Contents

cohomology

and

# Contents

## 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.

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 $2n - D - 1$:

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

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)
flavours of
Cohomotopy
cohomology theory
cohomology
(full or rational)
equivariant cohomology
(full or rational)
non-abelian cohomologyCohomotopy
(full or rational)
equivariant Cohomotopy
twisted cohomology
(full or rational)
twisted Cohomotopytwisted equivariant Cohomotopy
stable cohomology
(full or rational)
stable Cohomotopyequivariant stable Cohomotopy

### 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):

Review in:

### Rational equivariant Cohomotopy

Discussion of rational equivariant Cohomotopy:

### Rational twisted Cohomotopy

Discussion of rational twisted Cohomotopy: