# Timeline of manifolds

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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:

There are also related classes, such as homology manifolds and orbifolds, that resemble manifolds.

## 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.[1]
c.1825 Joseph Diez Gergonne, Jean-Victor Poncelet Geometric properties of the complex projective plane.[2]
1840 Hermann Grassmann General n-dimensional linear spaces.
1848 Carl Friedrich Gauss
Pierre 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.[3] 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.
1870s Sophus Lie The Lie group concept is developed, using local formulae.[4]
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.
from 1890s Élie Cartan Formulation of Hamiltonian mechanics in terms of the cotangent bundle of a manifold, the configuration space.[5]
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.[6]

## 1900 to the 1945 axioms for homology

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.[7]
1907 Henri Poincaré, Paul Koebe The uniformization theorem for simply connected Riemann surfaces.
1908 Ernst Steinitz, Heinrich Franz Friedrich 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.[8]
1913 Hermann Weyl Die Idee der Riemannschen Fläche gives a model definition of the idea of manifold, in the one-dimensional complex case.
1923 Hermann Künneth Künneth formula for homology of product of spaces.
1926 Élie Cartan Classification of symmetric spaces, a class of homogeneous spaces.
1926 Tibor Radó Two-dimensional topological manifolds have triangulations.[9]
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.
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.[10]
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.
1931–2 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.[11][12][7] This foundational point of view allows for a pseudogroup restriction on the transition functions, for example to introduce piecewise linear structures.[13]
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.[14]
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.[15]
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.
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

Year Contributors Event
1945 Saunders Mac LaneSamuel Eilenberg Foundation of category theory: axioms for categories, functors and natural transformations.
1945 Norman SteenrodSamuel 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.
1950 Henri Cartan In the sheaf theory notes from the Cartan seminar he defines: Sheaf space (étale space), support of sheaves axiomatically, sheaf cohomology with support. "The most natural proof of Poincaré duality is obtained by means of sheaf theory."[16]
1950 Samuel Eilenberg–Joe Zilber Simplicial sets as a purely algebraic model of well behaved topological spaces.
1951 Henri Cartan Definition of sheaf theory, with a sheaf defined using open subsets (rather than closed 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.
1970 John Conway Skein theory of knots: The computation of knot invariants by skein modules. Skein modules can be based on quantum invariants

## 1971–1980

Year Contributors Event
1974 Shiing-Shen ChernJames 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
1985 André JoyalRoss Street Braided monoidal categories
1985 André JoyalRoss Street Joyal–Street coherence theorem for braided monoidal categories
1986 Joachim Lambek–Phil Scott 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 (over the same topological space) which restricts to a dual equivalence of categories (or duality) between corresponding full subcategories of sheaves and of étale bundles
1986 Peter FreydDavid Yetter Constructs the (compact braided) monoidal category of tangles
1986 Vladimir Drinfel'dMichio 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, representation theory, q-deformation theory, conformal field theory, integrable systems. The invariants are constructed from braided monoidal categories that are categories of representations of quantum groups. The underlying structure of a TQFT is a modular category of representations of a quantum group
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. John Jardine has also given a model structure for the category of simplicial presheaves
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 Z:nCobC→Hilb satisfying some axioms
1988 Edward Witten Topological quantum field theory (TQFT): A monoidal functor Z:nCob→Hilb satisfying some axioms
1988 Edward Witten Topological string theory
1989 Edward Witten Understanding the Jones polynomial using Chern–Simons theory, leading to invariants for 3-manifolds
1990 Nicolai ReshetikhinVladimir TuraevEdward Witten Reshetikhin–Turaev-Witten invariants of knots from modular tensor categories of representations of quantum groups.

## 1991–2000

Year Contributors Event
1991 André JoyalRoss Street Formalization of Penrose string diagrams to calculate with abstract tensors in various monoidal categories with extra structure. The calculus now depends on the connection with low dimensional topology.
1992 John Greenlees–Peter May Greenlees–May duality
1992 Vladimir Turaev Modular tensor categories. Special tensor categories that arise in constructiong knot invariants, in constructing TQFTs and CFTs, as truncation (semisimple quotient) of the category of representations of a quantum group (at roots of unity), as categories of representations of weak Hopf algebras, as category of representations of a RCFT.
1992 Vladimir TuraevOleg Viro Turaev–Viro state sum models based on spherical categories (the first state sum models) and Turaev–Viro state sum invariants for 3-manifolds.
1993 Ruth Lawrence Extended TQFTs
1993 David YetterLouis Crane Crane–Yetter state sum models based on ribbon categories and Crane–Yetter state sum invariants for 4-manifolds.
1993 Kenji Fukaya A-categories and A-functors: Most commonly in homological algebra, a category with several compositions such that the first composition is associative up to homotopy which satisfies an equation that holds up to another homotopy, etc. (associative up to higher homotopy). A stands for associative.

Def: A category C such that
1) for all X,Y in Ob(C) the Hom-sets HomC(X,Y) are finite-dimensional chain complexes of Z-graded modules
2) for all objects X1,...,Xn in Ob(C) there is a family of linear composition maps (the higher compositions)
mn : HomC(X0,X1) ⊗ HomC(X1,X2) ⊗ ... ⊗ HomC(Xn−1,Xn) → HomC(X0,Xn) of degree n − 2 (homological grading convention is used) for n ≥ 1
3) m1 is the differential on the chain complex HomC(X,Y)
4) mn satisfy the quadratic A-associativity equation for all n ≥ 0.

m1 and m2 will be chain maps but the compositions mi of higher order are not chain maps, nevertheless they are Massey products. In particular it is a linear category. Examples are the Fukaya category Fuk(X) and loop space ΩX where X is a topological space and A-algebras as A-categories with one object. When there are no higher maps (trivial homotopies) C is a dg-category. Every A-category is quasiisomorphic in a functorial way to a dg-category. A quasiisomorphism is a chain map that is an isomorphism in homology.

The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of A-categories and A-functors. Many features of A-categories and A-functors come from the fact that they form a symmetric closed multicategory, which is revealed in the language of comonads. From a higher dimensional perspective A-categories are weak ω-categories with all morphisms invertible. A-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.

1993 John Barret-Bruce Westbury Spherical categories: Monoidal categories with duals for diagrams on spheres instead for in the plane.
1993 Maxim Kontsevich Kontsevich invariants for knots (are perturbation expansion Feynman integrals for the Witten functional integral) defined by the Kontsevich integral. They are the universal Vassiliev invariants for knots.
1993 Daniel Freed A new view on TQFT using modular tensor categories that unifies 3 approaches to TQFT (modular tensor categories from path integrals).
1994 Maxim Kontsevich Formulates homological mirror symmetry conjecture: X a compact symplectic manifold with first chern class c1(X) = 0 and Y a compact Calabi–Yau manifold are mirror pairs if and only if D(FukX) (the derived category of the Fukaya triangulated category of X concocted out of Lagrangian cycles with local systems) is equivalent to a subcategory of Db(CohY) (the bounded derived category of coherent sheaves on Y).
1994 Louis CraneIgor Frenkel Hopf categories and construction of 4D TQFTs by them.
```identifies k-tuply monoidal n-categories. It mirror the table of homotopy groups of the spheres.
```
1995 John BaezJames Dolan Outline a program in which n-dimensional TQFTs are described as n-category representations.
1995 John BaezJames Dolan Proposes n-dimensional deformation quantization.
1995 John BaezJames Dolan Tangle hypothesis: The n-category of framed n-tangles in n+k dimensions is (n + k)-equivalent to the free weak k-tuply monoidal n-category with duals on one object.
1995 John BaezJames Dolan Cobordism hypothesis (Extended TQFT hypothesis I): The n-category of which n-dimensional extended TQFTs are representations nCob is the free stable weak n-category with duals on one object.
1995 John BaezJames Dolan Extended TQFT hypothesis II: An n-dimensional unitary extended TQFT is a weak n-functor, preserving all levels of duality, from the free stable weak n-category with duals on one object to nHilb.
1995 Valentin Lychagin Categorical quantization
1997 Maxim Kontsevich Formal deformation quantization theorem: Every Poisson manifold admits a differentiable star product and they are classified up to equivalence by formal deformations of the Poisson structure.
1998 Richard Thomas Richard Thomas, a student of Simon Donaldson, introduces Donaldson–Thomas invariants which are systems of numerical invariants of complex oriented 3-manifolds X, analogous to Donaldson invariants in the theory of 4-manifolds. They are certain weighted Euler characteristics of the moduli space of sheaves on X and "count" Gieseker semistable coherent sheaves with fixed Chern character on X. Ideally the moduli spaces should be a critical sets of holomorphic Chern–Simons functions and the Donaldson–Thomas invariants should be the number of critical points of this function, counted correctly. Currently such holomorphic Chern–Simons functions exist at best locally.
1998 Maxim Kontsevich CalabivYau categories: A linear category with a trace map for each object of the category and an associated symmetric (with respects to objects) nondegenerate pairing to the trace map. If X is a smooth projective Calabi–Yau variety of dimension d then Db(Coh(X)) is a unital Calabi–Yau A-category of Calabi–Yau dimension d. A Calabi–Yau category with one object is a Frobenius algebra.
1999 Joseph BernsteinIgor FrenkelMikhail Khovanov Temperley–Lieb categories: Objects are enumerated by nonnegative integers. The set of homomorphisms from object n to object m is a free R-module with a basis over a ring R. R is given by the isotopy classes of systems of (|n| + |m|)/2 simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs |n| points on the bottom and |m| points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley–Lieb categories are categorized Temperley–Lieb algebras.
1999 Moira Chas–Dennis Sullivan Constructs String topology by cohomology. This is string theory on general topological manifolds.
1999 Mikhail Khovanov Khovanov homology: A homology theory for knots such that the dimensions of the homology groups are the coefficients of the Jones polynomial of the knot.
1999 Vladimir Turaev Homotopy quantum field theory HQFT
1999 Ronald Brown–George Janelidze 2-dimensional Galois theory
2000 Yakov EliashbergAlexander GiventalHelmut Hofer Symplectic field theory SFT: A functor Z from a geometric category of framed Hamiltonian structures and framed cobordisms between them to an algebraic category of certain differential D-modules and Fourier integral operators between them and satisfying some axioms.

## 2001–present

Year Contributors Event
2002 Dennis Gaitsgory–Kari Vilonen–Edward Frenkel Proves the geometric Langlands program for GL(n) over finite fields.
2004 Dennis Gaitsgory Extended the proof of the geometric Langlands program to include GL(n) over C. This allows to consider curves over C instead of over finite fields in the geometric Langlands program.
2004 Stephen StolzPeter Teichner Definition of nD quantum field theory of degree p parametrized by a manifold.
2004 Stephen StolzPeter Teichner Graeme Segal proposed in the 1980s to provide a geometric construction of elliptic cohomology (the precursor to tmf) as some kind of moduli space of CFTs. Stephan Stolz and Peter Teichner continued and expanded these ideas in a program to construct TMF as a moduli space of supersymmetric Euclidean field theories. They conjectured a Stolz–Teichner picture (analogy) between classifying spaces of cohomology theories in the chromatic filtration (de Rham cohomology, K-theory, Morava K-theories) and moduli spaces of supersymmetric QFTs parametrized by a manifold (proved in 0D and 1D).
2005 Peter OzsváthZoltán Szabó Knot Floer homology
2006 Aslak Buan–Robert Marsh–Markus Reineke–Idun Reiten–Gordana Todorov Cluster categories: Cluster categories are a special case of triangulated Calabi–Yau categories of Calabi–Yau dimension 2 and a generalization of cluster algebras.
2007 Dennis GaitsgoryJacob Lurie Presents a derived version of the geometric Satake equivalence and formulates a geometric Langlands duality for quantum groups.

The geometric Satake equivalence realized the category of representations of the Langlands dual group LG in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian GrG=G((t))/G[[t]] of the original group G.

2008 Bruce Bartlett Primacy of the point hypothesis: An n-dimensional unitary extended TQFT is completely described by the n-Hilbert space it assigns to a point. This is a reformulation of the cobordism hypothesis.
2008 Mike Hopkins–Jacob Lurie Sketch of proof of Baez–Dolan tangle hypothesis and Baez–Dolan cobordism hypothesis which classify extended TQFT in all dimensions.

## Notes

1. ^ Coxeter, H.S.M. (2012-12-06). The Real Projective Plane. Springer Science & Business Media. pp. 3–4. ISBN 9781461227342. Retrieved 16 January 2018.
2. ^ Buekenhout, Francis; Cohen, Arjeh M. (2013-01-26). Diagram Geometry: Related to Classical Groups and Buildings. Springer Science & Business Media. p. 366. ISBN 9783642344534. Retrieved 16 January 2018.
3. ^ García, Emilio Bujalance; Costa, A. F.; Martínez, E. (2001-06-14). Topics on Riemann Surfaces and Fuchsian Groups. Cambridge University Press. p. ix. ISBN 9780521003506. Retrieved 17 January 2018.
4. ^ Platonov, V. P. (2001) [1994], "Lie group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
5. ^ Stein, Erwin (2013-12-04). The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering. Springer Science & Business Media. pp. 70–1. ISBN 9783642399053. Retrieved 6 January 2018.
6. ^ Dieudonné, Jean (2009-09-01). A History of Algebraic and Differential Topology, 1900 - 1960. Springer Science & Business Media. p. 7. ISBN 9780817649074. Retrieved 4 January 2018.
7. ^ a b James, I.M. (1999-08-24). History of Topology. Elsevier. p. 47. ISBN 9780080534077. Retrieved 17 January 2018.
8. ^ Freudenthal, Hans (2014-05-12). L. E. J. Brouwer Collected Works: Geometry, Analysis, Topology and Mechanics. Elsevier Science. p. 435. ISBN 9781483257549. Retrieved 6 January 2018.
9. ^ James, I.M. (1999-08-24). History of Topology. Elsevier. p. 56. ISBN 9780080534077. Retrieved 17 January 2018.
10. ^ Hazewinkel, Michiel, ed. (2001) [1994], "De Rham theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
11. ^ James, I. M. (1999-08-24). History of Topology. Elsevier. p. 56. ISBN 9780080534077. Retrieved 17 January 2018.
12. ^ Wall, C. T. C. (2016-07-04). Differential Topology. Cambridge University Press. p. 34. ISBN 9781107153523. Retrieved 17 January 2018.
13. ^ James, I.M. (1999-08-24). History of Topology. Elsevier. p. 495. ISBN 9780080534077. Retrieved 17 January 2018.
14. ^ Postnikov, M. M.; Rudyak, Yu. B. (2001) [1994], "Morse theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
15. ^ Basener, William F. (2013-06-12). Topology and Its Applications. John Wiley & Sons. p. 95. ISBN 9781118626221. Retrieved 1 January 2018.
16. ^ Sklyarenko, E. G. (2001) [1994], "Poincaré duality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4