group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
and
Rational Cohomotopy theory $\pi^\bullet_{\mathbb{Q}}$ is the approximation in rational homotopy theory of Cohomotopy cohomology theory, i.e. the (non-abelian) generalized cohomology theory whose cocycle spaces are rationalizations of the cocycle spaces of plain Cohomotopy cohomology theory, hence which are spaces of maps into rational n-spheres $S^n_{\mathbb{Q}}$:
Discussion of rational cohomology and of Sullivan models for cocycle spaces in rational Cohomotopy.
(rational homotopy type of space of maps from n-sphere to itself)
Let $n \in \mathbb{N}$ be a natural number and $f\colon S^n \to S^n$ a continuous function from the n-sphere to itself. Then the connected component $Maps_f\big( S^n, S^n\big)$ of the space of maps which contains this map has the following rational homotopy type:
where $deg(f)$ is the degree of $f$.
Moreover, under the canonical morphism expressing the canonical action of the special orthogonal group $SO(n+1)$ on $S^n = S\big( \mathbb{R}^{n+1}\big)$ (regarded as the unit sphere in $(n+1)$-dimensional Cartesian space) we have that on ordinary homology
the generator in $\left\{ \array{ H_{2n+1}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{even} \\ H_{n}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{odd} } \right.$ maps to the fundamental class of the respective spheres in (1), all other generators mapping to zero.
(Møller-Raussen 85, Example 2.5, Cohen-Voronov 05, Lemma 5.3.5)
See at Sullivan model of a spherical fibration for more on this.
(rational cohomology of iterated loop space of the 2k-sphere)
Let
(hence two positive natural numbers, one of them required to be even and the other required to be smaller than the first) and consider the D-fold loop space $\Omega^D S^n$ of the n-sphere.
Its rational cohomology ring is the free graded-commutative algebra over $\mathbb{Q}$ on one generator $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:
(by this Prop. at Sullivan model of based loop space; see also Kallel-Sjerve 99, Prop. 4.10)
For the edge case $\Omega^D S^D$ the above formula does not apply, since $\Omega^{D-1} S^D$ is not simply connected (its fundamental group is $\pi_1\big( \Omega^{D-1}S^D \big) = \pi_0 \big(\Omega^D S^D\big) = \pi_D(S^D) = \mathbb{Z}$, the 0th stable homotopy group of spheres).
But:
The rational model for $\Omega^D S^D$ follows from Prop. by realizing the pointed mapping space as the homotopy fiber of the evaluation map from the free mapping space:
This yields for instance the following examples.
In odd dimensions:
In even dimensions:
(In the following $h_{\mathbb{K}}$ denotes the Hopf fibration of the division algebra $\mathbb{K}$, hence $h_{\mathbb{C}}$ denotes the complex Hopf fibration and $h_{\mathbb{H}}$ the quaternionic Hopf fibration.)
We discuss examples of cocycle spaces for rational 4-Cohomotopy, hence with coefficients in the rational 4-sphere, whose Sullivan model is
The Sullivan model for the space of maps $Maps(S^1, S^4)$ from the 1-sphere to the 4-sphere is
By this Prop., see FSS 16, Section 3.
The Sullivan model for the space of maps $Maps(S^3, S^4)$ from the 3-sphere to the 4-sphere is
By Møller-Raussen 85, Prop. 2.3.
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 |
See also
Discussion of cocycle spaces in rational Cohomotopy (see also at rational model for mapping spaces):
Sadok Kallel, Denis Sjerve, On Brace Products and the Structure of Fibrations with Section, 1999 (pdf, pdf)
Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)
J.-B. Gatsinzi, Rational Gottlieb Group of Function Spacesof Maps into an Even Sphere, International Journal of Algebra, Vol. 6, 2012, no. 9, 427 - 432 (pdf)
The observation that the equations of motion of the supergravity C-field and its dual in D=11 N=1 supergravity characterize it as a cocycle in rational 4-Cohomotopy:
Rational Cohomotopy of super-spaces (see also at geometry of physics – fundamental super p-branes):
Domenico Fiorenza, Hisham Sati, Urs Schreiber: Rational sphere valued supercocycles in M-theory and type IIA string theory, Journal of Geometry and Physics, Volume 114, Pages 91-108 (2017) (arXiv:1606.03206)
Domenico Fiorenza, Hisham Sati, Urs Schreiber: The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557)
Review in:
Discussion of rational equivariant Cohomotopy:
Discussion of rational twisted Cohomotopy:
Last revised on October 18, 2019 at 12:55:45. See the history of this page for a list of all contributions to it.