# nLab twisted Pontrjagin theorem

Contents

### Context

#### Cobordism theory

Concepts of cobordism theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

relative bordism theories:

algebraic:

# Contents

## Idea

A twisted Pontrjagin theorem should generalize the Pontrjagin theorem from plain homotopy theory/cobordism theory to parametrized homotopy theory/twisted cobordism theory:

Where the plain Pontrjagin theorem identifies the Cohomotopy of a differentiable manifold with its cobordism classes of normally framed submanifolds, the twisted Pontrjagin theorem should identify twisted Cohomotopy with cobordism classes of normally twisted-framed submanifolds (Cruickshank 03, Lemma 5.2)

## 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:

(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:

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

Review:

Discussion of the early history:

##### 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:

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:

Textbook accounts:

#### 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

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:

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)