nLab
twisted cohomotopy

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Manifolds and cobordisms

Spheres

Contents

Idea

Twisted Cohomotopy is the twisted cohomology-variant of the the non-abelian cohomology-theory Cohomotopy, represented by homotopy types of n-spheres.

The coefficients/twist for twisted Cohomotopy are spherical fibrations, and cocycles are sections of these. For those spherical fibrations arising as unit sphere bundles of real vector bundles the twist may be understood as given by the J-homomorphism.

Properties

Various classical theorems of differential topology are secretly theorems about twisted cohomotopy

table grabbed from FSS 19b

Twisted Pontrjagin-Thom theorem

The scanning map-equivalences on configuration spaces of points may be regarded as generalizations of the Pontryagin-Thom theorem from sets of Cohomotopy classes to homotopy types of twisted Cohomotopy cocycles:

Proposition

Let

  1. X nX^n be a smooth closed manifold of dimension nn;

  2. 1k1 \leq k \in \mathbb{N} a positive natural number.

Then the scanning map constitutes a weak homotopy equivalence

Maps /BO(n)(X n,S n def+k trivO(n))a J-twisted Cohomotopy spacescanning mapConf(X n,S k)aconfiguration spaceof points \underset{ \color{blue} { \phantom{a} \atop \text{ J-twisted Cohomotopy space}} }{ Maps_{{}_{/B O(n)}} \Big( X^n \;,\; S^{ \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}} } \!\sslash\! O(n) \Big) } \underoverset {\simeq} { \color{blue} \text{scanning map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }

between

  1. the J-twisted (n+k)-Cohomotopy space of X nX^n, hence the space of sections of the (n+k)(n + k)-spherical fibration over XX which is associated via the tangent bundle by the O(n)-action on S n+k=S( n× k+1)S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})

  2. the configuration space of points on X nX^n with labels in S kS^k.

(Bödigheimer 87, Prop. 2, following McDuff 75)

Remark

In the special case that the closed manifold X nX^n in Prop. is parallelizable, hence that its tangent bundle is trivializable, the statement of Prop. reduces to this:

Let

  1. X nX^n be a parallelizable closed manifold of dimension nn;

  2. 1k1 \leq k \in \mathbb{N} a positive natural number.

Then the scanning map constitutes a weak homotopy equivalence

Maps(X n,S n+k)a Cohomotopy spacescanning mapConf(X n,S k)aconfiguration spaceof points \underset{ \color{blue} { \phantom{a} \atop \text{ Cohomotopy space}} }{ Maps \Big( X^n \;,\; S^{ n + k } \Big) } \underoverset {\simeq} { \color{blue} \text{scanning map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }

between

  1. (n+k)(n+k)-Cohomotopy space of X nX^n, hence the space of maps from XX to the (n+k)-sphere

  2. the configuration space of points on X nX^n with labels in S kS^k.

Poincaré–Hopf theorem

See at

Equivariant Hopf degree theorem

On flat orbifolds, twisted Cohomotopy becomes equivariant Cohomotopy and the twisted Hopf degree theorem becomes the

flavours of
Cohomotopy
cohomology theory
cohomology
(full or rational)
equivariant cohomology
(full or rational)
non-abelian cohomologyCohomotopy
(full or rational)
equivariant Cohomotopy
twisted cohomology
(full or rational)
twisted Cohomotopytwisted equivariant Cohomotopy
stable cohomology
(full or rational)
stable Cohomotopyequivariant stable Cohomotopy

References

The concept is implicit in classical texts on differential topology, for instance

Discussion for twisted stable cohomotopy (framed cobordism cohomology theory):

Discussion of unstabilized twisted cohomotopy, with application to foundations of M-theory:

Last revised on December 1, 2019 at 14:12:02. See the history of this page for a list of all contributions to it.