and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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.
(Sullivan model for free loop space)
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
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
and on elements $s v$ of $s V$ by
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).
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 $X$ 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:
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
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 $X$ 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.
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
where
$\omega_2$ is in degree 2;
$d_{\mathcal{L}X/S^1}$ is defined on generators $w \in V\oplus s V$ by
Moreover, the canonical sequence of morphisms of dg-algebras
is a model for the rationalization of the homotopy fiber sequence
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).
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 $C^\bullet(X)$:
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:
(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
This is known as Jones' theorem (Jones 87)
An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.
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 $X = S^4$ be the 4-sphere. The corresponding rational n-sphere has minimal Sullivan model
with
Hence prop. gives for the rationalization of $\mathcal{L}S^4$ the model
with
and prop. gives for the rationalization of $\mathcal{L}S^4 \sslash S^1$ the model
with
(relation to twisted de Rham cohomology)
The equations (1) imply that dg-algebra homomorphisms of the form
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).
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
which is dually modeled by
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 $(A_X,d_X)$:
Then there is an isomorphism of hom-sets
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
to
where
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).
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
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:
as well as
Let $X = S^2$ be the 2-sphere. The corresponding rational n-sphere has minimal Sullivan model
with
Hence prop. gives for the rationalization of $\mathcal{L}S^2$ the model
with
and prop. gives for the rationalization of $\mathcal{L}S^2 \sslash S^1$ the model
with
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 $k$-fold iterated based loop space $\Omega^k S^n$ of the n-sphere for $k \lt n$ is
(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:
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:
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.)
Examples of Sullivan models in rational homotopy theory:
The original result is due to
Micheline Vigué-Poirrier, Dennis Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976) 633-644.
Micheline Vigué-Poirrier, Dan Burghelea, A model for cyclic homology and algebraic K-theory of 1-connected topological spaces, J. Differential Geom. Volume 22, Number 2 (1985), 243-253 (Euclid)
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:
Yves Félix, Steve Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.
Kathryn Hess, example 2.5 of Rational homotopy theory: a brief introduction (math.AT/0604626)
Yves Félix, John Oprea, Daniel Tanre, Algebraic models in geometry, Oxford graduate texts in mathematics 17 (2008)
A. Yu. Onishchenko and Th. Yu. Popelensky, Rational cohomology of the free loop space of a simply connected 4-manifold, J. Fixed Point Theory Appl. 12 (2012) 69–9 (DOI 10.1007/s11784-013-0100-0)
Luc Menichi, Sullivan models and free loop space, A short introduction to Sullivan models, with the Sullivan model of a free loop space and the detailed proof of Vigué-Sullivan theorem on the Betti numbers of free loop space. Workshop on free loop space à Strasbourg, November 2008 (scanned notes pdf)
Domenico Fiorenza, Hisham Sati, Urs Schreiber: Rational sphere valued supercocycles in M-theory and type IIA string theory, Journal of Geometry and Physics, Volume 114, Pages 91-108 (2017) (arXiv:1606.03206)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, T-Duality from super Lie n-algebra cocycles for super p-branes (arXiv:1611.06536)
General background on Hochschild homology and cyclic homology is in
John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) (pdf)
Jean-Louis Loday, Cyclic homology, Grundlehren Math.Wiss. 301, Springer (1998)
Jean-Louis Loday, Free loop space and homology (arXiv:1110.0405)
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.