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:

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 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

Year Contributors Event
18th centuryLeonhard EulerEuler'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.
1820–3János BolyaiDevelops non-Euclidean geometry, in particular the hyperbolic plane.
1822Jean-Victor PonceletReconstructs real projective geometry, including the real projective plane.[1]
c.1825Joseph Diez Gergonne, Jean-Victor PonceletGeometric properties of the complex projective plane.[2]
1840Hermann GrassmannGeneral n-dimensional linear spaces.
1848Carl Friedrich Gauss
Pierre Ossian Bonnet
Gauss–Bonnet theorem for the differential geometry of closed surfaces.
1851Bernhard RiemannIntroduction 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).
1854Bernhard RiemannRiemannian metrics give an idea of intrinsic geometry of manifolds of any dimension.
1861Folklore result since c.1850First conventional publication of the Kelvin–Stokes theorem, in three dimensions, relating integrals over a volume to those on its boundary.
1870sSophus LieThe Lie group concept is developed, using local formulae.[4]
1872Felix KleinKlein's Erlangen program puts an emphasis on the homogeneous spaces for the classical groups, as a class of manifolds foundational for geometry.
later 1870sUlisse DiniDini develops the implicit function theorem, the basic tool for constructing manifolds locally as the zero sets of smooth functions.[5]
from 1890sÉlie CartanFormulation of Hamiltonian mechanics in terms of the cotangent bundle of a manifold, the configuration space.[6]
1894Henri PoincaréFundamental group of a topological space. The Poincaré conjecture can now be formulated.
1895Henri PoincaréSimplicial homology.
1895Henri PoincaréFundamental 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.[7]

1900 to 1920

Year Contributors Event
1900David HilbertHilbert'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.
1902David HilbertTentative axiomatisation (topological spaces are not yet defined) of two-dimensional manifolds.[8]
1905Max DehnAs a conjecture, the Dehn-Somerville equations relating numerically triangulated manifolds and simplicial polytopes.[9]
1907Henri Poincaré, Paul KoebeThe uniformization theorem for simply connected Riemann surfaces.
1907Max Dehn, Poul HeegaardSurvey 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.[10][11][12] The work also contained a combinatorial definition of "topological manifold", a subject in definitional flux up to the 1930s.[13]
1908Heinrich Franz Friedrich TietzeHabilitationschrift for the University of Vienna, proposes another tentative definition, by combinatorial means, of "topological manifold".[13][14][15]
1908Ernst Steinitz, TietzeThe Hauptvermutung, a conjecture on the existence of a common refinement of two triangulations. This was an open problem, for manifolds, to 1961.
1910L. E. J. BrouwerBrouwer'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.[16] In the same year Brouwer gives the first example of a topological group that is not a Lie group.[17]
1912L. E. J. BrouwerBrouwer publishes on the degree of a continuous mapping, foreshadowing the fundamental class concept for orientable manifolds.[18][19]
1913Hermann WeylDie Idee der Riemannschen Fläche gives a model definition of the idea of manifold, in the one-dimensional complex case.
1915Oswald VeblenThe "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.[20]

1920 to the 1945 axioms for homology

Year Contributors Event
1923Hermann KünnethKünneth formula for homology of product of spaces.
1926Hellmuth KneserDefines "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.[21]
1926Élie CartanClassification of symmetric spaces, a class of homogeneous spaces.
1926Tibor RadóTwo-dimensional topological manifolds have triangulations.[22]
1926Heinz HopfPoincaré–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.
1926−7Otto SchreierDefinitions 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.[23]
1928Leopold VietorisDefinition of h-manifold, by combinatorial means, by proof analysis applied to Poincaré duality.[24]
1929Egbert van KampenIn his dissertation, by means of star-complexes for simplicial complexes, recovers Poincaré duality in a combinatorial setting.[25]
1930Bartel Leendert van der WaerdenPursuing 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).[26] 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.[27]
c.1930Emmy NoetherModule theory and general chain complexes are developed by Noether and her students, and algebraic topology begins as an axiomatic approach grounded in abstract algebra.
1931Georges de RhamDe Rham's theorem: for a compact differential manifold, the chain complex of differential forms computes the real (co)homology groups.[28]
1931Heinz HopfIntroduces the Hopf fibration, .
1931–2Oswald Veblen, J. H. C. WhiteheadWhitehead'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.[29][30][8] This foundational point of view allows for a pseudogroup restriction on the transition functions, for example to introduce piecewise linear structures.[31]
1932Eduard ČechČech cohomology.
1933Solomon LefschetzSingular homology of topological spaces.
1934Marston MorseMorse theory relates the real homology of compact differential manifolds to the critical points of a Morse function.[32]
1935Hassler WhitneyProof of the embedding theorem, stating that a smooth manifold of dimension n may be embedded in Euclidean space of dimension 2n.[33]
1941Witold HurewiczFirst 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.
1942Lev PontryaginPublishing 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.[34]
1943Werner GysinGysin sequence and Gysin homomorphism.
1943Norman SteenrodHomology with local coefficients.
1944Samuel Eilenberg"Modern" definition of singular homology and singular cohomology.
1945Beno EckmannDefines 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.[35] The term "separable manifold", to distinguish second countable manifolds, survived into the late 1950s.[36]

Year Contributors Event
1945Saunders Mac LaneSamuel EilenbergFoundation of category theory: axioms for categories, functors and natural transformations.
1945Norman SteenrodSamuel EilenbergEilenberg–Steenrod axioms for homology and cohomology.
1945Jean LerayFounds 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.
1945Jean LerayDefines sheaf cohomology.
1946Jean LerayInvents spectral sequences, a method for iteratively approximating cohomology groups.
1948Cartan seminarWrites up sheaf theory.
c.1949Norman SteenrodThe Steenrod problem, of representation of homology classes by fundamental classes of manifolds, can be solved by means of pseudomanifolds (and later, formulated via cobordism theory).[37]
1950Henri CartanIn 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."[38]
1950Samuel Eilenberg–Joe ZilberSimplicial sets as a purely algebraic model of well behaved topological spaces.
1950Charles EhresmannEhresmann's fibration theorem states that a smooth, proper, surjective submersion between smooth manifolds is a locally trivial fibration.
1951Henri CartanDefinition 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.
1952René ThomThe Thom isomorphism brings cobordism of manifolds into the ambit of homotopy theory.
1952Edwin E. MoiseMoise's theorem established that a 3-dimension compact connected topological manifold is a PL manifold (earlier terminology "combinatorial manifold"), having a unique PL structure. In particular it is triangulable.[39] This result is now known to extend no further into higher dimensions.
1956John MilnorThe 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.
1960John Milnor and Sergei NovikovThe ring of cobordism classes of stably complex manifolds is a polynomial ring on infinitely many generators of positive even degrees.

1961 to 1970

Year Contributors Event
1961Stephen SmaleProof of the generalized Poincaré conjecture in dimensions greater than four.
1962Stephen SmaleProof of the h-cobordism theorem in dimensions greater than four, based on the Whitney trick.
1963Michel KervaireJohn MilnorThe classification of exotic spheres: the monoid of smooth structures on the n-sphere is the collection of oriented smooth n-manifolds which are homeomorphic to , taken up to orientation-preserving diffeomorphism, with connected sum as the monoid operation. For , this monoid is a group, and is isomorphic to the group of h-cobordism classes of oriented homotopy n-spheres, which is finite and abelian.
1965Dennis BardenCompletes the classification of simply connected, compact 5-manifolds, started by Smale in 1962.
1967Friedhelm WaldhausenDefines and classifies 3-dimensional graph manifolds.
1968Robion Kirby and Laurent C. SiebenmannIn dimension at least five, the Kirby–Siebenmann class is the only obstruction to a topological manifold having a PL structure.[40]
1969Laurent C. SiebenmannExample of two homeomorphic PL manifolds that are not piecewise-linearly homeomorphic.[41]

The maximal atlas approach to structures on manifolds had clarified the Hauptvermutung for a topological manifold M, as a trichotomy. M might have no triangulation, hence no piecewise-linear maximal atlas; it might have a unique PL structure; or it might have more than one maximal atlas, and so more than one PL structure. The status of the conjecture, that the second option was always the case, became clarified at this point in the form that each of the three cases might apply, depending on M.

The "combinatorial triangulation conjecture" stated that the first case could not occur, for M compact.[42] The Kirby–Siebenmann result disposed of the conjecture. Siebenmann's example showed the third case is also possible.

1970John ConwaySkein theory of knots: The computation of knot invariants by skein modules. Skein modules can be based on quantum invariants.

1971–1980

Year Contributors Event
1974Shiing-Shen Chern–James SimonsChern–Simons theory: A particular TQFT which describe knot and manifold invariants, at that time only in 3D
1978Francois Bayen–Moshe Flato–Chris Fronsdal–Andre Lichnerowicz–Daniel SternheimerDeformation quantization, later to be a part of categorical quantization

1981–1990

Year Contributors Event
1984Vladimir Bazhanov–Razumov StroganovBazhanov–Stroganov d-simplex equation generalizing the Yang–Baxter equation and the Zamolodchikov equation
1986Joachim Lambek–Phil ScottSo-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 (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
1986Peter Freyd–David YetterConstructs the (compact braided) monoidal category of tangles
1986Vladimir Drinfel'd–Michio JimboQuantum 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.
1987Vladimir Drinfel'd–Gerard LaumonFormulates geometric Langlands program
1987Vladimir TuraevStarts 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.
1988Graeme SegalElliptic objects: A functor that is a categorified version of a vector bundle equipped with a connection, it is a 2D parallel transport for strings.
1988Graeme SegalConformal field theory: A symmetric monoidal functor satisfying some axioms
1988Edward WittenTopological quantum field theory (TQFT): A monoidal functor satisfying some axioms
1988Edward WittenTopological string theory
1989Edward WittenUnderstanding of the Jones polynomial using Chern–Simons theory, leading to invariants for 3-manifolds
1990Nicolai ReshetikhinVladimir TuraevEdward WittenReshetikhin–Turaev-Witten invariants of knots from modular tensor categories of representations of quantum groups.

1991–2000

Year Contributors Event
1991André JoyalRoss StreetFormalization 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.
1992John Greenlees–Peter MayGreenlees–May duality
1992Vladimir TuraevModular 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.
1992Vladimir TuraevOleg ViroTuraev–Viro state sum models based on spherical categories (the first state sum models) and Turaev–Viro state sum invariants for 3-manifolds.
1992Vladimir TuraevShadow world of links: Shadows of links give shadow invariants of links by shadow state sums.
1993Ruth LawrenceExtended TQFTs
1993David Yetter–Louis CraneCrane–Yetter state sum models based on ribbon categories and Crane–Yetter state sum invariants for 4-manifolds.
1993Kenji FukayaA-categories and A-functors. A-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.
1993John Barret-Bruce WestburySpherical categories: Monoidal categories with duals for diagrams on spheres instead for in the plane.
1993Maxim KontsevichKontsevich 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.
1993Daniel FreedA new view on TQFT using modular tensor categories that unifies 3 approaches to TQFT (modular tensor categories from path integrals).
1994Maxim KontsevichFormulates 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).
1994Louis Crane–Igor FrenkelHopf categories and construction of 4D TQFTs by them. Identifies k-tuply monoidal n-categories. It mirrors the table of homotopy groups of the spheres.
1995John Baez–James DolanOutline a program in which n-dimensional TQFTs are described as n-category representations.
1995John Baez–James DolanProposes n-dimensional deformation quantization.
1995John Baez–James DolanTangle 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.
1995John Baez–James DolanCobordism 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.
1995John Baez–James DolanExtended 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.
1995Valentin LychaginCategorical quantization
1997Maxim KontsevichFormal 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.
1998Richard ThomasThomas, 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.
1998Maxim KontsevichCalabi–Yau 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 is a unital Calabi–Yau A-category of Calabi–Yau dimension d. A Calabi–Yau category with one object is a Frobenius algebra.
1999Joseph BernsteinIgor FrenkelMikhail KhovanovTemperley–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 , where is given by the isotopy classes of systems of 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.
1999Moira Chas–Dennis SullivanConstructs string topology by cohomology. This is string theory on general topological manifolds.
1999Mikhail KhovanovKhovanov homology: A homology theory for knots such that the dimensions of the homology groups are the coefficients of the Jones polynomial of the knot.
1999Vladimir TuraevHomotopy quantum field theory HQFT
1999Ronald Brown–George Janelidze2-dimensional Galois theory.
2000Yakov EliashbergAlexander Givental–Helmut HoferSymplectic field theory SFT: A functor 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
2003Grigori PerelmanPerelman's proof of the Poincaré conjecture in dimension 3 using Ricci flow. The proof is more general.[43]
2004Stephen StolzPeter TeichnerDefinition of nD quantum field theory of degree p parametrized by a manifold.
2004Stephen StolzPeter TeichnerProgram to construct Topological modular forms 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).
2005Peter OzsváthZoltán SzabóKnot Floer homology
2008Bruce BartlettPrimacy 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.
2008Michael HopkinsJacob LurieSketch of proof of the Baez–Dolan tangle hypothesis and the Baez–Dolan cobordism hypothesis, which classify extended TQFT in all dimensions.
2016Ciprian ManolescuRefutation of the "triangulation conjecture", with the proof that in dimension at least five, there exists a compact topological manifold not homeomorphic to a simplicial complex.[44]

See also

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, Vladimir P. (2001) [1994], "Lie group", Encyclopedia of Mathematics, EMS Press
  5. James, Ioan M. (1999-08-24). History of Topology. Elsevier. p. 31. ISBN 9780080534077. Retrieved 30 June 2018.
  6. 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.
  7. 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.
  8. James, I.M. (1999-08-24). History of Topology. Elsevier. p. 47. ISBN 9780080534077. Retrieved 17 January 2018.
  9. Effenberger, Felix (2011). Hamiltonian Submanifolds of Regular Polytopes. Logos Verlag Berlin GmbH. p. 20. ISBN 9783832527587. Retrieved 15 June 2018.
  10. Dehn, Max; Heegaard, Poul (1907). "Analysis situs". Enzyklop. d. math. Wissensch. III. pp. 153–220. JFM 38.0510.14.
  11. O'Connor, John J.; Robertson, Edmund F., "Timeline of manifolds", MacTutor History of Mathematics archive, University of St Andrews.
  12. Peifer, David. "Max Dehn and the Origins of Topology and Infinite Group Theory" (PDF).
  13. James, Ioan M. (1999-08-24). History of Topology. Elsevier. p. 54. ISBN 9780080534077. Retrieved 15 June 2018.
  14. O'Connor, John J.; Robertson, Edmund F., "Timeline of manifolds", MacTutor History of Mathematics archive, University of St Andrews.
  15. Killy, Walther; Vierhaus, Rudolf (2011-11-30). Thibaut - Zycha. Walter de Gruyter. p. 43. ISBN 9783110961164. Retrieved 15 June 2018.
  16. 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.
  17. Dalen, Dirk van (2012-12-04). L.E.J. Brouwer – Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life. Springer Science & Business Media. p. 147. ISBN 9781447146162. Retrieved 30 June 2018.
  18. Mawhin, Jean (2001) [1994], "Brouwer degree", Encyclopedia of Mathematics, EMS Press
  19. Dalen, Dirk van (2012-12-04). L.E.J. Brouwer – Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life. Springer Science & Business Media. p. 171. ISBN 9781447146162. Retrieved 30 June 2018.
  20. Gallier, Jean; Xu, Dianna (2013). A Guide to the Classification Theorem for Compact Surfaces. Springer Science & Business Media. p. 156. ISBN 9783642343643.
  21. James, I.M. (1999-08-24). History of Topology. Elsevier. pp. 52–3. ISBN 9780080534077. Retrieved 15 June 2018.
  22. James, I.M. (1999-08-24). History of Topology. Elsevier. p. 56. ISBN 9780080534077. Retrieved 17 January 2018.
  23. Bourbaki, N. (2013-12-01). Elements of the History of Mathematics. Springer Science & Business Media. pp. 264 note 20. ISBN 9783642616938. Retrieved 30 June 2018.
  24. James, I. M. (1999-08-24). History of Topology. Elsevier. p. 54. ISBN 9780080534077. Retrieved 15 June 2018.
  25. James, I. M. (1999-08-24). History of Topology. Elsevier. p. 54. ISBN 9780080534077. Retrieved 15 June 2018.
  26. Fulton, W. (2013-06-29). Intersection Theory. Springer Science & Business Media. p. 128. ISBN 9783662024218. Retrieved 15 June 2018.
  27. James, I.M. (1999-08-24). History of Topology. Elsevier. p. 54. ISBN 9780080534077. Retrieved 15 June 2018.
  28. "De Rham theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  29. James, I. M. (1999-08-24). History of Topology. Elsevier. p. 56. ISBN 9780080534077. Retrieved 17 January 2018.
  30. Wall, C. T. C. (2016-07-04). Differential Topology. Cambridge University Press. p. 34. ISBN 9781107153523. Retrieved 17 January 2018.
  31. James, I.M. (1999-08-24). History of Topology. Elsevier. p. 495. ISBN 9780080534077. Retrieved 17 January 2018.
  32. Postnikov, M. M.; Rudyak, Yu. B. (2001) [1994], "Morse theory", Encyclopedia of Mathematics, EMS Press
  33. Basener, William F. (2013-06-12). Topology and Its Applications. John Wiley & Sons. p. 95. ISBN 9781118626221. Retrieved 1 January 2018.
  34. Society, Canadian Mathematical (1971). Canadian Mathematical Bulletin. Canadian Mathematical Society. p. 289. Retrieved 6 July 2018.
  35. James, I.M. (1999-08-24). History of Topology. Elsevier. p. 55. ISBN 9780080534077. Retrieved 15 June 2018.
  36. Milnor, John Willard; McCleary, John (2009). Homotopy, Homology, and Manifolds. American Mathematical Soc. p. 6. ISBN 9780821844755. Retrieved 15 June 2018.
  37. Rudyak, Yu. B. (2001) [1994], "Steenrod problem", Encyclopedia of Mathematics, EMS Press
  38. Sklyarenko, E. G. (2001) [1994], "Poincaré duality", Encyclopedia of Mathematics, EMS Press
  39. Spreer, Jonathan (2011). Blowups, Slicings and Permutation Groups in Combinatorial Topology. Logos Verlag Berlin GmbH. p. 39. ISBN 9783832529833. Retrieved 2 July 2018.
  40. Freed, Daniel S.; Uhlenbeck, Karen K. (2012-12-06). Instantons and Four-Manifolds. Springer Science & Business Media. p. 1. ISBN 9781461397038. Retrieved 6 July 2018.
  41. Rudyak, Yuli (2015-12-28). Piecewise Linear Structures On Topological Manifolds. World Scientific. p. 81. ISBN 9789814733809. Retrieved 6 July 2018.
  42. Ranicki, Andrew A.; Casson, Andrew J.; Sullivan, Dennis P.; Armstrong, M.A.; Rourke, Colin P.; Cooke, G.E. (2013-03-09). The Hauptvermutung Book: A Collection of Papers on the Topology of Manifolds. Springer Science & Business Media. p. 5. ISBN 9789401733434. Retrieved 7 July 2018.
  43. Morgan, John W.; Tian, Gang (2007). Ricci Flow and the Poincaré Conjecture. American Mathematical Soc. p. ix. ISBN 9780821843284.
  44. Manolescu, Ciprian (2016), "Pin(2)-equivariant Seiberg–Witten Floer homology and the Triangulation Conjecture", Journal of the American Mathematical Society, 29: 147–176, arXiv:1303.2354, doi:10.1090/jams829
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