme which gives us information about the combinatorial structure of the system of sets. What the simplexes look like — whether they are “straight” or “curved” — what the nature of the vertices is, is completely immaterial to us; the only thing that concerns us is the structure of the network of vertices of the complex, that is, the manner in which the system of all vertices of the complex decompose 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 set E of (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:

Ii) Each individual vertex is a frame

(ii) 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 each 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 simplex. The dimension of the frame is called the dimension of the simplex; the simplexes spanned by the sub-frames of a given simplex x^{n} are called the faces of x^{n}. 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. If this is done, then the concept of an abstract algebraic complex with respect to a definite domain of coefficients necessarily results.

To be continued in next blog article,

Cheers,

Nalin Pithwa

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