ABRIDGED HISTORY OF TOPOLOGY
  • Subject began with the 1736 paper, Solutio problematis ad geometriam situs pertenentis (The aolution of a problem relating to the geometry of position), by Leonard Euler (1707-83) on the Könisberg Bridge problem: a river flowing through the town of Könisberg was spanned by seven pridges, and the problem concerned the possibility of crossing all seven bridges without crossing one of them twice. The struture involved is now known as a graph, and Euler proved the following (in modern mathematical languge): a graph has a path traversing each edge exactly once if exactly two vertices of the graph have exactly two odd vertices. Since the above graph has four vertices, and each is odd (four odd vertices), the theorem explained why the problem has no solution. Euler's title showed that he had set forth a new subject, which was long known as "Analysis Situs" (analysis of position). (Note: If a graph has zero odd vertices, it can not only be traversed with repetition, but also by a path returnng to the starting point, that is, the path is unicursal. This is the case with the familiar five-point star (only two-vertices and four-vertices), which can be drawn without taking the pencil from the paper and the drawing returns to the starting point.)
  • Also, in advancing mathematics beyond measurement, Euler (1750) stated a formula for convex polygons ("Euler characteristic"), in terms of the number of polygonal vertices , V, the number of polygonal edges, E . amd the number of polygonal faces, F: V - E + F = 2. His 1752 proof dissected solids into tetraheadral slices, but assumed the polygons are convex (any line segment connecing two points within the polygon also lies within the polygon).
  • In 1813, Antoine-Jean Lhuilier (1750-1840) showed that Euler's formula is incorrect for a polygon with holes; given G holes: V - E + F = 2 - 2G. (This is the first statement of a topological invariant. A t.i. is a property of a topological space which is shared by any space topologically equivalent or homeomorphic to it, meaning any map from an open set of one space to the other space corresponds to a point in an open set of the latter. Familiar examples include non-algebraic invariants, such as compactness, coonnectedness, Haussdorfness. Euler characteristic, orientability, dimension and algebraic invariants such as homology groups, homotopy groups, and k-theoyr.)
  • In 1865, Augustin Möbius (1790-1848) showed that aa band of paper, when twisted and glued together, constitutes the one-sided nonorientable surface now known as a Möbius strip or band or Möboid. (Möbius proved it cannot be filled with compatibly oriented triangles.) Johann Benedict Listing (1802=1882) -- first to use the word "topology" -- discovered and described the möboid in 1847.
  • Bernard Riemann (x-y) studied conectivity of surfaces, introducing (1857) Riemann surfaces in complex function analysis: since, given the polynomial equation, f(w,z), its roots vary with w, z. on Riemann surfaces determined by f(w,z), the function w(z) = 0 is single-valued.
  • Camille Jordan (1838-1922) introduced another procedure for analyzing the connectivity of a surface:
    1. a simple closed curve ("Jordan curve") separates space into exactly two parts, namely, inside and outside the curve;
    2. a Jordan curve on a surface is irreducible if cannot be transformed into a point if a general circuit, c, can be transformed into a system of irreducible circuits, a1, a1, ..., an , so that mi counts occurrences of ai in circuit c, then c = m1a1 + m1a1 + ... + mnan.
    3. a circuit is irreducible if, and only if, c = m1a1 + m1a1 + ... + mnan = 0;
    4. a system of irreducible circuits, a1, a1, ..., an, is independent if, and only if, the previous vanishing condition does not exist;
    5. such a system is independent if, and only if, any cicuit can be expressed in terms of them;
    6. Jordan proved that the number of circuits in a complete independent set is a topological invariant of the surface.
  • In 1871, Enrico Betti (1823-92) published a memoir on topology which included what are now labeled "Betti numbers", so named by Henri Poincaré (x-y), who was inspired to work in topology by Betti's work. Betti introduced coonectivity numberts for each dimension, 1, 2, n - 1:
    1. 1-D connectivity number: number of nonbifurcateable closed curves which can be drawn in a structure;
    2. 2-D connectivity number: number of (2-D) closed surfaces in a structure which do not bound a 3D-region of the structure;
    3. higher dimensional connectivity numbers are similarly defined;
    4. the closed curves, surfaces, and higher-dimensinal figures involved in this analysis are called "cycles".
  • Poincaré assumed a space is triangulable, i.e., dissectable into triangles, entirely sharing each triangular side with two adjacent triangles. A triangle is an instance of a 1-simplex, a notion introduced by L. E. J. Brouwer (1881-1966). So Poincaré treated space as what is now called a simplicial complex (also from Brouwer). For each dimension of possible simplexes in a complex, P. defined the number of independent cycles of that dimension as the Betti number of that dimension. Given a manifold, M, with various manifolds, mi , mapped to M. In P.'s work:
    1. the manifolds, mi, form a homology when they form the oriented boundary of a higher-dimensional manifold inside ofthe k-th homology group M;
    2. given a complex, Ck, one can form Eki, a linear combination of its k -dimensional oriented simplexes s.t. Ck = c1 Ek1 + c2Ek2 + ... + cnE kn (for positive or negative integers, c i, labeled a chain, and a chain with boundary zero is a cycle , so some chains are cycles. P. also introducted torsion coefficients to provide for the condition that (say, in the projective plane) a cycle does not bound but twice the cycle does bound. (P.'s treatment resembles modern cobordism.) Eventually P.'s homology was replaced by the more general singular homology.
  • In 1890, Guiseppe Peano (1858-1932) showed that Jordan's curve theorem was "too broad" by showing that a curve fittting his definition can run throgh all points of a square at least once, thereby filling a plane section. This "Peano curve" is neither continuoous nor single-valued.
  • Oswald Veblen (1880-1960) and James W. Alexander (1888-1971) replaced an oriented simplex by an unoriented one with coefficients modulo 2; also coefficients modulo m for chains and cycles. Lev S. Pontrjagin (1908-1960) replaced these by elements of an Abelian group.
  • during 1925-30, many mathematicians, inluding Emmy Noether (x-y), recast tht thory of chains, cycles, bounding cycles in terms of group theory:
    1. chains and cycles can be added by summing coefficients of the same simplex, forming Abelian groups, Ck(K);
    2. the relation of chain to boundary becomes a homomorphism from Ck(K) (the group of k-dimensional chains) into a particular subgroup, Hk-1, of the group, Ck-1(K), of k-1- dimensional chain.
    3. the set of all k-cycles k > 0 is a subgroup, Zk(K), of Ck(K), and, under the above homomophism, this goes into the identity element (0) of Ck-1(K). Since every chain is a cycle, Hk-1 is a subgroup of Zk-1(K).
    4. then:
      • for any k 0, the factor group of Zk(k) -- i.e., the k-dimensional cycles, modulo the subgroup, Hk(k), of the bounding cycles -- is the k-th homology group of K and denoted, Bk(K);
      • the number of linearly independent generators of this factor group is the k-th Betti number of the complex, denoted pk (k);
      • the k-th homology group may also contain finite cyclic groups, corresponding to the torsion groups, and the orders of these finite groups are the torsion coefficient.
    TOPOLOGY IN CONVERGENCE PROCESS IN ANALYSIS AND POINT SET TOPOLOGY
    TOPOLOGY IN FUNCTIONAL ANALYSIS
    This originated in problems both in physics and astronomy.