nLab Pontryagin's theorem

Idea

The Pontryagin theorem (Pontryagin 38a, 50, 55 II.6) identifies, for a closed smooth manifold $M^d$

• the cobordism classes of normally framed closed submanifolds of codimension $n$ in $M^d$

with

• the $n$-Cohomotopy of $M^d$ – the homotopy classes of maps from $M^d$ into the n-sphere;

via

From SS 2021

• the Cohomotopy charge map, often known as the Pontryagin-Thom collapse construction, which sends a normally framed submanifold $\Sigma^{d-n} \subset M^d$ to the continuous function taking values in the one-point compactification $\big( \mathbb{R}^n \big)^{cpt} \simeq S^n$ that assigns to points in $M^d$ sufficiently close to $\Sigma$ their “directed distance” from $\Sigma$, namely their normal vector, and regards all other points as being “at infinity”:

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.

Related concepts

• equivariant Pontrjagin theorem, twisted Pontrjagin theorem

• Hopf degree theorem, equivariant Hopf degree theorem

• Poincaré–Hopf theorem

References

Pontrjagin-Thom construction

Pontrjagin’s construction

General

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:

• 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)

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:

• Lev Pontrjagin, Sur les transformations des sphères en sphères (pdf) in: Comptes Rendus du Congrès International des Mathématiques – Oslo 1936 (pdf)

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:

• Kosinski 93, Section IX.9

Twisted/equivariant generalizations

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:

• James Cruickshank, Lemma 5.2 using Sec. 5.1 in: Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

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:

• James Cruickshank, Thm. 5.0.6, Cor. 6.0.13 in: Twisted Cobordism and its Relationship to Equivariant Homotopy Theory, 1999 (pdf, pdf)

In negative codimension

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 construction

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:

• René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, digiz:GDZPPN002056259, pdf)

Textbook accounts:

• Robert Stong, Notes on Cobordism theory, 1968 (toc pdf, publisher page)

Lashof’s construction

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

• Richard Lashof, Poincaré duality and cobordism, Trans. AMS 109 (1963), 257-277 (doi:10.1090/S0002-9947-1963-0156357-4)

and the general statement that has come to be known as the Pontryagin-Thom isomorphism (identifying the stable cobordism classes of normally (B,f)-structured submanifolds with homotopy classes of maps to the Thom spectrum Mf) is really due to Lashof 63, Theorem C.

Textbook accounts:

• Theodor Bröcker, Tammo tom Dieck, Satz 3.1 & 4.9 in: Kobordismentheorie, Lecture Notes in Mathematics 178, Springer (1970) [ISBN:9783540053415]

• 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:

• Manifold Atlas, The Pontrjagin-Thom isomorphism

Cohomotopy in topological data analysis

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)