Rational Homotopy Type Research Articles

A differential algebra and the homotopy type of the complement of a toric arrangement

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of discrete data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of \mathcal A .

On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds

We answer a question of Oprea–Tralle on the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four, along with a question of Lupton–Oprea in all even dimensions. This will also allow us to address, in all even dimensions six and higher, another question of Oprea–Tralle on the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold.

Massey Products for Graph Homology

Abstract This paper shows that the operad encoding modular operads is Koszul. Using this result, we construct higher composition operations on (hairy) graph homology, which characterize its rational homotopy type.

Morphisms of Rational Motivic Homotopy Types

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

English

We propose a motivic generalization of rational homotopy types. The algebraic invariants we study are defined as algebra objects in the category of mixed motives. This invariant plays a role of Sullivan's polynomial de Rham algebras. Another main notion is that of cotangent motives. Our main objective is to investigate the topological realization of these invariants and study their structures. Applying these machineries and the Tannakian theory, we construct actions of a derived motivic Galois group on rational homotopy types. Thanks to this, we deduce actions of the motivic Galois group of pro-unipotent completions of homotopy groups.

Classification of rational homotopy type for 10-cohomological dimension elliptic spaces

Classification of rational homotopy type for 10-cohomological dimension elliptic spaces

The rational cohomology Hopf algebra of a generic Kac–Moody group

In this paper we determine the rational homotopy type of the classifying space of a generic Kac-Moody group by computing its rational cohomology ring. As an application we determine the rational homology Hopf algebra of the generic Kac-Moody group.

The rational homotopy type of (n−1)‐connected manifolds of dimension up to 5n−3

We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity.

Integral and rational mapping classes

Let X and Y be finite complexes. When Y is a nilpotent space, it has a rationalization Y → Y ( 0 ) which is well understood. Early on it was found that the induced map [ X , Y ] → [ X , Y ( 0 ) ] on sets of mapping classes is finite-to-one. The sizes of the preimages need not be bounded; we show, however, that, as the complexity (in a suitable sense) of a rational mapping class increases, these sizes are at most polynomial. This “torsion” information about [ X , Y ] is in some sense orthogonal to rational homotopy theory but is nevertheless an invariant of the rational homotopy type of Y in at least some cases. The notion of complexity is geometric, and we also prove a conjecture of Gromov regarding the number of mapping classes that have Lipschitz constant at most L .

On the rational homotopy type of intersection spaces

Banagl's method of intersection spaces allows to modify certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincare duality in analogy with Goresky-MacPherson's intersection homology. In the case of one isolated singularity, we show that the duality isomorphism comes from a nondegenerate intersection pairing which depends on the choice of a chain representative of the fundamental class of the regular stratum. On the technical side, we use piecewise linear polynomial differential forms due to Sullivan to define a suitable commutative cochain algebra model for intersection spaces. Our construction parallels Banagl's commutative cochain algebra of smooth differential forms modeling intersection space cohomology, and we show that both algebras are weakly equivalent.

Rational maps from Euclidean configuration spaces to spheres

In this note we give an algorithm to determine the rational homotopy type of the free and pointed mapping spaces $\map(F(\br^m,k),\bs^n)$ and $\map^*(F(\br^m,k),\bs^n)$. An explicit description of these spaces is given for $k=3$. The general case for $n$ odd is also presented as an immediate consequence of the rational version of a classical result of Thom.

Almost formality of manifolds of low dimension

In this paper we introduce the notion of Poincare DGCAs of Hodge type, which is a subclass of Poincare DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincare DGCA of Hodge type. Using these concepts, we investigate the equivalence class of $(r-1)$ connected $(r>1)$ Poincare DGCAs of Hodge type. In particular, we show that a $(r-1)$ connected Poincare DGCA of Hodge type ${\mathcal A}^\ast$ of dimension $n \le 5r-3$ is $A_\infty$-quasi-isomorphic to an $A_3$-algebra and prove that the only obstruction to the formality of ${\mathcal A}^\ast$ is a distinguished Harrison cohomology class $[\mu_3] \in {\mathsf{Harr}}^{3,-1} (H^*({\mathcal A}^\ast), H^*({\mathcal A}^\ast))$. Moreover, the cohomology class $[\mu_3]$ and the DGCA isomorphism class of $H^*({\mathcal A}^\ast)$ determine the $A_\infty$-quasi-isomorphism class of ${\mathcal A}^\ast$. This can be seen as a Harrison cohomology version of the Crowley-Nordstrom results [D. Crowley, J. Nordstrom, The rational homotopy type of $(n-1)$-connected manifolds of dimension up to $5n-3$, arXiv:1505.04184v2] on rational homotopy type of $(r-1)$-connected $(r>1)$ closed manifolds of dimension up to $5r-3$. We also derive the almost formality of closed $G_2$-manifolds, which have been discovered recently by Chan-Karigiannis-Tsang in [K.F. Chan, S. Karigiannis and C.C. Tsang, The ${\mathcal L}_B$-cohomology on compact torsion-free ${\rm G}_2$ manifolds and an application to `almost' formality, arXiv:1801.06410, to appear in Ann. Global Anal. Geom.], from our results and the Cheeger-Gromoll splitting theorem.

Rational homotopy of mapping spaces between complex Grassmannians

The complex Grassmann Gr(k, n) is the space of k dimensional subspaces of ℂ n . It is a complex manifold of complex dimension k(n − k). There is a natural inclusion ik ,n : Gr(k, n) ↪ Gr(k, n + r). In this paper, we use Sullivan models to compute the rational homotopy type of the component of the inclusion Gr(2, n) ↪ Gr(2, n + r) in the space of mappings from Gr(2, n) to Gr(2, n + r), r ≥ 1. We show in particular that map(Gr(2, n), Gr(2, n + 1); in ) has the rational homotopy type of a product of odd spheres.

De Rham theory and cocycles of cubical sets from smooth quandles

We show a de Rham theorem for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in [8, 9] produce cocycles of quandles.

A zoo of growth functions of mapping class sets

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which can be realized by [Formula: see text]-Lipschitz maps grows asymptotically as [Formula: see text], where [Formula: see text] is an integer determined by the rational homotopy type of [Formula: see text] and the rational cohomology of [Formula: see text]. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is [Formula: see text] but the true growth is [Formula: see text]. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number [Formula: see text], there is a pair [Formula: see text] for which the growth of [Formula: see text] is essentially [Formula: see text].

The extended rational homotopy theory of operads

Abstract 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 [B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups. Part 2. The Applications of (Rational) Homotopy Theory Methods, Math. Surveys Monogr. 217, American Mathematical Society, Providence, 2017]. 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-type rational cohomology groups).

Quantitative nullhomotopy and rational homotopy type

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$. More generally, we prove a similar result for maps $f:X \to Y$ for any compact Riemannian manifold $X$ and $Y$ a compact simply connected Riemannian manifold in a class which includes complex projective spaces, Grassmannians, and all other simply connected homogeneous spaces. Moreover, for all simply connected $Y$, asymptotic restrictions on the size of nullhomotopies are determined by rational homotopy type.

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, we use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology. For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (cdga's) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse cdga's, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PL-pseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in CP(4), is intersection-formal.

Rational Models of the Complement of a Subpolyhedron in a Manifold with Boundary

AbstractLet W be a compact simply connected triangulated manifold with boundary and let K ⊂ W be a subpolyhedron. We construct an algebraic model of the rational homotopy type of W\K out of a model of the map of pairs (K, K⋂∂W) ↪ (W, ∂W) under some high codimension hypothesis.We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

Spaces Realized and Non-Realized as Dold-Lashof Classifying Spaces

Let $X$ be a simply connected CW-complex of finite type. Denote by ${\rm Baut}_{1}(X)$ the Dold-Lashof classifying space of fibrations with fiber $X$. This paper is a survey about the problem of realizing Dold-Lashof classifying spaces. We will also present some new results: we show that not all rank-two rational $H$-spaces can be realized as ${\rm Baut}_{1}(X)$ for simply connected, rational elliptic space $X$. Moreover, we construct an infinite family of rational spaces $X,$ such that ${\rm Baut}_{1}(X)$ is rationally a finite $H$-space of rank-two (up to rational homotopy type).