nLab Lev Pontrjagin

Lev Semёnovič Pontrjagin (Лев Семёнович Понтрягин) was an influential Soviet mathematician working in Moscow. His main works till the late 1940s were in general topology, especially the study of topological groups (including study of Pontrjagin duality) and algebraic topology in which he also explored applications of differentiable manifolds for computation of homotopy groups (the Pontrjagin theorem); including the introduction of unstable framed cobordism (unstable Cohomotopy) in that work. After the success of the French mathematical school in the late 1940s in introducing new methods of sheaf theory, which Pontrjagin did not like as being less direct than the more intuitive geometric methods he was master of, and after some pressure from the government, Pontrjagin switched into applied mathematics where he did some fundamental work, especially in optimization theory.

Pontrjagin was a full member of the (Soviet) Academy of Sciences. He was blind since the age of 14.

• Wikipedia on Pontrjagin: in German, in English, in Russian,

• entry at joyalscatlab

Selected writings

Introducing Pontrjagin duality:

• Lev Pontrjagin, Theory of topological commutative groups, Annals of Mathematics Second Series, Vol. 35, No. 2 (Apr., 1934), pp. 361-388 (doi:10.2307/1968438)

  Russian translation: Uspekhi Mat. Nauk, 1936, no. 2, 177–195 (mathnet:umn8882)

Introducing Pontryagin's theorem – the bijection between cobordism classes of normally framed submanifolds of a closed smooth manifold and its Cohomotopy-sets, established via the Pontryagin-Thom collapse/Cohomotopy charge – and its application to the computation of

the first stable homotopy group of spheres:

• Lev Pontrjagin: Classification of continuous maps of a complex into a sphere, Communication I, Doklady Akademii Nauk SSSR 19 3 (1938) 147-149

and the second stable homotopy group of spheres:

• Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Communication II, Dokl. Akad. Nauk SSSR 19 (1938) 361-363

  (this article contains a famous mistake, see Hopkins‘s talk at Atiyah’s 80th Birthday conference, see slide 8, 9:45)

• Lev Pontryagin, Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959, (Russian) (pdf)

  (this article fixes the mistake)

all three of which are available in English translation in Gamkrelidze 86.

A comprehensive account of this cobordism theory/stable homotopy theory via Pontryagin's theorem (now mostly: “Pontryagin-Thom theorem”) is then given in:

• Lev Pontrjagin, Smooth manifolds and their applications in homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955, AMS Translation Series 2, Vol. 11, 1959 (doi:10.1142/9789812772107_0001, pdf)

Introducing the Pontrjagin product:

• Lev Pontrjagin, Homologies in compact Lie groups, Rec. Math. [Mat. Sbornik] N.S., 1939 Volume 6(48), Number 3, Pages 389–422 (mathnet:5835)

Introducing Pontrjagin classes:

• 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).

Selected articles:

• Revaz Gamkrelidze (ed.), L. S. Pontryagin – Selected Works, Volume 1: Selected Research Papers, Classics of Soviet Mathematics, Gordon and Breach 1986, CRC Press 2019 (ISBN:9782881241055)

Related $n$Lab entries

• Pontrjagin class

• Pontrjagin-Thom theorem

category: people