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.
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.
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.[10][11][12] The work also contained a combinatorial definition of "topological manifold", a subject in definitional flux up to the 1930s.[13]
Habilitationschrift for the University of Vienna, proposes another tentative definition, by combinatorial means, of "topological manifold".[13][11][14]
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.[15] In the same year Brouwer gives the first example of a topological group that is not a Lie group.[16]
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.[19]
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.[20]
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.
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.[22]
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.
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.[27][28][8] This foundational point of view allows for a pseudogroup restriction on the transition functions, for example to introduce piecewise linear structures.[29]
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.
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.[32]
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.[33] The term "separable manifold", to distinguish second countable manifolds, survived into the late 1950s.[34]
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.
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."[36]
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.
Moise'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.[37] This result is now known to extend no further into higher dimensions.
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.
The 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.
Example of two homeomorphic PL manifolds that are not piecewise-linearly homeomorphic.[39]
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.[40] The Kirby–Siebenmann result disposed of the conjecture. Siebenmann's example showed the third case is also possible.
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.
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
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.
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 (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
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.
Kronheimer and Mrowka introduce the idea of "canonical classes" in the cohomology of simple smooth 4-manifolds which hypothetically allow one to compute the Donaldson invariants of smooth 4-manifolds. See more under the Kronheimer–Mrowka basic class.
Nathan Seiberg and Edward Witten introduce new invariants for smooth oriented 4-manifolds. Like Donaldson, they are motivated by mathematical physics, but their invariants are easier to work with than the Donaldson invariants. See more under the Seiberg–Witten invariants and Seiberg–Witten theory.
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).
Tangle hypothesis: The n-category of framed n-tangles in dimensions is -equivalent to the free weak k-tuply monoidal n-category with duals on one object.
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.
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.
Calabi–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.
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 , 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.
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.
Symplectic 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.
Program 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).
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.
Refutation 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.[42]
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