Sullivan model of loop space




In rational homotopy theory, given a rational topological space modeled by a Sullivan model dg-algebra, there is an explicit description of the Sullivan model of its loop space, (free loop space or based loop space).

This is a special case of Sullivan models of mapping spaces.


For the free loop space


(Sullivan model for free loop space)

Let ( V,d X)(\wedge^\bullet V, d_X) be a semifree dg-algebra being a minimal Sullivan model of a rational simply connected space XX. Then a Sullivan model for the free loop space X\mathcal{L} X is given by

( (VsV),d X), (\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X}) \,,


  • sVs V is the graded vector space obtained from VV by shifting degrees down by one: deg(sv)=deg(v)1deg(s v) = deg(v)-1;

  • d Xd_{\mathcal{L}X} is defined on elements vv of VV by

    d Xvdv d_{\mathcal{L}X} v \coloneqq d v

    and on elements svs v of sVs V by

    d Xsvs(dv), d_{\mathcal{L}X} s v \coloneqq - s ( d v ) \,,

    where on the right s:VsVs \colon V \to s V is extended as a graded derivation s: 2V (VsV)s \colon \wedge^2 V \to \wedge^\bullet (V \oplus s V) .

This is due to (Vigué-Sullivan 76). Review includes (Felix-Halperin-Thomas 00, p. 206, Hess 06, example 2.5, Félix-Oprea-Tanre 08, theorem 5.11).


The formula in prop. is akin to that that for the Weil algebra of the L L_\infty -algebra of which ( V,d X)(\wedge^\bullet V,d_X) is the Chevalley-Eilenberg algebra, except that here ss shifts down, whereas for the Weil algebra it shifts up.

For the based loop space

For XX a pointed topological space and for the circle S 1S^1 regarded as pointed by any base point *S 1\ast \to S^1 there is the following homotopy fiber sequence which exhibits the based loop space as the homotopy fiber of the evaluation map out of the free loop space:

ΩXfib(ev *)Xev *X. \Omega X \overset{fib(ev_\ast)}{\longrightarrow} \mathcal{L}X \overset{ ev_\ast }{\longrightarrow} X \,.

With the dgc-algebra model from Prop. for X\mathcal{L}X it follows that the dgc-algebra model for the based loop space is the homotopy cofiber dgc-algebra ( (sV),d ΩX)(\wedge^\bullet( s V ), d_{\Omega X}) in

( (sV),d ΩX)cofib((ev *) *)( (VsV),d X)(ev *) *(V,d X). (\wedge^\bullet( s V ), d_{\Omega X}) \overset{ cofib\big( (ev_\ast)^\ast \big) }{\longleftarrow} (\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X}) \overset{ (ev_\ast)^\ast }{\longleftarrow} (\wedge\bullet V, d_X) \,.

Thus the inclusion on the right is manifestly a relative Sullivan algebra, its homotopy cofiber is represented by the ordinary cofiber, which is readily read off:


(Sullivan model for based loop space)

For XX a connected and simply connected topological space with Sullivan model (V,d X)(\wedge\bullet V, d_X), the Sullivan model ( (sV),d ΩX)(\wedge^\bullet( s V ), d_{\Omega X}) of its based loop space ΩX\Omega X is the dgc-algebra obtained from (V,d X)(\wedge\bullet V, d_X) by shifting down all generators in degree by 1, and by keeping only the co-unary componend of the differential.


Homotopy quotient by S 1S^1


Given a Sullivan model ( (VsV),d X)(\wedge^\bullet (V \oplus s V), d_{\mathcal{L}X}) for a free loop space as in prop. , then a Sullivan model for the cyclic loop space, i.e. for the homotopy quotient XS 1\mathcal{L} X \sslash S^1 with respect to the canonical circle group action that rotates loops (i.e. for the Borel construction X× S 1ES 1\mathcal{L}X \times_{S^1} E S^1) is given by

( (VsVω 2),d X/S 1) (\wedge^\bullet( V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1})


  • ω 2\omega_2 is in degree 2;

  • d X/S 1d_{\mathcal{L}X/S^1} is defined on generators wVsVw \in V\oplus s V by

    d X/S 1wd Xw+ω 2sw. d_{\mathcal{L}X/S^1} w \;\coloneqq\; d_{\mathcal{L}X} w + \omega_2 \wedge s w \,.

Moreover, the canonical sequence of morphisms of dg-algebras

(ω 2,d=0)( (VsVω 2),d X/S 1)( (VsV),d X) (\wedge \omega_2, d = 0) \longrightarrow (\wedge^\bullet( V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1}) \longrightarrow (\wedge^\bullet( V\oplus s V ), d_{\mathcal{L}X})

is a model for the rationalization of the homotopy fiber sequence

XXS 1BS 1 \mathcal{L}X \longrightarrow \mathcal{L}X \sslash S^1 \longrightarrow B S^1

which exhibits the infinity-action (by the discussion there) of S 1S^1 on X\mathcal{L}X.

This is due to (Vigué-Burghelea 85, theorem A).

Relation to Hochschild- and cyclic-homology

Let XX be a simply connected topological space.

The ordinary cohomology H H^\bullet of its free loop space is the Hochschild homology HH HH_\bullet of its singular chains C (X)C^\bullet(X):

H (X)HH (C (X)). H^\bullet(\mathcal{L}X) \simeq HH_\bullet( C^\bullet(X) ) \,.

Moreover the S 1S^1-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space XS 1\mathcal{L}X \sslash S^1 is the cyclic homology HC HC_\bullet of the singular chains:

H (XS 1)HC (C (X)) H^\bullet(\mathcal{L}X \sslash S^1) \simeq HC_\bullet( C^\bullet(X) )

(Jones 87, Thm. A, review in Loday 92, Cor. 7.3.14, Loday 11, Sec 4)

If the coefficients are rational, and XX is of finite type then by prop. and prop. , and the general statements at rational homotopy theory, the cochain cohomology of the above minimal Sullivan models for X\mathcal{L}X and 𝓁X/S 1\mathcal{l}X/S^1 compute the rational Hochschild homology and cyclic homology of (the cochains on) XX, respectively.

In the special case that the topological space XX carries the structure of a smooth manifold, then the singular cochains on XX are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

H (X)HH (Ω (X)). H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.
H (XS 1)HC (Ω (X)). H^\bullet(\mathcal{L}X \sslash S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,.

This is known as Jones' theorem (Jones 87)

An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.


Free loop space of the 4-sphere

We discuss the Sullivan model for the free and cyclic loop space of the 4-sphere. This may also be thought of as the cocycle space for rational 4-Cohomotopy, see FSS16, Section 3.


Let X=S 4X = S^4 be the 4-sphere. The corresponding rational n-sphere has minimal Sullivan model

( g 4,g 7,d) (\wedge^\bullet \langle g_4, g_7 \rangle, d)


dg 4=0,dg 7=12g 4g 4. d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,.

Hence prop. gives for the rationalization of S 4\mathcal{L}S^4 the model

( ω 4,ω 6,h 3,h 7,d S 4) ( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} )


d S 4h 3 =0 d S 4ω 4 =0 d S 4ω 6 =h 3ω 4 d S 4h 7 =12ω 4ω 4 \begin{aligned} d_{\mathcal{L}S^4} h_3 & = 0 \\ d_{\mathcal{L}S^4} \omega_4 & = 0 \\ d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 \\ \end{aligned}

and prop. gives for the rationalization of S 4S 1\mathcal{L}S^4 \sslash S^1 the model

( ω 2,ω 4,ω 6,h 3,h 7,d S 4S 1) ( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 \sslash S^1} )


(1)d S 4S 1h 3 =0 d S 4S 1ω 2 =0 d S 4S 1ω 4 =h 3ω 2 d S 4S 1ω 6 =h 3ω 4 d S 4S 1h 7 =12ω 4ω 4+ω 2ω 6. \begin{aligned} d_{\mathcal{L}S^4 \sslash S^1} h_3 & = 0 \\ d_{\mathcal{L}S^4 \sslash S^1} \omega_2 & = 0 \\ d_{\mathcal{L}S^4 \sslash S^1} \omega_4 & = h_3 \wedge \omega_2 \\ d_{\mathcal{L}S^4 \sslash S^1} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4 \sslash S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6 \end{aligned} \,.


(relation to twisted de Rham cohomology)
The equations (1) imply that dg-algebra homomorphisms of the form

CE(𝔩((S 4)S 1))AAΩ dR (X f) CE \Big( \mathfrak{l} \big( (\mathcal{L}S^4) \sslash S^1 \big) \Big) \xrightarrow{\;\;AA\;\;} \Omega^\bullet_{dR}(X^f)

into the de Rham dg-algebra of a smooth manifold X fX^f of dimension 7\leq 7 are equivalently cocycles in the degree-3 twisted de Rham complex of X 7X^7 (together with any 7-form, if dim=7dim = 7), for 3-twist given by the image of the general h 3h_3.

This suggests a relation between the cyclification of S 4S^4 to the twisted Chern character on twisted K-theory (a relation further explored in BMSS 2019).


Let 𝔤^𝔤\hat \mathfrak{g} \to \mathfrak{g} be a central Lie algebra extension by \mathbb{R} of a finite dimensional Lie algebra 𝔤\mathfrak{g}, and let 𝔤b\mathfrak{g} \longrightarrow b \mathbb{R} be the corresponding L-∞ 2-cocycle with coefficients in the line Lie 2-algebra bb \mathbb{R}, hence (FSS 13, prop. 3.5) so that there is a homotopy fiber sequence of L-∞ algebras

𝔤^𝔤ω 2b \hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R}

which is dually modeled by

CE(𝔤^)=( (𝔤 *e),d 𝔤^| 𝔤 *=d 𝔤,d 𝔤^e=ω 2). CE(\hat \mathfrak{g}) = ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\; d_{\hat \mathfrak{g}} e = \omega_2) \,.

For XX a space with Sullivan model (A X,d X)(A_X,d_X) write 𝔩(X)\mathfrak{l}(X) for the corresponding L-∞ algebra, i.e. for the L L_\infty-algebra whose Chevalley-Eilenberg algebra is (A X,d X)(A_X,d_X):

CE(𝔩X)=(A X,d X). CE(\mathfrak{l}X) = (A_X,d_X) \,.

Then there is an isomorphism of hom-sets

Hom L Alg(𝔤^,𝔩(S 4))Hom L Alg/b(𝔤,𝔩(S 4/S 1)), Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) ) \;\simeq\; Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) ) \,,

with 𝔩(S 4)\mathfrak{l}(S^4) from prop. and 𝔩(S 4S 1)\mathfrak{l}(\mathcal{L}S^4 \sslash S^1) from prop. , where on the right we have homs in the slice over the line Lie 2-algebra, via prop. .

Moreover, this isomorphism takes

𝔤^(g 4,g 7)𝔩(S 4) \hat \mathfrak{g} \overset{(g_4, g_7)}{\longrightarrow} \mathfrak{l}(S^4)


𝔤 (ω 2,ω 4,ω 6,h 3,h 7) 𝔩(X/S 1) ω 2 ω 2 b, \array{ \mathfrak{g} && \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow} && \mathfrak{l}( \mathcal{L}X / S^1 ) \\ & {}_{\mathllap{\omega_2}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && b \mathbb{R} } \,,


ω 4=g 4h 3e,h 7=g 7+ω 6e \omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e

with ee being the central generator in CE(𝔤^)CE(\hat \mathfrak{g}) from above, and where the equations take place in 𝔤^ *\wedge^\bullet \hat \mathfrak{g}^\ast with the defining inclusion 𝔤 * 𝔤 *\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast understood.

This is observed in (Fiorenza-Sati-Schreiber 16, FSS 16b), where it serves to formalize, on the level of rational homotopy theory, the double dimensional reduction of M-branes in M-theory to D-branes in type IIA string theory (for the case that 𝔤\mathfrak{g} is type IIA super Minkowski spacetime 9,1|16+16¯\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} and 𝔤^\hat \mathfrak{g} is 11d super Minkowski spacetime 10,1|32\mathbb{R}^{10,1\vert \mathbf{32}}, and the cocycles are those of The brane bouquet).


By the fact that the underlying graded algebras are free, and since ee is a generator of odd degree, the given decomposition for ω 4\omega_4 and h 7h_7 is unique.

Hence it is sufficient to observe that under this decomposition the defining equations

dg 4=0,dg 7=12g 4g 4 d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4

for the 𝔩S 4\mathfrak{l}S^4-valued cocycle on 𝔤^\hat \mathfrak{g} turn into the equations for a 𝔩(S 4/S 1)\mathfrak{l} ( \mathcal{L}S^4 / S^1 )-valued cocycle on 𝔤\mathfrak{g}. This is straightforward:

d 𝔤^(ω 4+h 3e)=0 d 𝔤(ω 4h 3ω 2)=0andd 𝔤h 3=0 d 𝔤ω 4=h 3ω 2andd 𝔤h 3=0 \begin{aligned} & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \end{aligned}

as well as

d 𝔤^(h 7ω 6e)=12(ω 4+h 3e)(ω 4+h 3e) d 𝔤h 7ω 6ω 2=12ω 4ω 4andd 𝔤ω 6=h 3ω 4 d 𝔤h 7=12ω 4ω 4+ω 6ω 2andd 𝔤h 6=h 3ω 4 \begin{aligned} & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e ) = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e) \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 \;\;\; and \;\;\; - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4 \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 + \omega_6 \wedge \omega_2 \;\;\; and \;\;\; d_\mathfrak{g} h_6 = h_3 \wedge \omega_4 \end{aligned}

Free loop space of the 2-sphere


Let X=S 2X = S^2 be the 2-sphere. The corresponding rational n-sphere has minimal Sullivan model

( g 3,g 2,d) (\wedge^\bullet \langle g_3, g_2 \rangle, d)


dg 2=0,dg 3=12g 2g 2. d g_2 = 0\,,\;\;\;\; d g_3 = -\tfrac{1}{2} g_2 \wedge g_2 \,.

Hence prop. gives for the rationalization of S 2\mathcal{L}S^2 the model

( ω 2 A,ω 2 B,h 1,h 3,d S 2) ( \wedge^\bullet \langle \omega^A_2, \omega^B_2, h_1, h_3 \rangle , d_{\mathcal{L}S^2} )


d S 2h 1 =0 d S 2ω 2 A =0 d S 2ω 2 B =h 1ω 2 A d S 2h 3 =12ω 2 Aω 2 A \begin{aligned} d_{\mathcal{L}S^2} h_1 & = 0 \\ d_{\mathcal{L}S^2} \omega^A_2 & = 0 \\ d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A \\ d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2 \end{aligned}

and prop. gives for the rationalization of S 2S 1\mathcal{L}S^2 \sslash S^1 the model

( ω 2 A,ω 2 B,ω 2 C,h 1,h 3,d S 2S 1) ( \wedge^\bullet \langle \omega^A_2, \omega^B_2, \omega^C_2, h_1, h_3 , d_{\mathcal{L}S^2 \sslash S^1} )


d S 2h 1 =0 d S 2ω 2 A =ω 2 Ch 1 d S 2ω 2 B =h 1ω 2 A d S 2ω 2 C =0 d S 2h 3 =12ω 2 Aω 2 A+ω 2 Cω 2 B. \begin{aligned} d_{\mathcal{L}S^2} h_1 & = 0 \\ d_{\mathcal{L}S^2} \omega^A_2 & = \omega^C_2 \wedge h_1 \\ d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A \\ d_{\mathcal{L}S^2} \omega^C_2 & = 0 \\ d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2 + \omega^C_2 \wedge \omega^B_2 \end{aligned} \,.

Iterated based loop spaces of nn-spheres

By iterating the Sullivan model construction for the based loop space from Prop. and using the Sullivan models of n-spheres we have that:


(Sullivan models for iterated loop spaces of n-spheres)

The Sullivan model of the kk-fold iterated based loop space Ω kS n\Omega^k S^n of the n-sphere for k<nk \lt n is

CE𝔩(Ω kS n)={(dω nk =0) | nis odd (dω nk =0 dω 2n1k =0) | nis evenAAAAfork<n. CE\mathfrak{l} \big( \Omega^k S^n \big) \;=\; \left\{ \array{ \left( \array{ d\,\omega_{n-k} & = 0 } \right) &\vert& n \;\text{is odd} \\ \left( \array{ d\,\omega_{n-k} & = 0 \\ d\,\omega_{2n-1-k} & = 0 } \right) &\vert& n \;\text{is even} } \right. \phantom{AAAA} \text{for}\; k \lt n \,.

(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).



The rational model for Ω DS D\Omega^D S^D follows from this 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:


The original result is due to


  • Bitjong Ndombol & M. El Haouari, The free loop space equivariant cohomology algebra of some formal spaces, Mathematische Zeitschrift 266 (2010) 863–875 (doi:10.1007/s00209-009-0602-z)

  • Kentaro Matsuo, The Borel cohomology of the loop space of a homogeneous space, Topology and its Applications 160 12 (2013) 1313-1332 (doi:10.1016/j.topol.2013.05.001)


General background on Hochschild homology and cyclic homology is in

The case of iterated based loop spaces of n-spheres is discussed also in

Last revised on August 20, 2021 at 13:05:05. See the history of this page for a list of all contributions to it.