Timeline of manifolds

This is a timeline of manifolds, one of the major geometric concepts of mathematics. For further background see history of manifolds and varieties.

Background Manifolds in contemporary mathematics come in a number of types. These include:

smooth manifolds, which are basic in calculus in several variables, mathematical analysis and differential geometry; piecewise-linear manifolds; topological manifolds.

There are also related classes, such as homology manifolds and orbifolds, that resemble manifolds. It took a generation for clarity to emerge, after the initial work of Henri Poincaré, on the fundamental definitions; and a further generation to discriminate more exactly between the three major classes. Low-dimensional topology (i.e., dimensions 3 and 4, in practice) turned out to be more resistant than the higher dimension, in clearing up Poincaré's legacy. Further developments brought in fresh geometric ideas, concepts from quantum field theory, and heavy use of category theory.

Participants in the first phase of axiomatization were influenced by David Hilbert: with Hilbert's axioms as exemplary, by Hilbert's third problem as solved by Dehn, one of the actors, by Hilbert's fifteenth problem from the needs of 19th century geometry. The subject matter of manifolds is a strand common to algebraic topology, differential topology and geometric topology.

Timeline to 1900 and Henri Poincaré Year Contributors Event 18th century Leonhard Euler Euler's theorem on polyhedra "triangulating" the 2-sphere. The subdivision of a convex polygon with n sides into n triangles, by means of any internal point, adds n edges, one vertex and n - 1 faces, preserving the result. So the case of triangulations proper implies the general result. 1820–3 János Bolyai Develops non-Euclidean geometry, in particular the hyperbolic plane. 1822 Jean-Victor Poncelet Reconstructs real projective geometry, including the real projective plane. c.1825 Joseph Diez Gergonne, Jean-Victor Poncelet Geometric properties of the complex projective plane. 1840 Hermann Grassmann General n-dimensional linear spaces. 1848 Carl Friedrich GaussPierre Ossian Bonnet Gauss–Bonnet theorem for the differential geometry of closed surfaces. 1851 Bernhard Riemann Introduction of the Riemann surface into the theory of analytic continuation. Riemann surfaces are complex manifolds of dimension 1, in this setting presented as ramified covering spaces of the Riemann sphere (the complex projective line). 1854 Bernhard Riemann Riemannian metrics give an idea of intrinsic geometry of manifolds of any dimension. 1861 Folklore result since c.1850 First conventional publication of the Kelvin–Stokes theorem, in three dimensions, relating integrals over a volume to those on its boundary. 1870s Sophus Lie The Lie group concept is developed, using local formulae. 1872 Felix Klein Klein's Erlangen program puts an emphasis on the homogeneous spaces for the classical groups, as a class of manifolds foundational for geometry. later 1870s Ulisse Dini Dini develops the implicit function theorem, the basic tool for constructing manifolds locally as the zero sets of smooth functions. from 1890s Élie Cartan Formulation of Hamiltonian mechanics in terms of the cotangent bundle of a manifold, the configuration space. 1894 Henri Poincaré Fundamental group of a topological space. The Poincaré conjecture can now be formulated. 1895 Henri Poincaré Simplicial homology. 1895 Henri Poincaré Fundamental work Analysis situs, the beginning of algebraic topology. The basic form of Poincaré duality for an orientable manifold (compact) is formulated as the central symmetry of the Betti numbers.

1900 to 1920 Year Contributors Event 1900 David Hilbert Hilbert's fifth problem posed the question of characterising Lie groups among transformation groups, an issue partially resolved in the 1950s. Hilbert's fifteenth problem required a rigorous approach to the Schubert calculus, a branch of intersection theory taking place on the complex Grassmannian manifolds. 1902 David Hilbert Tentative axiomatisation (topological spaces are not yet defined) of two-dimensional manifolds. 1905 Max Dehn As a conjecture, the Dehn-Somerville equations relating numerically triangulated manifolds and simplicial polytopes. 1907 Henri Poincaré, Paul Koebe The uniformization theorem for simply connected Riemann surfaces. 1907 Max Dehn, Poul Heegaard Survey article Analysis Situs in Klein's encyclopedia gives the first proof of the classification of surfaces, conditional on the existence of a triangulation, and lays the foundations of combinatorial topology. The work also contained a combinatorial definition of "topological manifold", a subject in definitional flux up to the 1930s. 1908 Heinrich Franz Friedrich Tietze Habilitationschrift for the University of Vienna, proposes another tentative definition, by combinatorial means, of "topological manifold". 1908 Ernst Steinitz, Tietze The Hauptvermutung, a conjecture on the existence of a common refinement of two triangulations. This was an open problem, for manifolds, to 1961. 1910 L. E. J. Brouwer Brouwer's theorem on invariance of domain has the corollary that a connected, non-empty manifold has a definite dimension. This result had been an open problem for three decades. In the same year Brouwer gives the first example of a topological group that is not a Lie group. 1912 L. E. J. Brouwer Brouwer publishes on the degree of a continuous mapping, foreshadowing the fundamental class concept for orientable manifolds. 1913 Hermann Weyl Die Idee der Riemannschen Fläche gives a model definition of the idea of manifold, in the one-dimensional complex case. 1915 Oswald Veblen The "method of cutting", a combinatorial approach to surfaces, presented in a Princeton seminar. It is used for the 1921 proof of the classification of surfaces by Henry Roy Brahana.

1920 to the 1945 axioms for homology Year Contributors Event 1923 Hermann Künneth Künneth formula for homology of product of spaces. 1926 Hellmuth Kneser Defines "topological manifold" as a second countable Hausdorff space, with points having neighbourhoods homeomorphic to open balls; and "combinatorial manifold" in an inductive fashion depending on a cell complex definition and the Hauptvermutung. 1926 Élie Cartan Classification of symmetric spaces, a class of homogeneous spaces. 1926 Tibor Radó Two-dimensional topological manifolds have triangulations. 1926 Heinz Hopf Poincaré–Hopf theorem, the sum of the indexes of a vector field with isolated zeroes on a compact differential manifold M is equal to the Euler characteristic of M. 1926−7 Otto Schreier Definitions of topological group and of "continuous group" (traditional term, ultimately Lie group) as a locally Euclidean topological group. He also introduces the universal cover in this context. 1928 Leopold Vietoris Definition of h-manifold, by combinatorial means, by proof analysis applied to Poincaré duality. 1929 Egbert van Kampen In his dissertation, by means of star-complexes for simplicial complexes, recovers Poincaré duality in a combinatorial setting. 1930 Bartel Leendert van der Waerden Pursuing the goal of foundations for the Schubert calculus in enumerative geometry, he examined the Poincaré-Lefschetz intersection theory for its version of intersection number, in a 1930 paper (given the triangulability of algebraic varieties). In the same year, he published a note Kombinatorische Topologie on a talk for the Deutsche Mathematiker-Vereinigung, in which he surveyed definitions for "topological manifold" so far given, by eight authors. c.1930 Emmy Noether Module theory and general chain complexes are developed by Noether and her students, and algebraic topology begins as an axiomatic approach grounded in abstract algebra. 1931 Georges de Rham De Rham's theorem: for a compact differential manifold, the chain complex of differential forms computes the real (co)homology groups. 1931 Heinz Hopf Introduces the Hopf fibration, . 1931–2 Oswald Veblen, J. H. C. Whitehead Whitehead's 1931 thesis, The Representation of Projective Spaces, written with Veblen as advisor, gives an intrinsic and axiomatic view of manifolds as Hausdorff spaces subject to certain axioms. It was followed by the joint book Foundations of Differential Geometry (1932). The "chart" concept of Poincaré, a local coordinate system, is organised into the atlas; in this setting, regularity conditions may be applied to the transition functions. This foundational point of view allows for a pseudogroup restriction on the transition functions, for example to introduce piecewise linear structures. 1932 Eduard Čech Čech cohomology. 1933 Solomon Lefschetz Singular homology of topological spaces. 1934 Marston Morse Morse theory relates the real homology of compact differential manifolds to the critical points of a Morse function. 1935 Hassler Whitney Proof of the embedding theorem, stating that a smooth manifold of dimension n may be embedded in Euclidean space of dimension 2n. 1941 Witold Hurewicz First fundamental theorem of homological algebra: Given a short exact sequence of spaces there exist a connecting homomorphism such that the long sequence of cohomology groups of the spaces is exact. 1942 Lev Pontryagin Publishing in full in 1947, Pontryagin founded a new theory of cobordism with the result that a closed manifold that is a boundary has vanishing Stiefel-Whitney numbers. From Stokes's theorem cobordism classes of submanifolds are invariant for the integration of closed differential forms; the introduction of algebraic invariants gave the opening for computing with the equivalence relation as something intrinsic. 1943 Werner Gysin Gysin sequence and Gysin homomorphism. 1943 Norman Steenrod Homology with local coefficients. 1944 Samuel Eilenberg "Modern" definition of singular homology and singular cohomology. 1945 Beno Eckmann Defines the cohomology ring building on Heinz Hopf's work. In the case of manifolds, there are multiple interpretations of the ring product, including wedge product of differential forms, and cup product representing intersecting cycles.

1945 to 1960 Terminology: By this period manifolds are generally assumed to be those of Veblen-Whitehead, so locally Euclidean Hausdorff spaces, but the application of countability axioms was also becoming standard. Veblen-Whitehead did not assume, as Kneser earlier had, that manifolds are second countable. The term "separable manifold", to distinguish second countable manifolds, survived into the late 1950s.