Topology

• 1. Introduction

  ›

  Topology asks whether a coffee mug and a doughnut are the same thing. To a topologist, they are. A pliable torus shaped like a doughnut can be reshaped into a coffee mug by pressing in a dimple, enlarging it, and shrinking the central hole into the mug's handle. No cutting. No gluing. This playful idea, sometimes called the Topologist's Breakfast, sits at the heart of a serious field. Topology is the branch of mathematics concerned with properties of geometric objects preserved under continuous deformations such as stretching, twisting, crumpling, and bending. What it forbids is closing holes, opening holes, tearing, gluing, or passing an object through itself. The word comes from the Greek words tthe deformations leave certain features untouched, and those features are what topologists chase. So what does it mean for two shapes to be the same when length no longer matters? Why can a square and a circle share deep properties? And how did an idea this abstract end up describing DNA, robots, and the shape of the universe?

• 2. The Bridges Of Konigsberg

  ›

  In a paper widely regarded as one of the first in the field, Leonhard Euler proved that no route through the town of Konigsberg, now Kaliningrad, could cross each of its seven bridges exactly once. The result ignored the lengths of the bridges and the distances between them. It depended only on connectivity: which bridges connect to which islands or riverbanks. From this single negative answer grew the branch of mathematics known as graph theory. Euler's 1736 paper on the Seven Bridges is regarded as one of the first practical applications of topology. The same indifference to exact shape appears in the hairy ball theorem of algebraic topology, which states that one cannot comb the hair flat on a hairy ball without creating a cowlick. The formal version says there is no nonvanishing continuous tangent vector field on the sphere. Like the bridges, the theorem cares nothing for the shape of the sphere. It applies to any smooth blob, as long as it has no holes. These problems forced mathematicians to name exactly what they relied on, and from that need arose the notion of homeomorphism.

• 3. When Shapes Are The Same

  ›

  Two spaces are homeomorphic if one can be deformed into the other without cutting or gluing, which is why the cube and the sphere count as the same, and why the coffee cup and the doughnut do too. Yet the sphere is not homeomorphic to the doughnut, because the hole cannot be removed without tearing. Homeomorphism is the most basic topological equivalence. A function from one topological space to another is called continuous if the inverse image of any open set is open. When a continuous function is one-to-one and onto, and its inverse is also continuous, it is a homeomorphism, and its domain is homeomorphic to its range. Spaces that are homeomorphic share identical topological properties and are considered topologically the same. Another equivalence, harder to describe without getting technical, is homotopy equivalence. The essential idea is that two objects are homotopy equivalent if they both result from squishing some larger object. These equivalences let mathematicians sort an endless variety of shapes into families, and certain features survive every allowed deformation.

• 4. Topological Properties

  ›

  Dimension is the property that distinguishes a line from a surface, and it is one of the basic examples of a topological property. Compactness separates a line from a circle, while connectedness tells a single circle apart from two non-intersecting circles. A property that stays fixed under homeomorphisms and homotopies is, by definition, a topological property. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. Words like nearby, arbitrarily small, and far apart sound vague, but each can be made precise using open sets. A topology on a set X is a family of subsets satisfying three rules: both the empty set and X belong to it, any union of its members belongs to it, and any intersection of finitely many of its members belongs to it. The members are called open sets. A subset is closed if its complement is open, and a set may be open, closed, both, which makes it clopen, or neither. The same set can carry different topologies. The real line, the complex plane, and the Cantor set can each be thought of as the same set wearing a different topology.

• 5. From Leibniz To Poincare

  ›

  Gottfried Wilhelm Leibniz, in the 17th century, envisioned what he called geometria situs and analysis situs, planting the ideas that underlie topology. The term itself came later, coined by Johann Benedict Listing, who introduced Topologie in his Vorstudien zur Topologie, written in German in 1847. Listing had used the word for ten years in correspondence before it appeared in print. The English form topology surfaced in 1883, in Listing's obituary in the journal Nature, used to distinguish qualitative geometry from the ordinary geometry concerned chiefly with quantitative relations. On the 14th of November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron, a realization that led to his polyhedron formula relating the numbers of vertices, edges, and faces. Some authorities regard this analysis as the first theorem, signaling the birth of topology. Further contributions came from Augustin-Louis Cauchy, Ludwig Schlafli, Bernhard Riemann, and Enrico Betti. Their work was corrected, consolidated, and greatly extended by Henri Poincare, whose ground-breaking 1895 paper on Analysis Situs introduced the concepts now known as homotopy and homology. Those concepts became foundations of algebraic topology.

• 6. Building The Modern Theory

  ›

  Maurice Frechet introduced the metric space in 1906, unifying earlier work on function spaces by Georg Cantor, Vito Volterra, Cesare Arzela, Jacques Hadamard, and Giulio Ascoli. A metric space is now seen as a special case of a general topological space, and a single topological space can give rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term topological space and defined what is now called a Hausdorff space. The version used today is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Set theory, developed by Georg Cantor in the later part of the 19th century, underpins all of this. Cantor studied point sets in Euclidean space as part of his work on Fourier series. The field continues to be honored at the highest level. The 2022 Abel Prize went to Dennis Sullivan for his groundbreaking contributions to topology in its broadest sense, including its algebraic, geometric, and dynamical aspects.

• 7. The Four Branches

  ›

  General topology, also called point-set topology, supplies the basic set-theoretic definitions and constructions on which the other branches rest. Its core objects are topological spaces, and from open sets it builds the fundamental notions of continuity, compactness, and connectedness. Metric spaces, where distance is given by a function called a metric, form an important class, and many familiar spaces such as the real line and the complex plane draw their topology from a metric. Algebraic topology uses tools from algebra to find invariants that classify spaces up to homeomorphism, or more often up to homotopy equivalence. Its most important invariants are homotopy groups, homology, and cohomology, and it can even prove that any subgroup of a free group is again a free group. Differential topology deals with differentiable functions on differentiable manifolds and pairs with differential geometry to form the geometric theory of differentiable manifolds. Geometric topology focuses on low-dimensional manifolds of dimensions 2, 3, and 4. There the uniformization theorem says every surface admits a constant curvature metric with one of three geometries: spherical, flat, or hyperbolic. The geometrization conjecture, now a theorem, says every 3-manifold can be cut into pieces, each carrying one of eight possible geometries. Manifolds are the connecting thread: a manifold is a topological space that resembles Euclidean space near each point, and two-dimensional manifolds are also called surfaces.

• 8. Topology In The Wild

  ›

  Knot theory, a branch of topology, is used in biology to study how certain enzymes act on DNA. These enzymes cut, twist, and reconnect DNA, causing knotting with observable effects such as slower electrophoresis. Circuit topology classifies folded molecular chains by the pairwise arrangement of their intra-chain contacts and chain crossings, and both methods help compare the topology of folded proteins and nucleic acids. In computer science, topological data analysis determines the large-scale structure of a set, such as whether a cloud of points is spherical or toroidal, by replacing data points with simplicial complexes and reading their persistent homology as a barcode. Physics leans on topology across condensed matter, quantum field theory, quantum computing, and cosmology, where topology describes the overall shape of the universe in a study known as spacetime topology. David Thouless, Duncan Haldane, and Michael Kosterlitz won the 2016 Nobel Prize in Physics for their work on topological orders. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. The reach extends even to motion: the possible positions of a robot form a manifold called configuration space, where motion planning finds paths that move the robot's joints into a desired pose. The same Eulerian path that defeated the Konigsberg bridges now guides fiber artists joining modular pieces, surrounding each piece and traversing each edge only once.