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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

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

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

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

Since the inclusion on the right is manifestly a relative Sullivan algebra, hence a cofibration in the projective model structure on dgc-algebras, and since the latter is left proper (see there), the 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 component of the differential.

###### Remark

While Prop. says that the rational Whitehead brackets of a loop space all vanish (as generally for any H-space), the ordinary homology of a loop space inherits another product, namely the Pontrjagin product, and this makes (see there) the homology form the universal enveloping algebra of the binary Whitehead bracket super Lie algebra of the original space.

## 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

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

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).

###### Example

(Sullivan model of cyclic loop space of EM-space)
For $n \geq 1$ consider the Eilenberg-MacLane space $X \,=\, B^{n+1} \mathbb{Q}$, whose Sullivan model of a classifying space is

$CE\big( \mathfrak{l} B^{n+1} \mathbb{Q} \big) \;\; = \;\; \mathbb{Q}[c_{n+1}] \big/ \big( \mathrm{d} \, c_{n+1} \;=\; 0 \big) \,.$

Notice – from this Prop at free loop space of classifying space – that its free loop space is the product

$\mathcal{L} \; B^{n+1} \mathbb{Q} \;\simeq\; B^{n} \mathbb{Q} \,\times\, B^{n+1} \mathbb{Q} \,.$

Now Prop. shows that the corresponding cyclic loop space is as in the middle item here:

$\begin{array}{ccc} B^n \mathbb{Q} &\longrightarrow& \Big( \overset{ B^{n} \mathbb{Q} \,\times\, B^{n+1} \mathbb{Q} }{ \overbrace{ \mathcal{L} \;B^{n+1} \mathbb{Q} } } \Big) \sslash S^1 &\longrightarrow& B^n \mathbb{Q} \\ \left( \begin{array}{lcl} d & c_{n} & = 0 \end{array} \right) & \longleftarrow & \left( \begin{array}{lcl} d & c_{n+1} & = \omega_2 \wedge c_n \\ d & c_{n} & = 0 \\ d & \omega_2 & = 0 \end{array} \right) & \longleftarrow & \left( \begin{array}{lcl} d & c_{n} & = 0 \end{array} \right) \end{array}$

Incidentally, as indicated by the full diagram, this readily shows that $\big(\mathcal{L}\, B^{n+1} \mathbb{Q}\big) \sslash S^1$ retracts onto $B^{n} \mathbb{Q}$.

### 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.)

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