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

Topology…bare facts…part 3

II. Algebraic Complexes

A. There is something artificial about considering a manifold as a polyhedron : the general idea of the manifold as a homogeneous structure of n-fold extent, an idea which goes back to Riemann, has nothing intrinsically to do with the simplicial decompositions which were used to introduce polyhedra. Poincare, 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 equations. However, within only four years after the appearance of the first pioneering work he took the point of view which today is known as combinatorial topology, and essentially amounts to the consideration of manifolds as polyhedra. The advantage of this viewpoint lies in the fact that with its help the difficult — 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 simplexes of a simplicial decomposition of the polyhedron (that is, the geometrical complex) — which opens 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 connected with the concept of manifold are by no means solved by the definition which we have adopted.

B. We shall now turn to briefly discuss the 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 algebraic complex.

An oriented one-dimensional simplex is a directed line segment (a_{0}a_{1}), that is, a line which is traversed from the vertex a_{0} to the vertex a_{1}. One can also say: an oriented one-dimensional simplex is one with a particular ordering of its endpoints. If we denote the oriented line (a_{0}a_{1}) by x^{1} (where the superscript 1 gives the dimension), the oppositely oriented simplex (a_{1}a_{0}) will be denoted by -x^{1}. This same line considered without orientation we denote by

|x^{1}|= |a_{0}a_{1}| = |a_{1}a_{0}|

An oriented two dimensional simplex — an oriented triangle — is a triangle with a particular sense of rotation or with a particular ordering of its vertices; at the same time, no distinction is made between orderings which differ from another by an even permutation, so that (a_{0}a_{1}a_{2}), (a_{1}a_{2}a_{0}) and (a_{2}a_{0}a_{1}) represent one permutation, and (a_{0}a_{2}a_{1}), (a_{1}a_{0}a_{2}), (a_{2}a_{1}a_{0}), the other is called -x^{2}. The triangle considered without orientation will again be denoted by |x^{2}|. The essential thing here is that in an oriented triangle the boundary is also to be understood as an oriented (directed) polygon. The boundary of an oriented triangle (a_{0}a_{1}a_{2}) is the collection of oriented lines (a_{0}a_{1}), (a_{1}a_{2}) and (a_{2}a_{0}). If one denotes the boundary of x^{2} by \dot{x}^{2}, then our last statement is expressed by the formula

(1^{'}) \dot{x}^{2} = (a_{0}a_{1}) + (a_{1}a_{2}) + (a_{2}a_{0})

or equivalently

(1) \dot{x}^{2} = (a_{1}a_{2}) - (a_{0}a_{2}) + (a_{0}a_{1})

We may also say that in the boundary of x^{2}, the sides (a_{1}a_{2}) and (a_{0}a_{1}) appear with the coefficient +1, and the side (a_{0}a_{2}) with the coefficient -1.

C. Consider now any decomposition into triangles (or, triangulation) of a two-dimensional polyhedron P^{2}. The system comprised of the triangles together with their edges and vertices forms what we called earlier a two-dimensional geometrical complex K^{2}. Now, we choose a particular (but completely arbitrary) orientation x_{i}^{2} of any one of the triangles |x_{i}^{2}|, 1 \leq i \leq \alpha_{2}, of our complex; in a similar way, we can choose any (Note: where \alpha_{0}, \alpha_{1}, \alpha_{2}) denote-, the number of two-, one-, or zero-dimensional elements of the geometrical complex.) (Note: if one imagines x^{2} = (a_{0}a_{1}a_{2}) as a symbolic product of three “variables”, a_{0}, a_{1}, a_{2}, one may write \dot{x}^{2} = \sum_{i=0}^{2} (-1)^{i} \frac{\partial {x^{2}}} {\partial {a_{i}}}) particular orientation x_{j}^{1} of one of the sides |x_{j}^{1}|, 1 \leq j \leq \alpha_{1}. The system of all x_{i}^{2} we call an oriented two-dimensional complex C^{2}, that is, an orientation of the geometrical complex K^{2}. For the oriented complex C^{2} we use the notation

C^{2} = \sum_{i=1}^{\alpha_{2}} x_{i}^{2}

In order to indicate that C^{2} is the result of orienting the complex K^{2}, we shall sometimes write |C^{2}|=K^{2}.

The boundary of each oriented triangle x_{i}^{2} can now be represented by a linear form

(2) \dot{x}_{i}^{2} = \sum_{j=1}^{\alpha_{i}}t_{i}^{j}x_{j}^{1}

where t^{j}=+1, -1, 0 according to whether the oriented line x_{j}^{1} occurs in the boundary of the oriented triangle x_{i}^{2} with the coefficient +1, -1 or not at all.

If one sums equation (2) over all i, 1 \leq i \leq \alpha_{2}, one obtains

\sum_{i=1}^{\alpha_{2}} \dot{x}_{i}^{2} = \sum_{i=1}^{\alpha_{2}} \sum_{j=1}^{\alpha_{1}}t_{i}^{j}x_{j}^{1} = \sum_{j=1}^{\alpha_{1}} u^{j}x_{j}^{1}, where u^{j} = \sum_{i=1}^{\alpha_{2}}t_{i}^{j}

The above expression

\sum_{j=1}^{\alpha_{1}} u^{j}x_{j}^{1}

is called the boundary of the oriented complex C^{2} and is denoted by \dot{C}^{2}.

Examples.

Eg.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 direction of arrows. (I am not able to produce the diagram in LaTeX here, you will have to do reverse engineering sort of thing and produce the diagram from the system of equations given):

The boundary of the resulting oriented complex is:

C^{2}=x_{1}^{2} + x_{2}^{2} + x_{3}^{2}+x_{4}^{2}

equals zero, because each edge of the tetrahedron appears in the two triangles of which it is a side with different signs. In formulae:

x_{2}^{2} = (a_{0}a_{1}a_{2})

x_{2}^{2} = (a_{1}a_{0}a_{3})

x_{3}^{2} = (a_{1}a_{3}a_{2})

x_{4}^{2} = (a_{0}a_{2}a_{3})

and

x_{1}^{1} = (a_{0}a_{1})

x_{2}^{1} = (a_{0}a_{2})

x_{3}^{1} = (a_{0}a_{3})

x_{4}^{1} = (a_{1}a_{2})

x_{5}^{1} = (a_{1}a_{3})

x_{6}^{1} = (a_{2}a_{3})

(PS: from the above, you will be able to reconstruct the diagram of the planar tetrahedron)

Thus,

\dot{x}_{1}^{2} = + x_{1}^{1} - x_{2}^{1} + x_{4}^{1}

\dot{x}_{2}^{2}= - x_{1}^{1} + x_{3}^{1} - x_{5}^{1}

\dot{x}_{3}^{2} = - x_{4}^{1} + x_{5}^{1} - x_{4}^{1}

\dot{x}_{4}^{2} = + x_{2}^{1} - x_{3}^{1} + x_{6}^{1}

Adding all the above, we get \dot{C}^{2} = \sum_{i=1}^{4}\dot{x}_{i}^{2}=0.

Eg. 2:

If one orients the ten triangles of the triangulation of the projective plane shown (again I am not able to produce in LaTeX) as indicated by the arrows, and puts:

C^{2} = \sum_{i=1}^{10} x_{i}^{2}

then

\dot{C}^{2} = 2x_{1}^{1}+2x_{2}^{1}+2x_{3}^{1}….equation (3)

The boundary of the oriented complex consists, therefore, of the projective line AX (composed of the three segments x_{1}^{1}, x_{2}^{1}, x_{3}^{1}) counted twice. With another choice of orientations x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, \ldots, x_{10}^{2} of the ten triangles of this triangulation one would obtain another oriented complex and its boundary would be different from equation (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 this figure, the boundary of the resulting complex

C^{2} = \sum_{i=1}^{10}x_{i}^{2}

is never zero. In fact, the following general result (which can be taken as the definition of orientability of a closed surface) holds:

A closed surface is orientable if and only if one can orient the triangles of any of its triangulations in such a way that the oriented complex thus arising has boundary zero.

Eg. 3.

In the triangulation and orientation for the Mobius band given in the figure (again not shown but can be reverse engineered) we have:

\dot{C}^{2} = 2x_{1}^{1}+x_{2}^{1} + x_{3}^{1} + x_{4}^{1} + x_{5}^{1}

C. 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, x_{i}^{2}; furthermore, as the boundary of the oriented complex C^{2}=\sum x_{i}^{2}, there appeared a linear form \sum u^{i}x_{j}^{1} whose coefficients are, in general, taken as arbitrary integers. Such linear forms are called algebraic complexes. The same considerations hold in the n-dimensional case, if we make the general definition:

Definition I.

An oriented r-dimensional simplex x^{r} is an r-dimensional simplex with an arbitrarily chosen ordering of its vertices,

x^{r} = (a_{0}a_{1}\ldots a_{r})

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^{r}| possesses two orientations, x^{r} and -x^{r}.

Note: A zero dimensional simplex has only one orientation, and thus it makes no sense to distinguish between x^{0} and |x^{0}|.

Remark:

Let x^{r} be an oriented simplex. Through the r+1 vertices of x^{r} passes a unique r-dimensional hyperplane R^{r} (the R^{r} in which x^{r} lies ), and to each r-dimensional simplex |y^{r}| of R^{r} there exists a unique orientation y^{r} such that one can map R^{r} onto itself by an affine mapping with a positive determinant in such a way that under this mapping the oriented simplex x^{r} goes over into the simplex y^{r}. Thus, the orientation x^{r} of |x^{r}| induces a completely determined orientation y^{r} for each simplex |y^{r}| which lies in the hyperplane R^{r} containing x^{r}. Under these circumstances, one says that the simplexes x^{r} and y^{r} are equivalently — or consistently — oriented simplexes of R^{r}. One says also that the whole coordinate space R^{r} is oriented by x^{r}, which means precisely that from the oriented simplex x^{r} all r-dimensional simplexes of R^{r} acquire a fixed orientation. In particular, one can orient each r-dimensional simplex lying in R^{r} so that it has an orientation equivalent to that of x^{r}.

Definition II. A linear form with integral coefficients C^{r} = \sum t^{i} x_{i}^{r} whose indeterminants x_{i} are oriented r-dimensional simplexes, is called an r-dimensional algebraic complex.

(the above definition also has meaning in the case r=0. A zero dimensional algebraic complex is a finite system of points with which some particular (positive, negative or vanishing) integers are associated as coefficients ; in modern terminology, C^{r} would be called an (integral) r-dimensional chain)).

Expressed otherwise: an algebraic complex is a system of oriented simplexes, each of which is to be counted with a certain multiplicity (that is, each one is provided with an integral coefficient). Here it will generally be assumed only that these simplexes lie in one and the same coordinate definite simplicial decomposition of a polyhedron (that is, a geometrical complex). On the contrary, the simplexes of an algebraic complex may, in general, intersect one another arbitrarily. In case, the simplexes of an algebraic complex C^{r} belong to a geometrical complex (that is, re obtained by orienting certain elements of a simplicial decomposition of a polyhedron ), C^{r} is called an alagebraic subcomplex of the geometrical complex (of the given simplicial decomposition) in question; here, self-intersections of simplexes, of course, cannot occur. This case is to be considered as the most important.

D. Algebraic complexes are to be considered as a higher dimensional generalization of ordinary directed polygonal paths; here, however, the concept of polygonal path is taken from the outset in the most general sense: the individual lines may intersect themselves, and there may also exist lines which are traversed many times; moreover, one should not forget that the whole thing is to be considered algebraically, and a line which is traversed twice in opposite directions no longer counts at all. Furthermore, polygonal paths may consist of several pieces (thus, no requirement of connectedness). Thus, the two figures 8 and 9 represent polygonal paths which, considered as algebraic complexes, have the same structure. (that is, represent the same linear form).

Since the r-dimensional algebraic complexes of R^{n} may as linear forms, be added and subtracted according to the usual rules of calculating with such symbols, they form an Abelian group L_{r}(R^{n}). One can also consider, instead of the whole of R^{n}, a subspace G of R^{n}, for example, the r-dimensional algebraic complexes lying in it then form the group L_{r}(G) — a subgroup of L_{r}(R^{n}).

Also, the r-dimensional algebraic subcomplexes of a geometric complex K form a group — the group L_{r}(K); it is the starting point for almost all further considerations. Before we continue with these considerations however, I would like to direct the attention of the reader to the fact that the concepts “polyhedron,” “geometrical complex,” and “algebraic complex” belong to entirely different logical categories: a polyhedron is a point set, thus a set whose elements are ordinary points of R^{n} ; a geometrical complex is a finite set whose elements are simplexes, and, indeed, simplexes 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.

E. The boundary C^{r} of the algebraic complex C^{r} = \sum t^{i} x_{i}^{r} is defined to be the algebraic sum of the boundaries of the oriented simplexes \sum t^{i} x_{i}^{r} where the boundary of the oriented simplex x^{r} = (a_{0}a_{1}\ldots a_{r}) is the (r-1)-dimensional algebraic complex

(Equation 4) \dot{x}^{r} = \sigma_{i=0}^{r}(-1)^{i}(a_{0}\ldots \hat{a}_{i}\ldots a_{r} )

where \hat{a}_{i} means that the vertex a_{i} is to be omitted. In case the boundary of C^{r} is zero, then C^{r} is called a cycle. Thus, in the group L_{r}(R^{n}), and analogously in L_{r}(K) and L_{r}(G), the subgroup of all r-dimensional cycles Z_{r}(R^{n}), or, respectively, Z_{r}(K) and Z_{r}(G), is defined.

We can now say : a closed surface is orientable if and only if one can arrange, by a suitably chosen orientation of any simplicial (that is, in this case, triangular) decomposition of the surface, that the oriented complex 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 simplicial decomposition of a manifold, actually expresses a property of the manifold itself, since it can be shown that if one simplicial decomposition of a manifold satisfies the condition of orientability, the same holds true for every simplicial decomposition of this manifold.

Remark: If x^{n} and y^{n} are two equivalently oriented simplexes of R^{n} which have the common face |x^{n-1}|, then the face x^{n-1} (with some orientation) appears in x^{n} and y^{n} with the same or different signs according to whether the simplexes |x^{n}| and |y^{n}| lie on the same side or or different sides of the hyperplane R^{n-1} containing |x^{n-1}|. The proof of this assertion is left to the reader as an exercise.

F. As is easily 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}, with r>0, in R^{n} there is an algebraic complex lying in this R^{n} which is bounded by Z^{r} (Note: on the other hand, a zero-dimensional cycle in R^{n} bounds if and only if the sum of its coefficients equals zero (the proof is by induction on the number of sides of the bounded polygon)) ; indeed, it suffices to choose a point O of the space different from all the vertices of the cycle Z^{r} and to consider the pyramid erected above the given cycle (with the apex at O). In other words, if

Z^{r} =\sum_{(i)}c^{i}x_{i}^{r} and x_{i}^{r} = (a_{0}^{i}, a_{1}^{i}, \ldots, a_{r}^{i}) then one defines the (r+1)- dimensional oriented simplex x_{i}^{r+1} as

x_{i}^{r+1} = (O, a_{0}^{1}, \ldots, a_{r}^{i})

and considers the algebraic complex

C^{r+1} = \sum_{(i)}c^{i}x_{i}^{r+1}.

The boundary of C^{r+1} is Z^{r}, since everything else cancels out.

If we consider, however, instead of the whole of R^{n}, some region G in R^{n} (or more generally an arbitrary open set in R^{n}), then the situation is no longer so simple: a cycle of R^{n} lying in G need not bound in G. Indeed, if the region G is a plane annulus, then it is easy to convince oneself that there are cycles which do not bound in G (in this case closed polygons which encircle the center hole). Similarly, in a geometrical complex, there are generally some cycles which do not bound in the complex.

Consequently, one distinguishes the subgroups B_{r}(G) of Z_{r}(G), and B_{r}(K) of Z_{r}(K) of bounding cycles: the elements of B_{r}(G) or B_{r}(K) are cycles which bound some (r+1)- dimensional algebraic complex in G, or respectively K.

In the example of the triangulation given (of the projective plane; again figure not shown here; but can be reconstructed or reverse engineered) we see that it can happen that a cycle z does not bound in K, while a certain fixed integral multiple of it (that is, a cycle of the form tx where t is an integer different from zero) does bound some algebraic subcomplex of K. We have, in fact, seen that the cycle 2x_{1}^{1}+2x_{2}^{1}+2x_{3}^{1}=2z^{1} (the doubly counted projections line) in the triangulation of the bounds of the figure, while in the same triangulation there is no algebraic complex which has the cycle z^{1} = (x_{1}+x_{2}+x_{3}) as its boundary. It is thus suitable to designate as boundary divisors all of those cycles z^{r} of K (of G) for which there exists a non-zero integer t such that tz bounds in K (in G). Since t may have the value 1, the true boundaries (that is, bounding cycles) are included among the boundary divisors. The boundary divisors form, as is easily seen, a subgroup of the group Z_{r}(K), which we denote by B_{r}^{'}(K) or B_{r}^{'}(K); (obviously), the group B_{r(K)} is contained in the group B_{r}^{'}(K).

G. If z^{r} bounds in K (in G) we also say that z^{r} is strongly homologous to zero in K (in G), and we write z^{r} \sim 0 (in K or in G); if x^{r} is a boundary divisor of K (of G), we say that z is weakly homologous to zero and write z \approx 0 (in K or in G).

If two cycles of a geometrical complex K (or of region G) have the property that the cycle z_{1}^{r} - z_{2}^{r} is homologous to zero, one says that the cycles z_{1}^{r} and z_{2}^{r} are homologous to one another; the definition is valid for strong as well as for weak homology, so that one has the relations z_{1}^{r} \sim z_{z}^{r} and z_{1}^{r} \approx z_{2}^{r}. Examples of these relations are given in Fig 12 (z_{1} \sim z_{2}) and in the following figures.

In the Figures 15 and 16, G is the region of three-dimensional space which is complementary to the closed Jordan curve S or, respectively, to the lemniscate A.

H. Thus, the group Z_{r}(K) falls into an so-called homology classes, that is, into classes of cycles which are homologous to one another; there are in general both weak and strong homology classes, according to whether the concept of homology is meant to be weak or strong. If one again takes for K the geometrical complex of Figure 6, then there are two strong homology classes of dimension one, for every one-dimensional cycle of K is either homologous to zero (that is, belongs to the zero-class) or homologous to the projective line(that is, say, the circle x_{1}+x_{2}+x_{3}). Since every one-dimensional cycle of K in our case is a boundary divisor, there is only one weak homology class — the zero class.

As for the one-dimensional homology classes of the complexes in Figures 12 and 13, they may be completely enumerated if one notices that in Figure 12 every one-dimensional cycle satisfies a homology of the form z \sim tz_{1}, and, in Figure 13, a homology of the form z \sim uz_{1} + vz_{2}, where t, u, and v are 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).

If \zeta_{1} and \zeta_{2} are two homology classes and z_{1} and z_{2} are arbitrarily chosen cycles in \zeta_{1} and \zeta_{2} respectively, then one denotes by \zeta_{1}+\zeta_{2} the homology class to which z_{1}+z_{2} belongs. This definition for the sum of two homology classes is valid because, as one may easily convince oneself, the homology class designated by \zeta_{1}+\zeta_{2} does not depend on the particular choice of the cycles z_{1} and z_{2} in \zeta_{1} and \zeta_{2}.

The r-dimensional topology classes of K therefore form a group — the so-called factor group of Z_{r}(K) modulo B_{r}(K), or modulo B_{r}^{'}(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 corresponds to the strong homology concept (it is, therefore, the factor group Z_{r}(K) modulo B_{r}(K), denoted H_{r}(K)), while the second is the group of the weak homology classes (the factor group Z_{r}(K) modulo latex B_{r}^{‘}(K)$$, denoted F_{r}(K). (Note : In fact, one may write H_{r}(K)=F_{r}(K) \bigoplus T_{r}(K) where T_{r}(K) is the subgroup of H_{r}(K) consisting of the elements of finite order; the so-called torsion subgroup of H_{r}(K). In current usage, the group H_{r}(K) is more often referred to as the r-dimensional homology group rather than the r-dimensional Betti group).

From the above discussion, it follows that the full one-dimensional Betti group of the triangulation of the projective plane given in Figure 6 is a 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 is the infinite cyclic group, while in Figure 13 the group of all linear forms u\zeta_{1}+v\zeta_{2} (with integral u and v) is the one-dimensional Betti group. In the latter two cases, the full and reduced Betti groups coincide.

From simple group-theoretic theorems it follows that the full and the reduced Betti groups (of any given dimension r) have the same rank, that is, the maximal number of linearly independent elements which can be chosen from each group is the same. This common rank is called the r-dimensional Betti number (you can easily prove that the zero-dimensional Betti number of a complex K equals the number of its components (that is, the number of disjoint pieces of which the corresponding polyhedron is composed))of the complex K. The one dimensional Betti number for the projective plane is zero; for Figures 12 and 13 it is, respectively, 1 and 2.

I. The same definitions are valid for arbitrary regions G contained in R^{n}. 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 R^{n}. Indeed, the region complementary to that consisting of infinitely many circles converging to a point (Figure 18) has, as one may easily see, an infinite one-dimensional Betti number (consequently, the one-dimensional Betti group does not have finite rank, therefore, certainly not a finite number of generators).

J. The presentation of the basic concepts of the so-called algebraic topology which have 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 problems of sign (which often leads to rather tedious calculations) disappears. The elimination of orientation throughout, wherever it is actually possible, leads to the so-called “modulo-2” theory in which all coefficients of the linear forms that we have previously considered are replaced by their residue classes modulo 2. Thus, one puts the digit 0 in place of any even number, the digit 1 in place of any odd number, and calculates with them in the following way:

0+0=0,

0+1=1+0=1

0-1=1-0=1

0-0=1-1=0

1+1=0

In particular, an algebraic complex mod 2 is a linear form whose indeterminants are simplexes considered without orientation and with coefficients 0 and 1. (Note: One can consider geometrical complexes as a special case of the algebraic complexes modulo 2, if one interprets the coefficient 1 as signifying the occurence, and the coefficient 0 as signifying the non-occurence, of a simplex in a complex. This remark allows us to apply to geometrical complexes theorems which are proved for algebraic complexes).

The boundary of a simplex x^{n} appears in the theory mod 2 as a complex mod 2 which consists of all (n-1)- dimensional faces of the simplex x^{n}. Hence, the boundary mod 2 of an arbitrary complex C^{n} is defined as the sum (always mod 2) of the boundaries of the individual n- dimensional simplexes of C^{n}. One can also say that the boundary mod 2 of C^{n} consists of those and only those (n-1)- dimensional simplexes of C^{n} with which an odd number of n-dimensional simplexes are incident. You may easily construct examples which illustrate what has been said.

The concepts of cycle, homology, and Betti group mod 2 can be introduced exactly as in the “oriented” case. It should be especially noticed that all of our groups L_{r}(K), Z_{r}(K), H_{r}(K) and so on, are now finite groups (which we shall denote by L_{r}(K;Z_{2}), Z_{r}(K;Z_{2}), H_{r}(K; Z_{2}) etc. where Z_{2} is the group of residue classes modulo 2), because we now have, throughout, linear forms in finitely many indeterminants whose coefficients take only the two values 0 and 1. The triangulation of the projective plane given in Figure 6 can serve as an example of a two-dimensional cycle mod 2, for — considered as an algebraic complex mod 2 — it obviously has vanishing boundary. In the case of an n-dimensional complex K^{n} (that is, a complex consisting of simplexes of dimension less than or equal to n), just as Z_{n}(K^{n}) is isomorphic to H_{n}(K^{n}; Z_{2}); therefore, the two-dimensional Betti group modulo 2 for the projective plane is different from zero (its order is 2); the one-dimensional Betti group modulo 2 in the case of the projective plane is also of order 2.

Finally, one can also introduce the concept of the r-dimensional Betti number modulo 2; this is the rank mod 2 of the group H_{r}(K; Z_{2}), that is, the greatest number of elements u_{1}, u_{2}, \ldots, u_{s} of this group such that a relation of the form

t_{1}u_{1}+t_{2}u_{2}+ \ldots + t_{s}u_{s}=0

is satisfied only if all t_{i} vanish (where the t_{i} take only the values 0 and 1).

The zero-, one-, and two-dimensional Betti numbers of the projective plane modulo 2 all have the same value 1.

(The theory modulo 2 is due to Veblen and Alexander; it plays a very important role in modern topology, and has also prepared the way for the most general formulation of the concept of “algebraic complex”: If J is any commutative ring with identity, we mean by an algebraic complex of coefficient domain J is linear form whose indeterminants are oriented simplexes whose coefficients are elements of the ring J. Then, one defines boundaries, cycles, homology, etc. exactly as before but with respect to the ring J; in particular, the coefficient 1 or -1 is now to be interpreted as an element of the ring (which, indeed, according to the hypothesis contains an identity). If J is the ring of residue classes modulo m we speak of algebraic complexes modulo m. These complexes are gaining more and more significance in topology. Of greater importance as a coefficient domain is the set R of rational numbers; in particular, the cycles which we have called boundary divisors are nothing else but the cycles with integer coefficients which bound in K with respect to R (but not necessarily with respect to the ring of integers).

K. We close our algebraic-combinatorial considerations with the concept of subdivision. If one decomposes each simplex of a geometrical complex K into (“smaller”) simplexes such that the totality of all simplexes thus obtained again forms a geometrical complex K_{1}, then K_{1} is called a subdivision of K. If K consists of a single simplex, then the elements of the subdivision which lie on the boundary of the simplex form a subdivision of the boundary. From this it follows that if K^{n} is a geometrical complex, K_{1}^{n} its subdivision, and K^{r} the complex consisting of all the r-dimensional elements of K^{n} (together with all of their faces) and r \leq n, the totality of those elements of K_{1}^{n} which lie on simplexes of K^{r} forms a subdivision of K^{r}.

One can speak of subdivisions of algebraic complexes; we shall do this for the most important special case, in which the algebraic complex C^{n} = \sum t^{i}x_{i}^{n} is an algebraic subcomplex of a geometrical complex. Then, it is also true that the totality of all simplexes (considered without coefficients or orientation) of C^{n} forms a geometrical complex K^{n}. Let K_{1}^{n} be a subdivision of K^{n}, and |y^{n}| some simplex of C^{n}; then |y^{n}| lies on some particular simplex |x_{i}^{n}| of C^{n}. We now orient the simplex |y^{n}| the same as x_{i}^{n} amd give it the coefficient t^{i}. In this way. we obtain an algebraic complex which is called a subdivision of the algebraic complex C^{n}. One can easily see that the boundary of the subdivision C_{1}^{n} of C^{n} is a subdivision of the boundary of C^{n}. (Considered modulo 2, the process gives nothing beyond the subdivisions of a geometrical complex).

Cheers,

Nalin Pithwa.

I have an advice worth two cents. If you have read the previous blog and this blog, you might like to build up more your grasp of topology with the following two articles from Quanta Magazine:

1.

https://www.quantamagazine.org/topology-101-how-mathematicians-study-holes-20210126/

2.

https://www.quantamagazine.org/how-mathematicians-use-homology-to-make-sense-of-topology-20210511/

Topology: bare facts…part 2

I. Polyhedra, Manifolds, Topological Spaces

A. 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 three dimensional simplex is a tetrahedron. It is known and easy to show that if one considers all possible distributions of non-negative point masses 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 dimensions. We assume here that r+1 vertices of an r dimensional simplex are not contained in an (r-1)-dimensional hyperplane of the \mathcal{R}^{n} we are considering.

Definition of a simplex: One could also define a simplex as a smallest closed convex set which contains the given vertices.

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Recall the following definitions:

Closed set: The complement of an open set. (Rudin says any open ball is an open set. He also says that a subset E of a metric space X is closed if every limit point of E is a point of E. Also, a subset E is open if every point of E is an interior point of E. A point p is an interior point of E if there is a neighbourhood N of p such that N \subset E \subset X where X is the metric space under consideration).

Bounded set: Again, Rudin says: E is bounded if there is a real number M and a point q \in X such that d(p,q)<M for all p \in E. Note here d is the distance/metric function under consideration.

Convex set: We call a set E \subset \mathcal{R}^{k} convex if : \lambda x + (1-\lambda)y \in E whenever x \in E, y \in E and 0 < \lambda <1. (Open balls are convex, closed balls are convex, k-cells are convex).

Open cover: By an open cover of a set E in a metric space X we mean a collection \{ G_{\alpha}\} of open subsets of X such that E \subset \bigcup_{\alpha}G_{\alpha}

Compact set: A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. More explicitly, the requirement is that if \{ G_{\alpha}\} is an open cover of K, then there are finitely many indices \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} such that

K \subset G_{\alpha_{1}} \bigcup \ldots G_{\alpha_{n}}.

The notion of compactness is of great importance in analysis, especially in connection with continuity.

Connected Sets: Two subsets A and B of a metric space X are said to be separated if both A \bigcap \overline{B} and \overline{A} \bigcup B are empty, that is, if no point of A lies in the closure of B and no point of B lies in the closure of A. A set E \subset X is said to be connected if E is not a union of two nonempty separated sets.

Hyperplane :

in geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined.

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Any s+1 of the r+1 vertices of an r-dimensional simplex (0 \leq s \leq r) 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 set of \mathcal{R}^{n} which can be decomposed into r-dimensional simplexes in such a way that two simplexes of this decomposition either have no points in common or have a common face (of arbitrary dimension) as their intersection. The system of all the simplexes (and their faces) which belong to a simplicial decomposition of a polyhedron is called a geometrical complex.

The dimension of a polyhedron is not only independent of the choice of simplicial decomposition, but indeed it expresses a topological invariant of the polyhedron; that is to say, two polyhedra have the same dimension if they are homeomorphic (if they can be mapped onto another in a one-to-one and bicontinuous fashion).

With this 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 polyhedron (defined in the above sense, that is, composed of ordinary “rectilinear” simplexes). Clearly, curved polyhedra admit decomposition into “curved” simplexes (that is, topological images of ordinary simplexes); the system of elements of such a decomposition is again called a geometrical complex. (Comment: Such decompositions are also partitions due to apt equivalence relations. So mutually exhaustive, exclusive, disjoint equivalent partititions).

B. 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^{n}. They are characterized by the following two properties. First, the polyhedra must be connected (that is, it must not be composed of several disjoint sub-polyhedra); second, it must be “homogeneously n-dimensional” in the sense that every point p of M^{n} possesses a neighbourhood which can be mapped onto the n-dimensional cube in a one-to-one and bicontinuous fashion, such that the point p under this mapping corresponds to the center of the cube.

C. In order to recognize the importance of the concept of the manifold, it suffices to remark that most geometrical forms whose points may be defined by n parameters are n-dimensional manifolds; to these structures belong, for example, phase-spaces of dynamical problems. These structures are, to be sure, only rarely defined directly as polyhedra; rather, they appear —- as the examples of phase-spaces or the structures of n-dimensional differential geometry already show — as abstract spatial construction, in which a concept of continuity is defined in one way or another; it turns out here (and it can be proved rigorously under suitable hypotheses) that in the sense of the above mentioned definition of continuity the “abstract” manifold in question can be mapped topologically onto a polyhedron, and thus falls under our definition of manifold. In this way, the projective plane, which is first defined as an abstract two-dimensional manifold, can be mapped topologically onto a polyhedron of four-dimensional space without singularities of self-intersections.

D. Just one step leads from our last remarks to one of the most important and at the same time most general concepts of the whole of modern topology —- the concept of topological space. A topological space is nothing other than a set of arbitrary elements (called “points” of the space) in which a concept of continuity is defined. Now this concept of continuity is based on the existence of relations, which may be defined as local or neighbourhood relations — it is precisely these relations which are preserved in a continuous mapping of one figure onto another. Therefore, in more precise wording, a topological space is a set in which certain subsets are defined and are associated to the points of the space as their neighbourhoods. Depending upon which axioms these neighbourhoods satisfy, one distinguishes between different types of topological spaces. The most important among them are the so-called Hausdorff spaces (in which the neighbourhoods satisfy the four well-known Hausdorff axioms).

These axioms are the following:

(a) To each point x there corresponds at least one neighbourhood U(x); each neighbourhood U(x) contains the point x.

(b) If U(x) and V(x) are two neighbourhoods of the same point x, then there are exists a neighbourhood W(x), which is a subset of both.

(c) If the point y lies in U(x), there exists a neighbourhood U(y), which is a subset of U(x).

(d) For two distinct points x, y there exists two neighbourhoods U(x), U(y) without common points.

Using the notion of neighbourhood, the concept of continuity can be immediately introduced: Definition: A mapping f of a topological space R onto a (proper or improper) subset of a topological space Y is called continuous at the point x, if for every neighbourhood U(y) of the point y=f(x) one can find a neighbourhood U(x) of x such that all points of U(x) are mapped into points of U(y) by means of f. If f is continuous at every point of R, it is called continuous in R.

E. The concept of topological space is only one link in the chain of abstract space constructions which forms an indispensable part of all modern geometric thought. All of these constructions are based on a common conception of space which amounts to considering one or more system of objects — points, lines, etc. — together with systems of axioms describing the relations between objects. Moreover, this idea of a space depends only on these relations and not on the nature of the respective objects. Perhaps this general standpoint found its most fruitful formulation in David Hilbert’s Grundlagen der Geometrie; however, I would especially emphasize that it is by no means only for investigations of the foundation that this concept is of decisive importance, but for all directions of present-day geometry — the modern construction of projective geometry as well as the concept of a many-dimensional Riemannian manifold (and, indeed, earlier still, the Gaussian intrinsic differential geometry of surfaces) may suffice as examples!

F. With the help of the concept of topological space we have at last found an adequate formulation for the general definition of a manifold: A topological space is called a closed n-dimensional manifold if it is homeomorphic to a connected polyhedron, and furthermore, if its points possess neighbourhoods which are homeomorphic to the interior of the n-dimensional spheres.

G. 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.

As examples of higher-dimensional manifolds — in addition to n-dimensional spherical or projective space — the following may be mentioned:

(i) The three-dimensional manifold of line elements lying on a closed surface F. (It can be easily proved that if the surface F is a sphere then the corresponding M^{p} is projective space). (Please refer to Wikipedia for projective space)

(ii) The four-dimensional manifold of lines of the three dimensional projective space.

(iii) The three-dimensional torus-manifold: it arises if one identifies the diametrically opposite sides of a cube pairwise. You 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 M^{p+q} = M^{p} \times M^{q} from the two manifolds M^{p} and M^{q} as follows: the points of M^{p+q} are the pairs z=(x,y), where x is an arbitrary point of M^{p} and y an arbitrary point M^{q}. A neighbourhood U(x_{0}) of the point z_{0} = (x_{0},y_{0}) is defined to be the collection of all points z=(x,y) such that x belongs to an arbitrarily chosen neighbourhood of x_{0} and y belongs to an arbitrarily chosen neighbourhood of y_{0}. It is natural to consider the two points x and y (of M^{p} and M^{q} respectively) as the two “coordinates” of the point (x,y) of M^{p+q}.

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) is the product of two straight lines, the torus is the product of two circles, and the three-dimensional 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^{2} \times S^{2} of the surface of a sphere with the circle, or the product of two projective planes, and so on. The particular manifold S^{2} \times S^{2} may also be obtained if one considers the spherical shell lying between two concentric spherical surfaces S^{2} and s^{2} and identifies the corresponding points (that is, those lying on the same radius) of S^{2} and s^{2}. Only slightly more difficult is the proof of the fact that, if one takes two congruent solid tori and (in accordance with the congruence mentioned) idenfities the corresponding points of their surfaces with one another, one likewise obtains the manifold S^{2} \times S^{2}. 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.

Cheers,

Nalin Pithwa

Ref: Elementary Concepts of Topology by Paul Alexandroff, Dover Publications, available Amazon India.

Topology: bare facts …

Reference: Elementary Concepts of Topology by Paul Alexandroff, Dover Publications. (Available Amazon India):

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. (Note one can also view topology as an abstract study of the concept of the limit point because so many fundamental mathematical ideas like continuity and connectedness depend on it. We can also say that algebra and topology are complementary mathematical operations. Further that, whereas in Euclidean geometry, we consider rigid motions of geometric objects and that lengths, areas, volumes, angles are to be invariant, that congruence and similarity are geometric properties; in topology, which can be considered rubber sheet geometry, a geometric object, example a polyhedron can be stretched, twisted, bent, pulled; cut and coalesced exactly same way as before; but no holes can be destroyed or made; topological properties of a geometric object are those which are invariant under these transformations. For example, that a point lies in the interior of a circle is a topological property). Professor Alexandroff had said further in the above reference:

A. It is impossible to map an n-dimensional cube onto a proper subset of itself by a continuous deformation in which the boundary remains point-wise fixed. (Comment: to “verify” or “visualize” this, just try to sketch a one-to-one correspondence of a Rubik’s cube onto any face of it. The cube is three dimensional whereas the face is 2 dimensional).

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 topological results:

B. The Tiling Theorem: If one “covers” an n-dimensional cube with finitely many sufficiently small (but otherwise entirely arbitrary) closed sets, then there are necesssarily points which belong to at least n+1 of these sets. (On the other hand, there exists arbitrarily fine coverings for which this number n+1 is not exceeded). (Comment: Recall definition of “covering”, “open covering” and a “compact set”). (Digressing, there is the well-known example of fixed point theorems or of the well-known topological properties of closed surfaces such as are described in Hilbert and Cohn-Vossen’s Geometry and the Imagination, Chelsea Publications) (NB: here, sufficiently small will mean with a sufficiently small diameter).

For n=2, the theorem asserts that if a country is divided into sufficiently small provinces,there necessarily exist points at which at least three provinces come together. Here these provinces may have entirely arbitrary shapes, in particular, they need not even be connected (that is, in the language of analysis or topology, they are “separated” sets); each one may consist of several pieces.

Recent(at that time circa. 1905) topological investigations have shown that the whole nature of the concept of dimensions is hidden in this covering or tiling property, and thus the tiling theorem has contributed in a significant way to the deepening of our understanding of space.

C. As the third example of an important and yet obvious-sounding theorem, we may choose the Jordan curve theorem. A simple closed curve (that is, the topological image of a circle) lying in the plane divides the plane into precisely two regions and forms their common boundary.

II. The question which naturally arises 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. Such pairs of points are said to be separated by the curve or “linked” with it.

In three-dimensional space there are certainly no pairs of points which are separated by our Jordan curve; but there are closed polygons which are linked with it in the natural sense that every piece of surface which is bounded by the polygon necessarily has points in common with the curve. Here the portion of the surface spanned by the polygon need not be simply connected, but may be chosen entirely arbitrarily.

The Jordan curve theorem can also be generalized in another way for three-dimensional space: in space there are not only closed curves but also closed surfaces, and every such surface divides the space into two regions — exactly as a closed curve did in the plane.

Supported by analogy, you can probably imagine what the relationships are in four-dimensional space: for every closed curve surface linked with it; for every closed three dimensional manifold a pair of points linked with it. These are special cases of the Alexander duality theorem which we will discuss later.

III. Perhaps the above examples leave you with the impression that in topology nothing at all but obvious things are proved; this impression will fade rather quickly as we go on. However, be that as it may, even these “obvious” things are to be taken much more seriously: one can easily give examples of propositions which sound as “obvious” as the Jordan curve theorem, but which may be proved false. Who would believe, for example, that in a plane there are three (in fact, four, five, six…infinitely many) simply connected bounded regions which all have the same boundary; or, that one can find in three-dimensional space a simple Jordan arc (that is, a topological image of a polygonal line) such that there are circles outside of it arc that cannot possibly be contracted to a point without meeting it? There are also closed surfaces of genus zero which possess an analogous property. In other words, one can construct a topological image of a sphere and an ordinary circle in its interior in such a way that the circle may not be contracted to a point wholly inside the surface.

IV. All of these phenomena were wholly unsuspected at the beginning of the century (20th century): the development of set theoretic methods in topology first led to their discovery and consequently,to a substantial extention 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” geometrical structures; on the contrary, they are concerned with something quite positive. Prof Alexandroff had formulated the basic problem of set theoretic topology as follows:

To determine which set theoretic structures have a connection 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 required type, particularly with reference to those of their properties which actually enable us to recognize the above mentioned connection and so bring about the geometrization of the most general set-theoretic-topological concepts.

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 theoretic topology are amenable to the methods which have been developed in polyhedral topology. Thus, it is justified in what follows we devote ourselves primarily to the topology of polyhedra. (Ref: next blog article).

Cheers,

Nalin Pithwa

Group theory: v v v brief historical origins

Reference: Abstract Algebra, Second Edition, Dummit and Foote. John Wiley and Sons Inc.

The modern treatment of abstract algebra begins with the disarmingly simple abstract definition of a group. This simple definition quickly leads to difficult questions involving the structure of such objects. There are many specific examples of groups and the power of the abstract point of view becomes apparent when results for all of these examples are obtained by proving a single result for the abstract group.

The notion of a group did not simply spring into existence, however, but is rather the culmination of a long period of mathematical investigation, the first formal definition of an abstract group in the form in which we use it appearing in 1882. The definition of an abstract group has its origins in extremely old problems in algebraic equations, number theory, and geometry, and arose because very similar techniques were found to be applicable in a variety of situations. As Otto Holder (1859-1937) observed, one of the essential characteristics of mathematics is that after applying a certain algorithm or method of proof one then considers the scope and limits of the method. As a result, properties possessed by a number of interesting objects are frequently abstracted and the question raised: can one determine all the objects possessing these properties ? Attempting to answer such a question also frequently adds considerable understanding of the original objects under consideration. It is in this fashion that the definition of an abstract group evolved into what is, for us, the starting point of abstract algebra.

We illustrate with a few of the disparate situations in which the ideas later formalized into the notion of an abstract group were used.

  1. In number theory the very object of study, the set of integers, is an example of a group. Consider, for example, what we refer to as “Euler’s Theorem”, one extremely simple example of which is that a^{40} has last two digits 01 if a is any integer not divisible by 2 nor by 5. This was proved in 1761 by Leonhard Euler (1707-1783) using “group-theoretic” ideas of Joseph Louis Lagrange (1736-1813), long before the first formal definition of a group. From our perspective, one now proves “Lagrange’s Theorem”, applying these techniques abstracted to an arbitrary group, and then recovers Euler’s theorem (and many others) as a special case.
  2. Investigations into the question of rational solutions to algebraic equatios of the form y^{2}=x^{3}-2x (there are infintely many, for example (0,0), (-1,1), (2,2), (\frac{9}{4}, \frac{-21}{8}), (\frac{-1}{169}, \frac{239}{2197})) showed that connecting any two solutions by a straight line and computing the intersection of this line with the curve y^{2}=x^{3}-2x produces another solution. Such “Diophantine equations,” among others, were considered by Pierre de Fermat (1601-1655) (this one was solved by him in 1644),by Euler, by Lagrange around 1777, and others. In 1730 Euler raised the question of determining the indefinite integral \int (\frac{dx}{\sqrt{1-x^{2}}}) of the “lemniscatic differential” \frac{dx}{\sqrt{1-x^{2}}}, used in determining the arc length along an ellipse (the question had also been considered by Wilhelm Gottfried Leibniz (1646-1716) and Johannes Bernoulli (1667-1748)). In 1752 Euler proved a “multiplication formula” for such elliptic integrals (using ideas of G. C. di Fagnano (1682-1766), received by Euler in 1751), which shows how two elliptic integrals give rise to a third, bringing into existence the theory of elliptic functions in analysis. In 1834 Carl Gustav Jacob Jacobi (1804-1851) observed that the work of Euler on solving certain Diophantine equations amounted to writing the multiplication formula for certain elliptic integrals. Today the curve above is referred to as an “elliptic curve” and these questions are viewed as two different aspects of the same thing — the fact that this geometric operation on points can be used to give the set of points on an elliptic curve the structure of a group. The study of the “arithmetic” of these groups is an active area of current research.
  3. By 1824 it was known that there are formulas giving the roots of quadratic, cubic and quartic equations (extending the familiar quadratic formula for the roots of ax^{2}+bx+c=0). In 1824, however, Niels Henrik Abel (1802-1829) proved that such a formula for the roots of a quintic is impossible. The proof is based on the idea of examining what happens when the roots are permuted amongst themselves (for example, interchanging two of the roots). The collection of such permutations has the structure of a group (called, naturally enough, a “permutation group”). This idea culminated in the beautiful work of Evariste Galois (1811-1832) in 1830-32, working with explicit groups of “substitutions.” Today this work is referred to as Galois Theory. Similar explicit groups were being used in geometry as collections of geometric transformations (translations, reflections, etc) by Arthur Cayley (1821-1895) around 1851. Camille Jordan (1838-1922) around 1867, Felix Klein (1849-1925) around 1870 etc. and the application of groups to geometry is still extremely active in current research into the structure of 3-space, 4-space, etc. The same group arising in the study of the solvability of the quintic arises in the study of the rigid motion of an icosahedron in geometry and in the study of elliptic functions in analysis.
  4. The precursors of todays group can be traced back many years, even before the groups of “substitutions” of Galois. the formal definitionof an abstract group which is our starting point appeared in 1882 in the work of Walter Dyck (1856-1934), an assistant to Felix Klein, and also in the work of Heinrich Weber (1842-1913) in the same year. It is freuently the case in mathematics research to find specific application of an idea before having that idea extracted and presented as an item of interest in its own right (for example, in 1830 Galois used the notion of quotient group implicitly in his investigations in 1830 and the definition of an abstract quotient group is due to Holder in 1889). It is important to realize, with or without the historical context, that the reason the abstract definitions are made is because it is useful to isolate specific characteristics and consider what structure is imposed on an object having these characteristics. The notion of the structure of an algebraic object (which is made more precise by the concept of an isomorphism —- which considers when two apparently different objects are in some sense the same) is a major theme that recurs in algebra.

Hope you enjoyed !

Cheers,

Nalin Pithwa