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


  • 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) \,.

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:


(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 component of the differential.


While Prop. says that the rational Whitehead brackets of a loop space all vanish (as generally for any H-space, see there), 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.


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


  • 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).


(Sullivan model of cyclic loop space of EM-space)
For n1n \geq 1 consider the Eilenberg-MacLane space X=B n+1X \,=\, B^{n+1} \mathbb{Q}, whose Sullivan model of a classifying space is

CE(𝔩B n+1)=[c n+1]/(dc n+1=0). 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

B n+1B n×B n+1. \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:

B n (B n+1B n×B n+1)S 1 B n (d c n =0) (d c n+1 =ω 2c n d c n =0 d ω 2 =0) (d c n =0) \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 (B n+1)S 1\big(\mathcal{L}\, B^{n+1} \mathbb{Q}\big) \sslash S^1 retracts onto B nB^{n} \mathbb{Q}.

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 December 21, 2023 at 17:13:41. See the history of this page for a list of all contributions to it.