The Many Faces of Topology

Abstract

Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see what all these branches have to do with each other or why they are all called topology. This article aims to shed light on this question and briefly summarize the content of the many branches of topology. We start with a historical review and move from pure set topology through the various analytical and geometric aspects of topology to algebraic varieties and buildings with apartments of Coxeter complexes and Weyl chambers. It should be noted that the transitions between some sub-areas such as topological analysis and differential topology or differential topology and algebraic topology or combinatorial and geometric topology are often fluid, and the categorization made here can only be fundamental.

Modal Geometry and Philosophy

When asked about the difference between geometry and topology we could simply answer by 2,400 years of mathematics. Whereas geometry – accounting aside – is the oldest branch of mathematics, topology is one of the youngest if not the youngest among the major disciplines – computer science aside. First results date back to Leibniz, Euler, and Möbius. A systematic approach that established the area as a separate branch of mathematics, however, began in the late 19th and early 20th century.

The other obvious answer is to look at the names. Geometry implies that we have a metric and can measure lengths and angles whereas $\lambda o \gamma o \varsigma \text{ of } \tau o \pi o \varsigma$ is the science of position, location, and shape. The name was coined by Johann Benedict Listing (1808-1882) [1] in a letter 1836 [2] and in a paper 1848 Vorstudien zur Topologie (Preliminaries to Topology) [3] in which he investigated the Möbius strip [4]. The picture of the sculpture of a Möbius strip was taken from [5]. We can find a more eloquent and detailed description on Wikipedia [6]:

A first approach to the geometric description of a location in space is the Cartesian coordinate system. Topological approaches were developed in contrast to Descartes’ analytical geometry in order to find a coordinate-free representation. Instead of just calculating something, the aim was to find the intrinsic structure and movement possibilities. The aim was therefore to replace quantitative geometry with modal geometry. In the 17th and 19th centuries, the term Geometria situs ‘geometry of position’ or Analysis situs ‘analysis of position’ was used instead of topology. Leibniz investigated the relationship between spatial points independently of the metric relationships in his paper De analysi situs. An example of the application of Geometria situs is the investigation of properties of geometric bodies, such as in the polyhedra formula, $\chi=V-E+F$ [7], which is attributed to both René Descartes [8] and Leonhard Euler [9].

The Seven Bridges of Königsberg [10] Euler (1736), is regarded as the first topological problem. The quotations are from a philosophical Wikipedia page on topology!

The topologische Wende (spatial turn) [11] in the humanities has particularly drawn attention to the consideration of place, field, and space categories in philosophy. This also creates a connection to Japanese philosophy, in which place (basho) has played a central role since the beginning of the 20th century. The term bashoron (doctrine of place) used there describes this type of philosophical topology. [6]

Topology is not only the youngest of major branches in mathematics, it is also the area that has branched out the fastest and most. Many of its sub-areas therefore carry additional adjectives like geometric, combinatorial, algebraic, analytical, differential, homological. There is even a difference between the words topology and a topology and the category TOP of topological spaces. The sentence: A topological space in topology carries a topology. makes perfect sense.

General Topology

General topology actually owes its origins in part to a misunderstanding. When Georg Cantor was investigating the representability of functions of a real variable using trigonometric series or Fourier series around 1870, he was interested in characterizing the sets of points at which the value of the function can be changed without the series changing. This prompted him to investigate the properties of subsets of real lines. He introduced (in addition to the concept of accumulation points, which we owe to Weierstraß) some basic concepts of topology, including that of the derived set as the set of accumulation points. [2]

Cantor, together with Dedekind, can be viewed as the father of modern set theory. Hence, it is not surprising that Cantor also looked at the general properties of certain sets, particularly their cardinalities. Felix Hausdorff  [12] introduced an axiomatic concept of neighborhoods in 1914 [13]. The development of general topology has largely occurred parallel to the development of set theory.

It should also be mentioned here that the term general topology (in contrast to algebraic topology) is only partially justified because the latter is just as general as the former. [2]

Dieudonné suggests that the term fundamental topology might have been a better choice. Maybe even the term set topology would have been closer to what we understand as general topology nowadays. The reader should keep in mind that Cantor thought of subsets of the real number line, sets of accumulation points, and Hausdorff of neighborhoods. This needs to be mentioned before we have a look at the definitions that allow exotic topological spaces which the founding fathers had not in mind when they made their observations.

The category TOP of topological spaces consists of sets as objects that carry a topology and continuous functions as morphisms. A set $\mathrm{X}$ carries a topology $\mathcal{T}$ means that we consider a subset of the powerset $\mathcal{T}\subseteq \mathcal{P}(X)$ with the following properties:

1. $\emptyset\, , \,X\, \in \,\mathcal{T}$
2. $U_1,\ldots,U_n\in \mathcal{T}$ for a finite $n\in \mathbb{N}$ implies that $\displaystyle{\bigcap_{k=1}^n U_k\in\mathcal{T}}.$
3. $U_\iota \in \mathcal{T}$ for arbitrary many $\iota \in \mathcal{I}$ implies that $\displaystyle{\bigcup_{\iota\in\mathcal{I}} U_k\in\mathcal{T}}.$

The elements of $\mathcal{T}$ are called open sets, the elements $X\setminus U$ with $U\in \mathcal{T}$ are called closed sets. A function between two topological spaces is called continuous if every preimage under the function of an open set is open.

The definition of a topology states that arbitrary unions and finite intersections of open sets are open, and arbitrary intersections and finite unions of closed sets are closed. Why do we have this asymmetry in the cardinalities of the index sets? Right now, we just have two names, open and closed, without meaning. To answer this question we need to give them a meaning.

Inspired by the real number line, we call open sets those whose elements are only surrounded by elements of the same set and closed sets if we include the boundaries, i.e. the points that also have neighboring points outside the set. This leads us to arbitrary unions of open intervals as open sets, and finite unions of closed intervals as closed sets. It is the topology that we get from the Euclidean metric. Points that are strictly closer to a specific point than a given distance build an open interval, and if we include the boundaries, i.e. include the points that are exactly distanced, we get a closed interval. This topology also has the properties we expect. The metric is a continuous function. Measurements of distances have no sudden gaps if the lengths are varied a little bit. A limit point of a sequence is a singleton and a unique accumulation point. Singletons are closed. Furthermore, we cannot have arbitrary unions of singletons as closed sets since this would make every subset closed and the distinction obsolete. On the other hand, we do not want arbitrary intersections of open sets to be open since this would make singletons open and thus every subset open. There is of course a topology where all subsets are open, the discrete topology, or a topology where only the empty set and the entire set are open, the trivial or indiscrete topology, however, we do not want to require that in general.

Topologies, i.e. topological structures on sets vary a lot between the discrete as the finest topology and the indiscrete as the coarsest topology. I like to speak of a zoo of topological spaces. Mathematicians speak of separation axioms to distinguish them. They describe the way how points and sets are separated. E.g. two different points in T2 spaces, so-called Hausdorff spaces that are named after Felix Hausdorff [12], can be separated by two disjoint open sets containing them respectively. Not only sets but also functions are used to separate sets and distinguish topologies. Topologies can have unexpected properties. E.g. there are Hausdorff spaces on which there are no continuous functions except the constants, see Pavel Urysohn [14],[15]. The notation of the separation axioms T1-T4 in remembrance of Andrey Tikhonov [16] sometimes called Tikhonov separation axioms are the classical criteria. The letter “T” stands for the German word Trennung which means separation. The list has been significantly expanded ever since and different types of topological spaces have been named, see [17]. A short list of mathematicians who contributed to the development of general topology can be found e.g. in [18].

Analytical Topology

Analysis is deeply interwoven with the Euclidean metric that automatically brings its own topology so that it is difficult to see where analysis ends and topology begins; the more topology has been developed from the investigation of the real number line and series expansions (Georg Cantor [19]) as we have seen in the previous section. We have also learned that an open set of real numbers is a set in which every element is surrounded only by elements of the same set. This means we can find to every point of an open set a neighborhood of that point that is completely contained in the open set. If $f$ is a continuous function and $U_\varepsilon$ an open set containing a point $f(x_0)$ then the preimage $f^{-1}(U_\varepsilon)$ is open and therefore contains an open neighborhood $U_\delta$ of $x_0$ such that $f(U_\delta)\subseteq U_\varepsilon.$ Together with the fact that such open sets of the real number line are always open intervals, defined by the Euclidean metric, we end up with the $\varepsilon-\delta$ definition of continuity of real functions.

A topological space in which two disjoint closed sets can be separated by two disjoint open sets containing the closed sets respectively is called a T4 space and its topology is called normal. Real intervals fulfill this condition. Urysohn has shown that there is always a continuous function from normal spaces to the unit interval that is identically zero on one of the closed sets and identically one on the other closed set [14]. This statement known as Urysohn’s lemma about the separation of closed sets by a continuous function has far-reaching consequences especially in physics since it allows the approximation of indicator functions by continuous functions. It is furthermore the core of Tietze’s [20] extension theorem [21] and closely related to the concept of the partition of unity [22] which is an important tool in differential topology.

The consideration of continuous, real functions leads automatically to the question about the points of discontinuities since they particularly disrupt Riemann integrability.

According to N. Bourbaki [23], the form that Riemann gave to the integrability condition suggests introducing a measure for the set of discontinuities of a function in an interval. However, to be able to define the concept of measure exactly, the first elements of set theory and general topology had to have permeated mathematical research. [2]

Hence, there was quite a way to go from classical analysis to Lebesgue integrability and Borel’s measure theory.

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. [24],[25],[26]

Measure theory provides an approach to analysis that is closer related to topology than classical analysis which focuses on the Euclidean metric. It reflects the step from Riemann-Darboux to Lebesgue integrability. A mathematical treatment can be found in Hewitt, Stromberg [27], and an entire chapter about its history in Dieudonné [2].

Differential Topology

Differential topology deals with differentiable manifolds and Lie groups. So what is the difference to differential geometry? Local properties are a matter of (differential) analysis, and metrics and symplectic forms are a matter of differential geometry. Global invariants such as compactness, closeness, and boundaries of topological manifolds with an additional differentiable structure are subject to differentiable topology. Its founding fathers are considered to be Bernhard Riemann [28] and Henri Poincaré [29].

We find in Riemann’s habilitation lecture [30] also [besides topics of differential geometry, ed.] other germs of concepts of modern differential topology, such as that of foliation … or the distinction between compact (boundaryless) and non-compact submanifolds. [2]

Differential topology is closely related to algebraic topology, e.g. by the de Rham cohomology [31].

Differential topology considers differentiable manifolds and smooth mappings between them. Typical questions are, for example: are two given manifolds diffeomorphic? Are two given mappings homotopic? Do all self-mappings of a manifold into themselves have fixed points in a given homotopy class? [32]

Topology of Vector Spaces

A topological vector space is a vector space with a compatible topological structure, e.g. induced by a metric. Compatibility means that scalar multiplication and vector addition are continuous functions. This implies that the underlying scalar field is a topological field. Those vector spaces are of particular interest when they are function spaces of typically infinite dimension. And again, we have to refer to Riemann.

Since Riemann, non-Euclidean geometry has been approached from a new perspective. Before him, differential geometry was limited to the study of curves and surfaces embedded in three-dimensional Euclidean space. Under the influence of mechanics (systems with n degrees of freedom) and physics, Riemann first introduced what he called “n-fold extended quantities” and “n-fold extended manifolds”, an idea that gave rise to the modern concept of n-dimensional manifolds. He even suggested the idea of “manifold” of infinite dimension (e.g. the set of continuous functions on a subset) and thus outlined the modern concept of a function space for the first time. [2]

Hilbert spaces which are topological vector spaces play a central role in physics. [33],[34],[35]

Topology of Fields

Topological vector spaces are usually real or complex and both of these scalar fields carry a norm topology. What other options do we have? Finite fields are discrete and as such not suited for analytical considerations. The rational numbers with the induced subspace topology of the real numbers are a dense subset but also totally disconnected, i.e. all of its connected components are the empty set or singletons. Before we conclude that totally disconnected topological spaces are inappropriate for analysis, we have to note that the field of p-adic numbers [36] is complete with respect to its non-Archimedean metric, locally compact, totally disconnected, and allows analysis, e.g. [37],[38]. The p-adic numbers, however, are still a stepchild in mathematics despite Hasse’s [39] principle [40]. The real numbers are the topological completion of the rational numbers with respect to the Euclidean norm, i.e. we take the rational numbers and add all limit points of rational Cauchy sequences. This is why real numbers and complex numbers as their algebraic completion are the fields we usually associate with a topological vector space.

Geometric Topology

The apparent contradiction between the request of measurements in geometry and unmeasurable shapes in topology is resolved if we look at the history of geometric topology. It reveals that the adjective geometric in this context refers to geometric objects like curves, spheres, or tori, or generally topological manifolds rather than measurements. The first results in this sub-area of topology were the Schoenflies [41] theorem in 1910 [42] about the homeomorphisms and embeddings of Jordan curves, based on the Jordan curve theorem [43] by Camille Jordan [44] in 1887. These seeds already describe what geometric topology is about: topological manifolds and their embeddings, low-dimensionality, and a close relation to combinatorial topology, e.g. to the four-color-theorem [45] and crystallography. A short historical overview can be found on [46], a not-so-short list of twenty-one subjects from geometric group theory to homology manifolds in Handbook of Geometric Topology [47]. It’s not possible to summarize its more than a thousand pages here, so we simply state that the Poincaré Conjecture [48].

Every simply connected, compact, unbounded, 3-dimensional manifold

is homeomorphic to the 3-sphere.

is possibly the most prominent example that has driven geometric topology like Fermat’s theorem has driven number theory.

Combinatorial Topology and Physics

We already mentioned the seven bridges of Königsberg [10] from 1736 as an example of combinatorial topology which is actually considered to be its origin. It is a graph theoretic problem that asks to find a certain path in a graph. A graph is a set of vertices and edges. It is called directed if the edges have a direction, and undirected if not. The seven bridges of Königsberg are an instance of finding an Eulerian path where every edge, the bridges, is visited exactly once. A Hamilton [49] path is a path that visits each vertex exactly once. Determining whether a graph has a Hamilton path (or Hamilton cycle) is NP-complete [50].

Graph theory has since then undergone a very remarkable development. In particular, it has found applications in various fields, the first of which (if we leave aside those which have more the character of games, as was the case, for example, with Hamilton in 1859) was undoubtedly provided by Kirchhoff’s [51] rules [52] (1847), which concern the flow of electric current through branched conductors. [2]

The four-color theorem [45] has to be seen in this realm, too. If we dig deeper into the area of combinatorial topology, we will find knot theory [53], and objects such as links [54], braids [55], strings [56], tangles [57], and of course the Klein bottle [58].

Mathematicians always look for invariants in the categories they examine. This is particularly true for topology with its sparse general requirements. An important invariant of knot theory is the Jones polynomial [59] named after Vaughan Jones [60] who received the Fields medal for his work. Edward Witten [61] has shown that the Jones polynomial can be defined by a topological quantum field theory [62], the Chern-Simons-Theory [63].

Dieudonné [2] combines certain methods of algebraic topology, especially homology theory with combinatorial topology so that, in addition to Kirchhoff and Witten, further connections to physics can be created via this detour. This shows once again that the transitions between the sub-areas of topology are fluid. Both sub-areas investigate simplicial complexes [64],[65]. There are many ways to represent a topological space and splitting it into parts is one of them. It is a method that is known from complex analysis where domains (Gebiete) are triangulated before integration. Since we are only interested in continuity and not in differentiability, we are allowed to use polytopes instead of curved manifolds, i.e. we concentrate on flattened connected components.

Algebraic Topology

The central idea of algebraic topology is to link topological invariants with algebraic objects so that the investigation of the latter allows conclusions about the former. Those objects are mainly groups, but e.g. algebras and chain complexes [66] occur, too. For a detailed view of algebraic topology, see Albrecht Dold [67] Lectures on Algebraic Topology [68]. It is also the appropriate framework for categoric and functor considerations.

Algebraic topology does not have a long history, but since it grew up in an era in which mathematics as a whole was developing at a very rapid pace never seen before and in which the curiosity of mathematicians was entirely directed towards exploring new areas, its history illustrates some aspects that are found to varying degrees in other branches of mathematics. [2]

Algebraic topology evolved from combinatorial topology and it was once more the works of Riemann [28] and Poincaré [29] that can be seen as its origins. It was the investigation of complex functions in one variable that led to the topological properties of surfaces.

Simplicial Complexes

Brouwer [69] is the originator of the method of simplicial approximation [70], which not only provided new results but also made it possible to define in a strict and precise way the concepts and proofs that had often remained unclear and incomplete since the beginnings of combinatorial topology. If a continuous mapping of one polyhedron into another is given, this can be replaced by a sufficiently adjacent mapping via a simplicial approximation (after possible subdivision of the polyhedron) that is simplicial, i.e. that maps simplices to (possibly degenerate) simplices by an affine transformation. (Degenerate simplices are those whose vertices do not all have to be different.) This allows in particular the introduction of the degree of a mapping [71] i.e. the algebraic sum of the number of coverings of an arbitrary point by the image. [2]

Simplicial complexes are a set of polytopes and the corresponding algebraic objects are the free abelian groups generated by the vertices ordered by the dimension of the according polytopes.

Homotopy and Covering

A homotopy is a continuous path between two continuous functions in the compact-open topology [72]. It is the origin of the joke that a cup and a ring are the same in the eyes of a topologist. A homotopy establishes an equivalence relation between continuous functions, the morphisms of topological objects. Homotopy theory considers closed paths, loops, and asks whether they can be continuously shrunk to a single basis point. E.g. a loop around a cup can be shrunk, but a loop around its handle can not. We therefore have to investigate the discontinuities, holes, and gaps. The equivalence classes build a group, the fundamental group [73], and establish thus the linkage to abstract algebra. The Whitehead [74] theorem [75] states that homotopy equivalence is equivalent to isomorphic fundamental groups.

If the topological space satisfies certain local conditions (which are always satisfied in the case of polyhedrons and especially manifolds), every subgroup of the fundamental group corresponds (up to isomorphism) to a covering that has exactly this subgroup as fundamental group. [2]

Homology

The study of surfaces (simplicial complexes) and holes (homotopies) leads both to varieties of (algebraic) groups. The next step is to bring order into them. A natural order, a grading [76] is provided by the dimension of the polytopes, the dimensions of the holes, or the cardinalities of coverings. Dieudonné writes about the beginning of homology theory:

As Poincaré [29] wrote in an analysis of his scientific work [77], his work on mathematical analysis (as well as on mechanics) led him to deal with problems of a topological nature, in particular through the study of curves defined by differential equations and through the study of functions of a complex variable; for these, in 1908, after long competition with the German mathematicians of the school of Felix Klein [78], he [and Paul Koebe [79], ed.] was able to give the solution to the uniformization problem [80]. [2]

The ingenious key, however, wasn’t the graded variety of those groups but rather the homomorphisms between them: the differentials [81] or boundary maps [82]. These homomorphisms satisfy the condition
$$
\partial^2=0
$$
turning the variety of graded groups into a chain complex [66] with
$$
\operatorname{im}\partial \subseteq \operatorname{ker}\partial .
$$
allowing a refinement to a long exact sequence [83] of group homomorphisms. (We left out the grading here to emphasize the idea, not the indices. The given links provide details.) The factor groups are called homology groups.

In Riemann, and sometimes even 40 years later in Poincare, the terms homology and homotopy are not always clearly distinguished, because it is not specified whether the surfaces whose edges are considered are simply connected [84]. [i.e. null-homotopic, the equivalence class of the neutral element in the fundamental groups, ed.] [2]

The detour from algebraic topology to physics mentioned earlier is by the Chevalley [85] Eilenberg [86] calculus [87] for finite-dimensional Lie [88] algebras [89]. We are defining for a given representation [90] of a Lie algebra a chain complex of Graßmann [91] algebras [92] with coefficients in the representation vector space, and a dual cochain complex [93] of algebra homomorphisms into the representation vector space [94]. Homology theory is defined by the Tor-functor [95] derived from tensor products and cohomology theory is defined by the Ext-functor [96] derived from homomorphisms. The cohomology groups of semisimple Lie algebras of irreducible representations are all trivial, and the first two cases are known as Whitehead lemmas [97]. This means applied to the adjoint representation, that all derivations of a semisimple Lie algebra are inner derivations [98].

Algebraic Geometry

Algebraic geometry examines geometric properties of zero sets of complex algebraic equations and links them with algebraic objects. It has an algebraic version of local coordinates and uses mainly commutative rings and prime ideals [99]. The space of complex points that contains the zero sets carries the Zariski [100] topology [101] which defines the sets of zeros of multivariate complex polynomials as closed, and the set of non-zeros as open sets. The spectrum [102] of a unitary commutative ring consists of its prime ideals as points. Closed sets under the also called Zariski topology are the sets of prime ideals that contain a given ideal of the ring, the ideal generated by the algebraic equations whose zeros we consider in the ring of multivariate complex polynomials. The connection between the two has been established by Hilbert’s [103] Nullstellensatz [104] that relates sets of zero points to ideals. It is a generalization of the fundamental theorem of algebra to multivariate polynomials.

Buildings

Let us finally answer the question about chambers, apartments, and buildings. Simple Lie groups [105] which are central to our standard model are also algebraic groups, i.e. matrix groups. Their classification is basically of a geometric nature. The Killing-form defines a metric, and root systems which in turn define graphs [90], the Coxeter [106] diagrams, which are similar to the Dynkin [107] diagrams. The symmetries defined by the hyperplanes in these diagrams build the Weyl [108] groups and Weyl chambers. They also build a simplicial complex called a building with apartments as their subcomplexes [109].

 

Everything is connected to everything.
Alexander von Humboldt (1769-1859)}

References