We describe a collection of higher homotopy operations which determine the rational homotopy type of a simply-connected space X. These are described in terms of simplicial resolutions of successive approximations (L^k,α} to the Quillen DGL model for X. The operations lie in suitable cohomology groups H^*{L^{(k,α)}; π_* X\otimes Q} of these DGLs. To facilitate the recovery of an integral version of the operations from the rational description, we also define a differential graded non-associative algebra model for rational spaces.
In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type
This survey contains the main results in rational homotopy, from the beginning to the most recent ones. It makes the status of the art, gives a short presentation of some areas where rational homotopy has been used, and contains a lot of important open problems
In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.
We generalize theorems of McGibbon-Roitberg, Iriye, and Meier on the relations between phantom maps and rational homotopy, and apply them to provide new calculational examples of the homotopy sets Ph(X, Y ) of phantom maps and the subsets SPh(X, Y ) of special phantom maps.
We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In particular, this gives explicit formulas for the rational homotopy groups of all classical compact symmetric spaces.
We calculate the higher homotopy groups of the Deligne-Getzler infinity-groupoid associated to a nilpotent L-infinity algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.
In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and Grothendieck-Teichmüller groups". In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite…
The notion of a cyclic map g: A -> X is a natural generalization of a Gottlieb element in pi_n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups.
We calculate the rational homotopy and the K(1)-local homotopy of the K(2)-local sphere at the prime 3 and level 2. We use this to verify the chromatic splitting conjecture in this case.
Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X,X;1)->X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gamma_ev: S_ev -> X where S_ev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gamma_ev on rational ho…
We construct Lie algebras of derivations (and identify their geometrical realization) whose Maurer-Cartan sets provide moduli spaces describing the classes of homotopy types of rational spaces sharing either the same homotopy Lie algebra, homology or cohomology.
Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of the identity component of Aut(p) as the homology of a certain DG Lie algebra of derivations arising from the Koszul-Sullivan model of p. We obtain related identifications for the rational homotopy groups of fibrewise mapping spaces and for the rationalization of a natural nilpotent subgroup of Aut(p).
These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings.
We generalize some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $\lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.
For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex $X_{n}$ such that $\mathcal{E}(X_{n})\cong \underset{2^{n+1}\mathrm{. times}}{\underbrace{\Bbb Z_{2}\oplus... \Bbb \oplus \Bbb Z_{2}}}$.
We develop an obstruction theory for homotopy of homomorphisms f,g : M -> N between minimal differential graded algebras. We assume that M = Lambda V has an obstruction decomposition given by V = V_0 oplus V_1 and that f and g are homotopic on Lambda V_0. An obstruction is then obtained as a vector space homomorphism V_1 -> H^*(N). We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes [M, N]. This enables us to give a fairly complete answer to a…
Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of connected and pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps $π_1$-rational homotopy equivalences. In this paper, we compare these two notions and show that $π_1$-rational homotop…
In this paper, we focus on some models in rational homotopy theory, Sullivan model, Quillen model, C_\infty model, and L_\infty model. We give some connections between them. As an application, we prove the Torus Rank Conjecture.
We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by "generalized linking/intersection invariants" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogene…
It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (See Theorem 21.3 of \cite{Fuchs:abelian-group}). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say $\barρ : [S_{\Q}^{n},X] \to H_n(X;\Z)$ and generalized Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, $T$-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.
Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Felix-Oprea-Tanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give a criterion of representability of classes by submanifolds, generalising results of Papadima. Along the way, we study issues of formality and give formulas for Massey products of Thom spaces. Las…
In this paper we study the rational homotopy of the space of immersions, $Imm\left(M,N\right)$, of a manifold $M$ of dimension $m\geq 0$ into a manifold $N$ of dimension $m+k$, with $k\geq 2$. In the special case when $N=\mathbb{R}^{m+k}$ and $k$ is odd we prove that each connected component of $Imm\left(M,\mathbb{R}^{m+k}\right)$ has the rational homotopy type of product of Eilenberg Mac Lane space. We give an explicit description of each connected component and prove that it only depends on $m$, $k$ and the rational Betti numbers of $M$. For a more general manifold $N$, we prove that the pat…
Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.
We define a simplicial enrichment on the category of differential graded Hopf cooperads (the category of dg Hopf cooperads for short). We prove that our simplicial enrichment satisfies, in part, the axioms of a simplicial model category structure on the category of dg Hopf cooperads. We use this simplicial model structure to define a model of mapping spaces in the category of dg Hopf cooperads and to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. The rational homotopy theory of operads implies that the…
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.
We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher …
Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and sho…
This is a (slightly edited) version of the PhD dissertation of the author, submitted to Brown University in July 2005. We construct a homotopy calculus of functors in the sense of Goodwillie for the categories of rational homotopy theory. More precisely, given a homotopy functor between any of the categories of differential graded vector spaces (DG), reduced differential graded vector spaces, differential graded Lie algebras (DGL), and differential graded coalgebras (DGC), we show that there is an associated approximating "rational Taylor tower" of excisive functors. The fibers in this tower a…
In this paper we describe explicit $L_\infty$ algebras modeling the rational homotopy type of any component of the spaces $\map(X,Y)$ and $\map^*(X,Y)$ of free and pointed maps between the finite nilpotent CW-complex $X$ and the finite type nilpotent CW-complex $Y$. When $X$ is of finite type, non necessarily finite, we also show that the algebraic covers of these $L_\infty$ algebras model the universal covers of the corresponding mapping spaces.
By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $C$ is a coalgebra model of a space $X$, then the $A_\infty$-coalgebra structure in $H_*(X;\mathbb{Q})\cong H_*(C)$ induced by the higher Massey coproducts provides the construction of the Quillen minimal model of $X$. We also describe an explicit $L_\infty$-structure on the complex of linear maps ${\rm Hom}(H_*(X; \mathbb{Q}), π_*(ΩY)\otimes\mathbb{Q})$, where $X$ is a finite nilpotent CW-complex and $Y$ is a nilpotent CW-complex of finite type, modeling the rational homotopy type of the mapping …
We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in ${\mathbb R}^d$, where $d\ge 4$, collapses at the $E^2$ page. The main ingredients in the proof are Sinha's cosimplicial model for the space of long knots and a coformality result for the little balls operad.
We classify thick tensor ideals of finite objects in the category of rational torus-equivariant spectra, showing that they are completely determined by geometric isotropy. This is essentially equivalent to showing that the Balmer spectrum is the set of closed subgroups under cotoral inclusion. Corresponding statements are deduced for toral spectra for general groups.
Let M be a homogeneous space admitting a left translation by a connected Lie group G. The adjoint to the action gives rise to a map from G to the monoid of self-homotopy equivalences of M.The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy. In particular, the visible degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik.
Greenlees defined an abelian category A whose derived category is equivalent to the rational S^1-equivariant stable homotopy category whose objects represent rational S^1-equivariant cohomology theories. We show that in fact the model category of differential graded objects in A (dgA) models the whole rational S^1-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S^1-equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structur…
In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra $H^*(X; \Q)$ and the rational homotopy type of $Y$. From this, we deduce the existence of H-structures on a component of the mapping space $\map(X, Y)$, assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we will show that $\map(X, Y; f)\simeq\map(X, Y; f')$ if the corresponding \emph{Maurer-Cartan elements} are connected by an algebra autom…
We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage.
We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give dg Lie algebra models for the homotopy automorphisms and the block diffeomorphisms of such manifolds. Moreover, we use these models to calculate the rational cohomology of the classifying spaces of the homotopy automorphisms and block diffeomorphisms of the manifold #^g S^d x S^d relative to an embedded disk as g tends to infinity. The answer is expressed in terms of stable cohomology of arithmetic groups and invariant Lie algebra cohomology…
In this paper an algebraic model for unbased rational homotopy theory from the perspective of curved Lie algebras is constructed. As part of this construction a model structure for the category of pseudo-compact curved Lie algebras with curved morphisms will be introduced; one which is Quillen equivalent to a certain model structure of unital commutative differential graded algebras, thus extending the known Quillen equivalence of augmented algebras and differential graded Lie algebras.
Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_\infty \colon UE \to \mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ ω\colon \mathrm{aut}_1(\mathrm{Baut}_1(X_{\mathbb Q})) \to \mathrm{Baut}_1(X_{\mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_\infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $\mathrm{Baut}_1(X_{\mathbb Q})$ as a consequence. We also prove tha…
For a simply connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalence of $X$ and let $\mathcal{E}_{\sharp}(X)$ be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group $\frac{\mathcal{E}(X)}{\mathcal{E}_{\sharp}(X)}$ can be identified with a subgroup of $Aut(π_{*}(X))$. The aim of this work is to determine this subgroup for rational spaces. We construct the Whitehead exact sequence associated with the minimal Sullivan model of $X$ which allows us to define the subgroup $\mathrm{Coh.A…
In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of arXiv:1910.14608, which are based on Quillen's rational homotopy theory. According to Sullivan, the rational homotopy type of a nilpotent space $X$ with finite Betti numbers is completely determined by a commutative differential graded algebra $A$ modelling the cup product on rational cohomology. In this article we extend this correspondence between topology and algebra to parametrised stable homotopy …
This chapter is a short introduction to Sullivan models. In particular, we find the Sullivan model of a free loop space and use it to prove the Vigué-Poirrier-Sullivan theorem on the Betti numbers of a free loop space.
We give details of models for rational torus equivariant homotopy theory based on (a) all subgroups, connected subgroups or dimensions of subgroups and (b) on pairs or general flags. We provide comparison functors and show the models are equivalent. This is used in (A) An algebraic model for rational torus equivariant spectra (with B.Shipley; the new version of 1101.2511), (B) An algebraic model for the toral part of rational equivariant G-spectra for an arbitrary compact Lie group (1501.03425) and (C) construction of torus equivariant spectra from algebraic geometric data. It also highlights …
Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the homotopy fixed point set $X^{hG}$, and the natural injection $k\colon X^G\hookrightarrow X^{hG}$. We show that if $X$ is elliptic, that is, it has finite dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, tha…
Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pi_n(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1_X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also al…
Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply-connected space $X$ there is a dg Lie algebra $Λ_X$ and a (coassociative cocommutative) dg coalgebra $C_X$ that model the rational homotopy type. In this article, we prove that the rational homotopy type of an $X$-parametrised spectrum is completely encoded by a $Λ_X$-representation and also by a $C_X$-comodule. The correspondence between rational parametrised spectra and algebraic data is obtained by means of symmetric monoidal equivalences of hom…
Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of (generalized) derivation spaces and adjoint maps. As a consequence, we obtain a unified description of the rational homotopy theory of function spaces, at the level of rational homotopy groups, in terms of derivations of Quillen models and adjoints. In particular, as a natural extension of a result of Tanre, we identify the rationalization of the evaluation …
Spaces of holomorphic maps from the Riemann sphere to various complex manifolds (holomorphic curves ) have played an important role in several area of mathematics. In a seminal paper G. Segal investigated the homotopy type of holomorphic curves on complex projective spaces and M. Guest on compact smooth toric varieties.. Recently Mostovoy and Villanueva, obtained a far reaching generalisation of these results, and in particular (for holomorphic curves) improved the stability dimension obtained by Guest. In this paper, we generalize their result to holomorphic curves, on certain non-compact smo…
Let $M$ be a simply connected closed manifold of dimension $n$. We study the rational homotopy type of the configuration space of 2 points in $M$, $F(M,2)$. When $M$ is even dimensional, we prove that the rational homotopy type of $F(M,2)$ depends only on the rational homotopy type of $M$. When the dimension of $M$ is odd, for every $x\in H^{n-2} (M, \mathbb{Q})$, we construct a commutative differential graded algebra $C(x)$. We prove that for some $x \in H^{n-2} (M; \mathbb{Q})$, $C(x)$ encodes completely the rational homotopy type of $F(M,2)$. For some class of manifolds, we show that we can…
Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.
We explain how to relate the problem of finding a mirror manifold for a Calabi-Yau manifold to the problem of characterizing the rational homotopy types of closed Kähler manifolds.
We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing a new $A_\infty$ model for simply connected rational homotopy types, and uncovering a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.
