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The Pontryagin theorem (Pontryagin 38a, 50, 55 II.6) identifies, for a closed smooth manifold $M^d$
with
via
In this form, with the assumption that $M^d$ is closed, hence compact, the statement appears for instance in Kosinski 93, Sec. IX Prop. 5.5.
More generally, if the smooth manifold $M^d$ is not assumed to be compact, essentially the same Pontryagin-Thom construction still gives an identification of the cobordism classes of its normally framed submanifolds with the reduced Cohomotopy of its one-point compactification:
This form of Pontryagin’s theorem seems to be folklore (e.g. here). It is made fully explicit in Csépai 20, p. 12-13.
An analogous statement, identifying cobordism classes of normally oriented submanifolds with homotopy classes of maps into the universal special orthogonal Thom space $M SO(n)$, is Thom's theorem (Thom 54):
(Now the notion of asymptotic directed distance depends on the normal tangent spaces, along $\Sigma$, which themselves vary now in the Grassmannian $Gr_n$, hence in the classifying space $B SO(n) \subset M SO(n)$.)
Both statements, Pontryagin’s and Thom’s, as well as their joint generalization to other tangential structures (besides framing and orientation structure) and notably their stabilization to Whitehead-generalized Cobordism cohomology theory, have all come to be widely known as the Pontryagin-Thom construction, or similar, a term commonly used also for rather more involved cases, such as in MUFr-theory. This type of construction constitutes the basis of modern cobordism theory and its application in stable homotopy theory.
The Pontryagin theorem, i.e. the unstable and framed version of the Pontrjagin-Thom construction, identifying cobordism classes of normally framed submanifolds with their Cohomotopy charge in unstable Borsuk-Spanier Cohomotopy sets, is due to:
Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Communication I, Doklady Akademii Nauk SSSR 19(3) (1938), 147-149
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 (pdf)
(both available in English translation in Gamkrelidze 86),
as presented more comprehensively in:
The Pontrjagin theorem must have been known to Pontrjagin at least by 1936, when he announced the computation of the second stem of homotopy groups of spheres:
Review:
Daniel Freed, Karen Uhlenbeck, Appendix B of: Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)
Antoni Kosinski, chapter IX of: Differential manifolds, Academic Press 1993 (pdf, ISBN:978-0-12-421850-5)
John Milnor, Chapter 7 of: Topology from the differentiable viewpoint, Princeton University Press, 1997. (ISBN:9780691048338, pdf)
Mladen Bestvina (notes by Adam Keenan), Chapter 16 in: Differentiable Topology and Geometry, 2002 (pdf)
Michel Kervaire, La méthode de Pontryagin pour la classification des applications sur une sphère, in: E. Vesentini (ed.), Topologia Differenziale, CIME Summer Schools, vol. 26, Springer 2011 (doi:10.1007/978-3-642-10988-1_3)
Rustam Sadykov, Section 1 of: Elements of Surgery Theory, 2013 (pdf, pdf)
András Csépai, Stable Pontryagin-Thom construction for proper maps, Period Math Hung 80, 259–268 (2020) (arXiv:1905.07734, doi:10.1007/s10998-020-00327-0)
Discussion of the early history:
The (fairly straightforward) generalization of the Pontrjagin theorem to the twisted Pontrjagin theorem, identifying twisted Cohomotopy with cobordism classes of normally twisted-framed submanifolds, is made explicit in:
A general equivariant Pontrjagin theorem – relating equivariant Cohomotopy to normal equivariant framed submanifolds – remains elusive, but on free G-manifolds it is again straightforward (and reduces to the twisted Pontrjagin theorem on the quotient space), made explicit in:
In negative codimension, the Cohomotopy charge map from the Pontrjagin theorem gives the May-Segal theorem, now identifying Cohomotopy cocycle spaces with configuration spaces of points:
Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)
c Generalization of these constructions and results is due to
Dusa McDuff, Configuration spaces of positive and negative particles, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)
Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)
Thom's theorem i.e. the unstable and oriented version of the Pontrjagin-Thom construction, identifying cobordism classes of normally oriented submanifolds with homotopy classes of maps to the universal special orthogonal Thom space $M SO(n)$, is due to:
Textbook accounts:
The joint generalization of Pontryagin 38a, 55 (framing structure) and Thom 54 (orientation structure) to any family of tangential structures (“(B,f)-structure”) is first made explicit in
and the general statement that has come to be known as Pontryagin-Thom isomorphism (identifying the stable cobordism classes of normally (B,f)-structure submanifolds with homotopy classes of maps to the Thom spectrum Mf) is Lashof 63, Theorem C.
Textbook accounts:
Stanley Kochman, section 1.5 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Yuli Rudyak, On Thom spectra, Orientability and Cobordism, Springer 1998 (pdf)
Lecture notes:
John Francis, Topology of manifolds course notes (2010) (web), Lecture 3: Thom’s theorem (pdf), Lecture 4 Transversality (notes by I. Bobkova) (pdf)
Cary Malkiewich, Section 3 of: Unoriented cobordism and $M O$, 2011 (pdf)
Tom Weston, Part I of An introduction to cobordism theory (pdf)
See also:
Introducing persistent cohomotopy as a tool in topological data analysis, improving on the use of well groups from persistent homology:
Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom 56 (2016) 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)
Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, 19 2 (2017) (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)
Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology 1 (2018) 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)
Review:
Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 $[$pdf$]$
Urs Schreiber on joint work with Hisham Sati: New Foundations for TDA – Cohomotopy, (May 2022)
Last revised on November 28, 2021 at 13:27:48. See the history of this page for a list of all contributions to it.