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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 theorems of differential topology are secretly theorems about twisted cohomotopy
table grabbed from FSS 19b
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:
Let
$X^n$ be a smooth closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the scanning map constitutes a weak homotopy equivalence
between
the J-twisted (n+k)-Cohomotopy space of $X^n$, hence the space of sections of the $(n + k)$-spherical fibration over $X$ which is associated via the tangent bundle by the O(n)-action on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$
the configuration space of points on $X^n$ with labels in $S^k$.
(Bödigheimer 87, Prop. 2, following McDuff 75)
In the special case that the closed manifold $X^n$ in Prop. is parallelizable, hence that its tangent bundle is trivializable, the statement of Prop. reduces to this:
Let
$X^n$ be a parallelizable closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the scanning map constitutes a weak homotopy equivalence
between
$(n+k)$-Cohomotopy space of $X^n$, hence the space of maps from $X$ to the (n+k)-sphere
the configuration space of points on $X^n$ with labels in $S^k$.
See at
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 cohomology | Cohomotopy (full or rational) | equivariant Cohomotopy |
twisted cohomology (full or rational) | twisted Cohomotopy | twisted equivariant Cohomotopy |
stable cohomology (full or rational) | stable Cohomotopy | equivariant stable Cohomotopy |
The concept is implicit in classical texts on differential topology, for instance
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, Section 2 of: Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)
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.