nLab
rational model of mapping space

Contents

Contents

Idea

Given two topological spaces XX, YY one may ask for the rational homotopy type of their mapping space Maps(X,Y)Maps(X,Y). Under good conditions this is a nilpotent space of finite type and hence admits a Sullivan model from which its rationalized homotopy groups and its rational cohomology groups may be read off.

Examples

Free loop spaces

See at Sullivan model of free loop space.

Rational Cohomotopy spaces

We discuss results on the rational homotopy type of spaces of maps into an n-sphere, hence rational Cohomotopy cocycle spaces.

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 of Sullivan models in rational homotopy theory:

References

General

Discussion of Sullivan models and models via L-∞ algebra for spaces of maps:

Rational Cohomotopy cocycle spaces

Discussion of rational Cohomotopy cocycle spaces:

  • 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 the rational cohomology of iterated loop spaces of n-spheres:

Spectral sequence for rational homotopy of mapping spaces

A spectral sequence computing the rational homotopy of mapping spaces:

  • Samuel B. Smith, A based Federer spectral sequence and the rational homotopy of function spaces, Manuscripta Math (1997) 93: 59 (doi:10.1007/BF02677458)

based on

  • Herbert Federer, A Study of Function Spaces by Spectral Sequences, Transactions of the American Mathematical Society Vol. 82, No. 2 (Jul., 1956), pp. 340-361 (jstor:1993052)

Last revised on October 19, 2019 at 03:57:36. See the history of this page for a list of all contributions to it.