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 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.

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–	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.

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–André Lichnerowicz–Daniel Sternheimer	Deformation quantization, later to be a part of categorical quantization

1981–1990

Year	Contributors	Event
circa 1983	Simon Donaldson	Simon Donaldson introduces self-dual connections into the theory of smooth 4-manifolds, revolutionizing the 4-dimensional geometry, and relating it to mathematical physics. Many of his results were later published in his joint monograph with Kronheimer in 1990. See more under the Donaldson theory.
circa 1983	William Thurston	William Thurston proves that all Haken 3-manifolds are hyperbolic, which gives a proof of the Thurston's Hyperbolization theorem, thus starting a revolution in the study of 3-manifolds. See also under Hyperbolization theorem, and Geometrization conjecture
1984	Vladimir Bazhanov–Razumov Stroganov	Bazhanov–Stroganov d-simplex equation generalizing the Yang–Baxter equation and the Zamolodchikov equation
circa 1985	Andrew Casson	Andrew Casson introduces the Casson invariant for homology 3-spheres, bringing the whole new set of ideas into the 3-dimensional topology, and relating the geometry of 3-manifolds with the geometry of representation spaces of the fundamental group of a 2-manifold. This leads to a direct connection with mathematical physics. See more under Casson invariant.
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 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.
circa 1988	Andreas Floer	Andreas Floer introduces instanton homology.
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.