nLab rational Cohomotopy

Idea

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 $S^n_{\mathbb{Q}}$:

$$\pi^n_{\mathbb{Q}}(-) \;=\; \pi_0 Maps\big(-, S^n\big)_{\mathbb{Q}} \;\simeq\; \pi_0 Maps\big(-, S^n_{\mathbb{Q}}\big) \;.$$

Properties

Sullivan models for cocycle spaces

Discussion of rational cohomology and of Sullivan models for cocycle spaces in rational Cohomotopy.

Proposition

(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:

(1)$$Maps_f\big( S^n, S^n\big) \;\simeq_{\mathbb{Q}}\; \left\{ \array{ S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0 \\ S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0 \\ S^n &\vert& n\, \text{odd} } \right.$$

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

$$\array{ H_\bullet\Big( SO\big( n+ 1 \big) \Big) &\longrightarrow& H_\bullet\Big( Maps_{f = id}\big( S^n, S^n \big) \Big) }$$

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.

Proposition

(rational cohomology of iterated loop space of the 2k-sphere)

Let

$$1 \leq D \lt n = 2k \in \mathbb{N}$$

(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 $2n - D - 1$:

$$H^\bullet \big( \Omega^D S^n , \mathbb{Q} \big) \;\simeq\; \mathbb{Q}\big[ \omega_{n - D}, \omega_{2n - 1 - D} \big] \,.$$

(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:

Example

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:

$$\array{ \mathllap{ \Omega^D S^D \simeq \;} Maps^{\ast/\!}\big( S^D, S^D\big) \\ \big\downarrow^{\mathrlap{fib(ev_\ast)}} \\ Maps(S^D, S^D) \\ \big\downarrow^{\mathrlap{ev_\ast}} \\ S^D }$$

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.)

Examples

We discuss examples of cocycle spaces for rational 4-Cohomotopy, hence with coefficients in the rational 4-sphere, whose Sullivan model is

$$CE \big( S^4 \big) \;=\; \left( \begin{aligned} d\,g_4 & = 0 \\ d\,g_7 & = -\tfrac{1}{2} g_4 \wedge g_4 \end{aligned} \right) \,.$$

The cocycle space for $\pi^4\big( S^1\big)_{\mathbb{Q}}$

The Sullivan model for the space of maps $Maps(S^1, S^4)$ from the 1-sphere to the 4-sphere is

$$CE \Big( \mathfrak{l} Maps \big( S^1, S^4 \big) \Big) \;=\; \left( \begin{aligned} d\,h_3 & = 0 \\ d\, \omega_4 & = 0 \\ d\, \omega_6 & = h_3 \wedge \omega_4 \\ d\, h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 \\ \end{aligned} \right)$$

By this Prop., see FSS 16, Section 3.

The cocycle space for $\pi^4\big( S^3\big)_{\mathbb{Q}}$

The Sullivan model for the space of maps $Maps(S^3, S^4)$ from the 3-sphere to the 4-sphere is

$$CE \Big( \mathfrak{l} Maps \big( S^3, S^4 \big) \Big) \;=\; \left( \begin{aligned} d\, b_1 & = 0 \\ d\, \omega_4 & = 0 \\ d\, v_{{}_{4}} & = \omega_4 \wedge b_1 \\ d\, \omega_7 & = - \tfrac{1}{2} \, \omega_4 \wedge \omega_4 \end{aligned} \right)$$

By Møller-Raussen 85, Prop. 2.3.

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

See also

• rational cohomology

References

Cocycle spaces

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)

On super-spaces

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:

• Hisham Sati, Section 2.5 of: Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018), 062304 (arXiv:1310.1060)

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)

  (arXiv:1506.07557)

Review in:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: The rational higher structure of M-theory, in: Proceedings of the LMS-EPSRC Durham Symposium: Higher Structures in M-Theory, August 2018m Fortschritte der Physik, 2019 (arXiv:1903.02834)

Rational equivariant Cohomotopy

Discussion of rational equivariant Cohomotopy:

• John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes, Comm. Math. Phys. 2019 (arXiv:1805.05987)

Rational twisted Cohomotopy

Discussion of rational twisted Cohomotopy:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3 of Twisted Cohomotopy implies M-theory anomaly cancellation (arXiv:1904.10207)

• 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)