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

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.

Various classical theorem of differential topology are secretly theorems about twisted cohomotopy, including:

cohomologyequivariant cohomology
non-abelian cohomologycohomotopyequivariant cohomotopy
twisted cohomologytwisted cohomotopy
stable cohomologystable cohomotopyequivariant stable cohomotopy

References

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

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

Last revised on June 19, 2019 at 04:27:29. See the history of this page for a list of all contributions to it.