nLab twisted cohomotopy

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 theorem

The twisted Pontrjagin theorem relates twisted cohomotopy with cobordism classes of normally twisted-framed submanifolds.

Twisted May-Segal theorem

The May-Segal theorem has a twisted generalization:

Proposition

Let

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

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

Then the scanning map constitutes a weak homotopy equivalence

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

between

1. the J-twisted (n+k)-cohomotopy space of $X^n$, 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 $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$

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

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

Remark

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

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

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

Then the scanning map constitutes a weak homotopy equivalence

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

between

1. $(n+k)$-cohomotopy space of $X^n$, hence the space of maps from $X$ to the (n+k)-sphere

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

Poincaré–Hopf theorem

See at

• Poincaré–Hopf theorem

Equivariant Hopf degree theorem

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

• equivariant Hopf degree theorem

Related concepts

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
differential cohomology	differential Cohomotopy	equivariant differential cohomotopy
persistent cohomology	persistent Cohomotopy	persistent equivariant Cohomotopy

• twisted Pontrjagin theorem

References

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 of the twisted Pontrjagin theorem and of twisted stable cohomotopy (framed cobordism cohomology theory):

• James Cruickshank, Lemma 5.2 and Section 7 of: 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)

Discussion of unstable twisted Cohomotopy, with application to flux quantization of supergravity C-field (“Hypothesis H”):

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds, Comm Math. Phys. 377 (2020) 1961-2025 [arXiv:1904.10207, doi:10.1007/s00220-020-03707-2]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane (arXiv:1906.07417)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5-brane anomaly cancellation (arXiv:2002.07737)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies twisted String structure on M5-branes (arXiv:2002.11093)