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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 19th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these pointers on the first Pontrjagin class as counting [[gravitational instanton number]]: * [[Tohru Eguchi]], [[Peter Freund]], _Quantum Gravity and World Topology_, Phys. Rev. Lett. 37, 1251 (1967) ([doi:10.1103/PhysRevLett.37.1251](https://doi.org/10.1103/PhysRevLett.37.1251)) * [[Alexander Belavin]], D. Burlankov, _The renormalisable theory of gravitation and the einstein equations_, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (<a href="https://doi.org/10.1016/0375-9601(76)90530-2">doi:10.1016/0375-9601(76)90530-2</a>) <a href="https://ncatlab.org/nlab/revision/diff/Pontryagin+class/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/Pontryagin+class/24">v24</a>, <a href="https://ncatlab.org/nlab/show/Pontryagin+class">current</a>

   added these pointers on the first Pontrjagin class as counting gravitational instanton number:

   • Tohru Eguchi, Peter Freund, Quantum Gravity and World Topology, Phys. Rev. Lett. 37, 1251 (1967) (doi:10.1103/PhysRevLett.37.1251)

   • Alexander Belavin, D. Burlankov, The renormalisable theory of gravitation and the einstein equations, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (doi:10.1016/0375-9601(76)90530-2)

   diff, v24, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeNov 13th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: The original definition is in * [[Л. С. Понтрягин]], _Характеристические циклы многообразий_, ДАН, XXXV, № 2 (1942). * [[Л. С. Понтрягин]], _Характеристические циклы дифференцируемых многообразий_, Матем. сб., 21(63):2 (1947), 233–284. [MathNet.Ru PDF](http://mi.mathnet.ru/msb6237 ). English translation: [[Lev Pontrjagin]], _Characteristic cycles on differentiable manifolds_, Mat. Sbornik N. S. 21(63) (1947), 233-284; A.M.S. Translation 32 (1950). <a href="https://ncatlab.org/nlab/revision/diff/Pontryagin+class/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/Pontryagin+class/29">v29</a>, <a href="https://ncatlab.org/nlab/show/Pontryagin+class">current</a>

   Added:

   The original definition is in

   • Л. С. Понтрягин, Характеристические циклы многообразий, ДАН, XXXV, № 2 (1942).

   • Л. С. Понтрягин, Характеристические циклы дифференцируемых многообразий, Матем. сб., 21(63):2 (1947), 233–284. MathNet.Ru PDF. English translation: Lev Pontrjagin, Characteristic cycles on differentiable manifolds, Mat. Sbornik N. S. 21(63) (1947), 233-284; A.M.S. Translation 32 (1950).

   diff, v29, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, that's good to have. I have copied these two items also to the author page, [here](https://ncatlab.org/nlab/show/Lev+Pontrjagin#IntroducingPontrjaginClasses).

   Thanks, that’s good to have. I have copied these two items also to the author page, here.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 2nd 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded references to two more translations of Pontrjagin's original articles. <a href="https://ncatlab.org/nlab/revision/diff/Pontryagin+class/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/Pontryagin+class/30">v30</a>, <a href="https://ncatlab.org/nlab/show/Pontryagin+class">current</a>

   Added references to two more translations of Pontrjagin’s original articles.

   diff, v30, current