Timeline of bordism

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This is a timeline of

homology theory using only (smooth) manifolds.[1]

Integral theorems

Year Contributors Event
Late 17th century Gottfried Wilhelm Leibniz and others The
definite integral
over an interval as a signed combination of the antiderivative at the endpoints. A corollary is that if the derivative of a function is zero, the function is constant.
1760s Joseph-Louis Lagrange Introduces a transformation of a
sound propagation.[2]
1889 Vito Volterra Version of Stokes' theorem in n dimensions, using anti-symmetry.[3]
1899 Henri Poincaré In Les méthodes nouvelles de la mécanique céleste, he introduces a version of Stokes' theorem in n dimensions using what is essentially differential form notation.[4]
1899 Élie Cartan Definition of the exterior algebra of differential forms in Euclidean space.[4]
c.1900 Mathematical folklore The situation at the end of the 19th century is that a geometric form of the fundamental theorem of calculus is available, if everything was smooth enough when rigour is required, and in Euclidean space of n dimensions.

The result corresponding to setting the derivative equal to zero is to apply it to

closed forms, and as such is "mathematical folklore". It is in the nature of a remark that there are integral theorems for submanifolds linked by cobordism
. The analogue of the theorem on derivative zero would be for submanifolds and that jointly form the boundary of a manifold N, and a form defined on N with . Then the integrals and of over the are equal. The signed sum seen in the case of a boundary of dimension 0 reflects the need to use orientations on the manifolds, to define integrals.

1931–2 W. V. D. Hodge The
codifferential adjoint to the exterior derivative is the general form of divergence operator. Closed forms are dual to forms of divergence 0.[5]

Cohomology

Year Contributors Event
1920s Élie Cartan and Hermann Weyl Topology of
Lie groups
.
1931 Georges de Rham
De Rham's theorem: for a compact differential manifold, the chain complex of differential forms computes the real homology groups.[6]
1935–1940 Group effort The
smooth manifolds
.
1942 Lev Pontryagin Publishing in full in 1947, Pontryagin founded a new theory of
closed differential forms; the introduction of algebraic invariants gives the opening for computing with the equivalence relation as something intrinsic.[7]
1940s Theories of
Stiefel-Whitney class and Pontryagin class
.
1945 Samuel Eilenberg and Norman Steenrod
homology theory
and cohomology, on a class of spaces.
1946 Norman Steenrod The Steenrod problem. Stated as Problem 25 in a list by Eilenberg compiled in 1946, it asks, given an integral homology class in degree n of a simplicial complex, is it the image by a continuous mapping of the fundamental class of an oriented manifold of dimension n? The preceding question asks for the spherical homology classes to be characterised. The following question asks for a criterion from algebraic topology for an orientable manifold to be a boundary.[8]
1958 Frank Adams
stable homotopy
groups from cohomology groups.

Homotopy theory

Year Contributors Event
1954 René Thom Formal definition of cobordism of oriented manifolds, as an equivalence relation.[9] Thom computed, as a ring under disjoint union and cartesian product, the cobordism ring of unoriented smooth manifolds; and introduced the ring of oriented smooth manifolds.[10] is a polynomial algebra over the field with two elements, with a single generator in each degree, except degrees one less than a power of 2.[1]
1954 René Thom In modern notation, Thom contributed to the Steenrod problem, by means of a homomorphism , the Thom homomorphism.[11] The Thom space construction M reduced the theory to the study of mappings in cohomology .[12]
1955 Michel Lazard Lazard's universal ring, the ring of definition of the universal formal group law in one dimension.
1960 Michael Atiyah Definition of cobordism groups and bordism groups of a space X.[13]
1969 Daniel Quillen The formal group law associated to complex cobordism is universal.[14]

Notes

  1. ^
    ISBN 978-0-8176-4907-4
    .
  2. ISBN 978-0-7190-1756-8
    .
  3. ISBN 978-3-642-22421-8
    .
  4. ^
    JSTOR 2690275
  5. ISBN 978-0-19-853275-0
    .
  6. ^ "De Rham theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  7. ^ Canadian Mathematical Bulletin. Canadian Mathematical Society. 1971. p. 289. Retrieved 6 July 2018.
  8. ^ Samuel Eilenberg, On the Problems of Topology, Annals of Mathematics Second Series, Vol. 50, No. 2 (Apr., 1949), pp. 247–260, at p. 257. Published by: Mathematics Department, Princeton University
    JSTOR 1969448
  9. ISBN 978-2-04-010012-4
    .
  10. ISBN 978-0-691-04938-0
    .
  11. ^ "Steenrod problem – Manifold Atlas". www.map.mpim-bonn.mpg.de.
  12. ^ Rudyak, Yu. B. (2001) [1994], "Steenrod problem", Encyclopedia of Mathematics, EMS Press
  13. ^ Anosov, D. V. (2001) [1994], "Bordism", Encyclopedia of Mathematics, EMS Press
  14. ^ Rudyak, Yu. B. (2001) [1994], "Cobordism", Encyclopedia of Mathematics, EMS Press
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