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.
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.
Participants in the first phase of axiomatisation 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
1900 to 1920
1920 to the 1945 axioms for homology
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.| Year | Contributors | Event |
| 1945 | Saunders Mac Lane–Samuel Eilenberg | Foundation of category theory: axioms for categories, functors and natural transformations. |
| 1945 | Norman Steenrod–Samuel Eilenberg | Eilenberg–Steenrod axioms for homology and cohomology. |
| 1945 | Jean Leray | Founds sheaf theory. For Leray a sheaf was a map assigning a module or a ring to a closed subspace of a topological space. The first example was the sheaf assigning to a closed subspace its p-th cohomology group. |
| 1945 | Jean Leray | Defines sheaf cohomology. |
| 1946 | Jean Leray | Invents spectral sequences, a method for iteratively approximating cohomology groups. |
| 1948 | Cartan seminar | Writes up sheaf theory. |
| c.1949 | Norman Steenrod | The Steenrod problem, of representation of homology classes by fundamental classes of manifolds, can be solved by means of pseudomanifolds. |
| 1950 | Henri Cartan | In the sheaf theory notes from the Cartan seminar he defines: Sheaf space, support of sheaves axiomatically, sheaf cohomology with support. "The most natural proof of Poincaré duality is obtained by means of sheaf theory." |
| 1950 | Samuel Eilenberg–Joe Zilber | Simplicial sets as a purely algebraic model of well behaved topological spaces. |
| 1950 | Charles Ehresmann | Ehresmann's fibration theorem states that a smooth, proper, surjective submersion between smooth manifolds is a locally trivial fibration. |
| 1951 | Henri Cartan | Definition of sheaf theory, with a sheaf defined using open subsets of a topological space. Sheaves connect local and global properties of topological spaces. |
| 1952 | René Thom | The Thom isomorphism brings cobordism of manifolds into the ambit of homotopy theory. |
| 1952 | Edwin E. Moise | Moise's theorem established that a 3-dimension compact connected topological manifold is a PL manifold, having a unique PL structure. In particular it is triangulable. This result is now known to extend no further into higher dimensions. |
| 1956 | John Milnor | The first exotic spheres were constructed by Milnor in dimension 7, as -bundles over. He showed that there are at least 7 differentiable structures on the 7-sphere. |
| 1960 | John Milnor and Sergei Novikov | The ring of cobordism classes of stably complex manifolds is a polynomial ring on infinitely many generators of positive even degrees. |
1961 to 1970
1971–1980
| Year | Contributors | Event |
| 1974 | Shiing-Shen Chern–James Simons | Chern–Simons theory: A particular TQFT which describe knot and manifold invariants, at that time only in 3D |
| 1978 | Francois Bayen–Moshe Flato–Chris Fronsdal–Andre Lichnerowicz–Daniel Sternheimer | Deformation quantization, later to be a part of categorical quantization |
1981–1990
| Year | Contributors | Event |
| 1984 | Vladimir Bazhanov–Razumov Stroganov | Bazhanov–Stroganov d-simplex equation generalizing the Yang–Baxter equation and the Zamolodchikov equation |
| 1986 | Joachim Lambek–Phil Scott | So-called Fundamental theorem of topology: The section-functor Γ and the germ-functor Λ establish a dual adjunction between the category of presheaves and the category of bundles which restricts to a dual equivalence of categories between corresponding full subcategories of sheaves and of étale bundles |
| 1986 | Peter Freyd–David Yetter | Constructs the monoidal category of tangles |
| 1986 | Vladimir Drinfel'd–Michio Jimbo | Quantum groups: In other words quasitriangular Hopf algebras. The point is that the categories of representations of quantum groups are tensor categories with extra structure. They are used in construction of quantum invariants of knots and links and low dimensional manifolds, among other applications. |
| 1987 | Vladimir Drinfel'd–Gerard Laumon | Formulates geometric Langlands program |
| 1987 | Vladimir Turaev | Starts quantum topology by using quantum groups and R-matrices to giving an algebraic unification of most of the known knot polynomials. Especially important was Vaughan Jones and Edward Witten's work on the Jones polynomial. |
| 1988 | Graeme Segal | Elliptic objects: A functor that is a categorified version of a vector bundle equipped with a connection, it is a 2D parallel transport for strings. |
| 1988 | Graeme Segal | Conformal field theory: A symmetric monoidal functor satisfying some axioms |
| 1988 | Edward Witten | Topological quantum field theory : A monoidal functor satisfying some axioms |
| 1988 | Edward Witten | Topological string theory |
| 1989 | Edward Witten | Understanding of the Jones polynomial using Chern–Simons theory, leading to invariants for 3-manifolds |
| 1990 | Nicolai Reshetikhin–Vladimir Turaev–Edward Witten | Reshetikhin–Turaev-Witten invariants of knots from modular tensor categories of representations of quantum groups. |