nLab
rational Cohomotopy

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Rational homotopy theory

Contents

Idea

Rational Cohomotopy theory π \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 nS^n_{\mathbb{Q}}:

π n()=π 0Maps(,S n) π 0Maps(,S n). \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 nn \in \mathbb{N} be a natural number and f:S nS nf\colon S^n \to S^n a continuous function from the n-sphere to itself. Then the connected component Maps f(S n,S n)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(S n,S n) {S n×S n1 | neven,deg(f)=0 S 2n1 | neven,deg(f)0 S n | nodd 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)deg(f) is the degree of ff.

Moreover, under the canonical morphism expressing the canonical action of the special orthogonal group SO(n+1)SO(n+1) on S n=S( n+1)S^n = S\big( \mathbb{R}^{n+1}\big) (regarded as the unit sphere in (n+1)(n+1)-dimensional Cartesian space) we have that on ordinary homology

H (SO(n+1)) H (Maps f=id(S n,S n)) \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 {H 2n+1(SO(n+1),) | neven H n(SO(n+1),) | nodd\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

1D<n=2k 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 Ω DS n\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 nDe_{n-D} of degree nDn - D and one generator a 2nD1a_{2n - D - 1} of degree 2nD12n - D - 1:

H (Ω DS n,)[ω nD,ω 2n1D]. 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 Ω DS D\Omega^D S^D the above formula does not apply, since Ω D1S D\Omega^{D-1} S^D is not simply connected (its fundamental group is π 1(Ω D1S D)=π 0(Ω DS D)=π D(S D)=\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 Ω DS D\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:

Ω DS DMaps */(S D,S D) fib(ev *) Maps(S D,S D) ev * S D \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 𝕂h_{\mathbb{K}} denotes the Hopf fibration of the division algebra 𝕂\mathbb{K}, hence h h_{\mathbb{C}} denotes the complex Hopf fibration and h 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(S 4)=(dg 4 =0 dg 7 =12g 4g 4). 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 π 4(S 1) \pi^4\big( S^1\big)_{\mathbb{Q}}

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

CE(𝔩Maps(S 1,S 4))=(dh 3 =0 dω 4 =0 dω 6 =h 3ω 4 dh 7 =12ω 4ω 4 ) 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 π 4(S 3) \pi^4\big( S^3\big)_{\mathbb{Q}}

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

CE(𝔩Maps(S 3,S 4))=(db 1 =0 dω 4 =0 dv 4 =ω 4b 1 dω 7 =12ω 4ω 4) 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.

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

See also

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):

Review in:

Rational equivariant Cohomotopy

Discussion of rational equivariant Cohomotopy:

Rational twisted 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.