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Contents

Idea

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.

Construction

For the free loop space

Proposition

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

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

where

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

• $d_{\mathcal{L}X}$ is defined on elements $v$ of $V$ by

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

and on elements $s v$ of $s V$ by

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

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

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).

Remark

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

For the based loop space

For $X$ a pointed topological space and for the circle $S^1$ regarded as pointed by any base point $\ast \to S^1$ 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:

$\Omega X \overset{fib(ev_\ast)}{\longrightarrow} \mathcal{L}X \overset{ ev_\ast }{\longrightarrow} X \,.$

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

$(\wedge^\bullet( s V ), d_{\Omega X}) \overset{ cofib\big( (ev_\ast)^\ast \big) }{\longleftarrow} (\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X}) \overset{ (ev_\ast)^\ast }{\longleftarrow} (\wedge\bullet V, d_X) \,.$

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:

Proposition

(Sullivan model for based loop space)

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

Properties

Homotopy quotient by $S^1$

Proposition

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

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

where

• $\omega_2$ is in degree 2;

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

$d_{\mathcal{L}X/S^1} w \;\coloneqq\; d_{\mathcal{L}X} w + \omega_2 \wedge s w \,.$

Moreover, the canonical sequence of morphisms of dg-algebras

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

is a model for the rationalization of the homotopy fiber sequence

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

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

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

Relation to Hochschild- and cyclic-homology

Let $X$ be a simply connected topological space.

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

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

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

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

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

If the coefficients are rational, and $X$ 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 $\mathcal{L}X$ and $\mathcal{l}X/S^1$ compute the rational Hochschild homology and cyclic homology of (the cochains on) $X$, respectively.

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

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

This is known as Jones' theorem (Jones 87)

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

Examples

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.

Example

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

$(\wedge^\bullet \langle g_4, g_7 \rangle, d)$

with

$d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,.$

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

$( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} )$

with

\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 $\mathcal{L}S^4 \sslash S^1$ the model

$( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 \sslash S^1} )$

with

(1)\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} \,.

Remark

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

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

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

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

Proposition

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

$\hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R}$

which is dually modeled by

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

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

$CE(\mathfrak{l}X) = (A_X,d_X) \,.$

Then there is an isomorphism of hom-sets

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

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

Moreover, this isomorphism takes

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

to

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

where

$\omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e$

with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ 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 $\mathfrak{g}$ is type IIA super Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d super Minkowski spacetime $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of The brane bouquet).

Proof

By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.

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

$d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4$

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

\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

\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

Example

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

$(\wedge^\bullet \langle g_3, g_2 \rangle, d)$

with

$d g_2 = 0\,,\;\;\;\; d g_3 = -\tfrac{1}{2} g_2 \wedge g_2 \,.$

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

$( \wedge^\bullet \langle \omega^A_2, \omega^B_2, h_1, h_3 \rangle , d_{\mathcal{L}S^2} )$

with

\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 $\mathcal{L}S^2 \sslash S^1$ the model

$( \wedge^\bullet \langle \omega^A_2, \omega^B_2, \omega^C_2, h_1, h_3 , d_{\mathcal{L}S^2 \sslash S^1} )$

with

\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 $n$-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:

Proposition

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

The Sullivan model of the $k$-fold iterated based loop space $\Omega^k S^n$ of the n-sphere for $k \lt n$ is

$CE\mathfrak{l} \big( \Omega^k S^n \big) \;=\; \left\{ \array{ \left( \array{ d\,\omega_{n-k} & = 0 } \right) &\vert& n \;\text{is odd} \\ \left( \array{ d\,\omega_{n-k} & = 0 \\ d\,\omega_{2n-1-k} & = 0 } \right) &\vert& n \;\text{is even} } \right. \phantom{AAAA} \text{for}\; k \lt n \,.$

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 this 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.)

References

The original result is due to

Examples:

• 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)

Review:

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.