nLab rational model of mapping space

Idea

Given two topological spaces $X$, $Y$ one may ask for the rational homotopy type of their mapping space $Maps(X,Y)$. 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.

General formula

Proposition

For $X, \mathcal{A}$ a pair of connected nilpotent topological spaces of rational finite type, the mapping space from $X$ into the rationalization $L^{\mathbb{Q}} \mathcal{A}$ has a (simplicial) weak homotopy equivalence

$$Map\big( X, L^{\mathbb{Q}}\mathcal{A} \big) \,\simeq\, \Omega^1_{cl}\big( X \times \Delta^\bullet; \mathfrak{l}\mathcal{A} \big) \mathrlap{\,,}$$

where we denote by

• $\mathfrak{l}(-)$ the rational Whitehead $L_\infty$-algebra,

• $\Omega^1_{cl}(-;\mathfrak{g}) \coloneqq Hom_{dgcAlg}\big(CE(\mathfrak{g}), \Omega^\bullet_{PL}(-)\big)$ the set of closed $L_\infty$-algebra valued differential forms, where

  • $\Omega^\bullet_{PL}\big(X \times \Delta^n\big)$ is the polynomial de Rham complex of the product space of $X$ with the n-simplex,

  • $CE(-)$ the Chevalley-Eilenberg algebra, so that $CE\big(\mathfrak{l}(-)\big)$ gives the minimal Sullivan model,

and we understand the topological space on the left as its homotopy type incarnated by its singular simplicial complex.

(This is, in paraphrase, Berglund 2015 Thm. 1.4.)

Over the real numbers and in the case that $X$ is a smooth manifold, there is a quick re-proof of Prop. using cohesive homotopy theory:

Proposition

For

• $X$ a smooth manifold,

• $\mathcal{A}$ a connected nilpotent topological space of rational finite type,

there is an equivalence

$$\mathbf{Map}\big( X, L^{\mathbb{R}} \mathcal{A} \big) \simeq \esh \mathbf{\Omega}^1_{cl}\big(X; \mathfrak{l}\mathcal{A}\big) \,,$$

of (geometrically discrete) smooth $\infty$-groupoids, where we denote by

• $\mathbf{Map}(-,-)$ the mapping stack (internal hom),

• $L^{\mathbb{R}}(-)$ the $\mathbb{R}$-rationalization (rationalization followed by derived extension of scalars),

• $\mathbf{\Omega}^1_{cl}\big(X; \mathfrak{g}\big) \,\colon\, U \mapsto Hom_{dgcAlg}\big( CE(\mathfrak{g}), \Omega^\bullet_{dR}(U \times X) \big)$ the moduli smooth set of closed $L_\infty$-algebra valued differential forms,

• $\esh$ the shape modality.

Proof

This follows readily as:

$$\begin{aligned} \esh \mathbf{\Omega}^1_{cl}\big( X; \mathfrak{l}\mathcal{A} \big) & \simeq \esh \mathbf{Map}\Big( X, \mathbf{\Omega}^1_{cl}\big( \ast; \mathfrak{l}\mathcal{A} \big) \Big) \\ & \simeq \mathbf{Map}\Big( X, \esh \mathbf{\Omega}^1_{cl}\big( \ast; \mathfrak{l}\mathcal{A} \big) \Big) \\ & = \mathbf{Map}\Big( X, L^{\mathbb{R}} \mathcal{A} \Big) \mathrlap{\,,} \end{aligned}$$

where we used, in order of appearance:

1. the formula for the plots of mapping stacks (cf. here),

2. the smooth Oka principle to pass the shape modality into the mapping stack,

3. the formula for shape via cohesive path ∞-groupoids to identify

   $$\esh \mathbf{\Omega}^1_{cl}(\ast; \mathfrak{g}) \simeq Dsc\, \Omega^1_{cl}\big(\mathbf{\Delta}^\bullet; \mathfrak{g}\big) \mathrlap{\,,}$$

   together with the fundamental theorem of dg-algebraic rational homotopy theory:

   $$L^{\mathbb{R}}\mathcal{A} \simeq \Omega^1_{cl}\big(\mathbf{\Delta}^\bullet; \mathfrak{l}\mathcal{A}\big) \mathrlap{\,,}$$

   here implemented not with piecewise linear but with smooth differential forms on smooth manifolds and extended simplices $\mathbf{\Delta}^n$ (which still satisfy the relevant extension lemma for differential forms, see there).

The global model of Prop. induces analog models for each of the connected components of the mapping space:

Proposition

Consider:

• $X, \mathcal{A}$ a pair of connected nilpotent CW-complexes of rational finite type,

  with $X$ a finite CW complex or $\mathcal{A}$ having a finite Postnikov tower,

• $\phi \colon X \to \mathcal{A}$ a continuous map.

Then a dg-Lie algebra model for the rational homotopy type of the mapping space $Map\big(X; \mathcal{A}\big)$ in the connected component of $\phi$ is given by the twisted tensor product

$$\big( \Omega^\bullet_{PL}(X) \textstyle{ \otimes_{\mathbb{Q}} } \mathfrak{l} \mathcal{A} \big)^{\xi_\phi} \langle 0 \rangle$$

where :

• $\Omega^\bullet_{PL}(-)$ denotes the PL de Rham dgc-algebra,

• $\mathfrak{l}(-)$ denotes the dg-Lie algebra model.

• $\xi_\phi \in MC\big( \Omega^\bullet_{PL}(X) \otimes \mathfrak{l}\mathcal{A}\big)$ is the Maurer-Cartan element corresponding to the pullback of differential forms

  $$\phi^\ast \,\colon\, CE(\mathfrak{l}\mathcal{A}) \longrightarrow \Omega^\bullet_{PL}(X) \mathrlap{\,,}$$

• $(-)^{\xi_\phi}$ means that the differential of the tensor dg-Lie algebra is twisted as

  $$d^{\mathfrak{l}\phi}(-) \coloneqq \mathrm{d}(-) + \big[\xi_\phi, -\big] \mathrlap{\,,}$$

• $(-)\langle-\rangle$ denotes the connective cover (discarding negative degrees and restricting to cycles in degree=0).

(Lazarev 2013 Thm. 8.1, cf. Berglund 2015 Thm. 1.5, Buijs, Felix & Murillo 2012 Thm. 3.1)

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 $n \in \mathbb{N}$ 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.)

Related concepts

Examples of Sullivan models in rational homotopy theory:

• Sullivan model of an n-sphere

• Sullivan model of a spherical fibration

• Sullivan model of complex projective space

• Sullivan model of a classifying space

• Sullivan model of mapping space

• Sullivan model of free loop space

• Sullivan model of a suspension

• Sullivan model of a finite G-quotient

References

General

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

• Samuel Bruce Smith, Rational evaluation subgroups, Math Z (1996) 221: 387 (doi:10.1007/BF02622121)

• Edgar H. Brown, Jr., Robert H. Szczarba: On the Rational Homotopy Type of Function Spaces, Transactions of the American Mathematical Society 349 12 (1997) 4931-4951 [jstor:2155493]

• Gregory Lupton, Samuel Bruce Smith, Rationalized Evaluation Subgroups of a Map and the Rationalized G-Sequence (arXiv:math/0309432)

• Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)

• Urtzi Buijs, Aniceto Murillo, Basic constructions in rational homotopy theory of function spaces, Annales de l’Institut Fourier, Volume 56 (2006) no. 3, p. 815-838 (doi:10.5802/aif.2201)

• Micheline Vigué-Poirrier, Rational formality of function spaces, Journal of Homotopy and Related Structures 2.1 (2007): 99-108 (arXiv:0706.2977)

• Gregory Lupton, Samuel Bruce Smithm, Rationalized evaluation subgroups of a map I: Sullivan models, derivations and $G$-sequences, Journal of Pure and Applied Algebra, Volume 209, Issue 1, April 2007, Pages 159-171 (doi:10.1016/j.jpaa.2006.05.018)

• Urtzi Buijs, Aniceto Murillo, The rational homotopy Lie algebra of function spaces, Comment. Math. Helv. 83 (2008), 723–739 (pdf)

• Gregory Lupton, Samuel Bruce Smith, Whitehead products in function spaces: Quillen model formulae, J. Math. Soc. Japan, Volume 62, Number 1 (2010), 49-81. (arXiv:0812.1829, euclid:jmsj/1265380424)

• Andrey Lazarev: Maurer-Cartan moduli and models for function spaces, Advances in Mathematics 235 (2013) 296–320 [arxiv:1109.3715 math.AT, doi:10.1016/j.aim.2012.11.009]

• J.-B. Gatsinzi, A model for function spaces, Topology and its Applications, Volume 168, 15 May 2014, Pages 153-158 (doi:10.1016/j.topol.2014.02.021)

• 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)

• Alexander Berglund: Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras, Homology, Homotopy and Applications 17 2 (2015) [arXiv:1110.6145 math.AT, doi:10.4310/HHA.2015.v17.n2.a16]

• Urtzi Buijs, Yves Félix, Aniceto Murillo: $L_\infty$-rational homotopy of mapping spaces, published as: $L_\infty$-models of based mapping spaces, J. Math. Soc. Japan 63 2 (2011) 503–524 [arXiv:1209.4756 math.AT, doi:10.2969/jmsj/06320503]

• Hisham Sati, Alexander A. Voronov: Section 2 of: Mysterious Triality and the Exceptional Symmetry of Loop Spaces [arXiv:2408.13337]

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:

• Sadok Kallel, Denis Sjerve, On Brace Products and the Structure of Fibrations with Section, 1999 (pdf, pdf)

Also

• Sati & Voronov 2024, Section 2

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)