Rational Homotopy Type Research Articles

Classification and characterization of rationally elliptic manifolds in low dimensions

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give partial results in dimensions 8 and 9.

Almost nonnegative curvature operator and cohomology rings

We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author.

Homogeneous spaces, curvature and cohomology

We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromov's Betti number theorem. Our counterexamples are homogeneous Riemannian manifolds, in contrast to previous ones. They consist of two families in dimensions 13 and 22. Both families are nonnegatively curved with an additional upper curvature bound and differ already by the ring structure of their cohomology rings with complex coefficients. The 22-dimensional examples also admit almost nonnegative curvature operator with respect to homogeneous metrics.

Realizing spaces as classifying spaces

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 C P 2 \mathbb {C} P^2 and S 4 S^4 are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.

Smooth manifolds with prescribed rational cohomology ring

The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincare duality algebra \({\mathcal {A}}\), does there exist a manifold M such that \(H^*(M;\mathbb {Q})={\mathcal {A}}\)? When \({\mathcal {A}}\) is the truncated polynomial algebra \(\mathbb {Q}[x]/\langle x^3\rangle \), we prove there exists a realizing closed smooth manifold \(M^n\) only if \(n=8(2^a+2^b)\). We also eliminate any existence between dimension 32 and 128. For \(n=32\), we show that such a realizing manifold does not admit a Spin structure, and therefore is not 2-connected. In the case that \({\mathcal {A}}=\mathbb {Q}[x]/\langle x^{m+1}\rangle , |x|=8\), we apply the rational surgery realization theorem to conclude that a rational octonionic projective space exists for m odd. Similar technique is applied to study if the Milnor \(E_8\) manifold has the rational homotopy type of a smooth manifold. The “Appendix” presents a recursive algorithm for efficiently computing the coefficients of the \({\mathcal {L}}\)-polynomials, which arise in the signature formula.

The rational homotopy type of elliptic spaces up to cohomological dimension 8

Our goal in this paper is to give a full classification of the rational homotopy type of any elliptic and simply connected space when the sum of its Betti numbers is less or equal than 8.

Homotopy transfer and rational models for 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 ${\rm Hom}(H_*(X; \mathbb{Q}), \pi_*(\Omega Y)\otimes\mathbb{Q})$, 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 space ${\rm map}(X, Y)$. As an application we give conditions on the source and target in order to detect rational $H$-space structures on the components.

Rational model of the configuration space of two points in a simply connected closed manifold

Let $M$ be a simply connected closed manifold of dimension $n$. We study the rational homotopy type of the configuration space of two 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 take $x=0$.

The rational homotopy type of the complement of the graph of a map

We prove that the rational homotopy type of the complement of the graph of a continuous map from a simply connected closed manifold to a 2-connected closed manifold of the same dimension depends only on the rational homotopy class of the map. We give a commutative differential graded algebra model (in the sense of Sullivan) of the complement of the graph and study its formality.

E1–formality of complex algebraic varieties

Let X be a smooth complex algebraic variety. Morgan [Mor78] showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term of its weight spectral sequence. In the present work we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. The result for algebraic morphisms generalizes the Formality Theorem of [DGMS75] for compact K\"ahler varieties, filling a gap in Morgan's theory concerning functoriality over the rational numbers. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams.

Polynomial invariants of Weyl groups for Kac–Moody groups

In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on $A$ is symmetrizable or not. The result was conjectured by Moody and assumed by Kac. As applications we discuss the rational homotopy types of Kac-Moody groups and their flag manifolds.

Rigid rational homotopy types

In this paper, we define a rigid rational homotopy type associated to any variety X over a perfect field k of positive characteristic. We prove comparison theorems with previous definitions in the smooth and proper, and log-smooth and proper case. Using these, we can show that if k is a finite field, then the Frobenius structure on the higher rational homotopy groups is mixed. We also define a relative rigid rational homotopy type, and use it to define a homotopy obstruction for the existence of sections

Rational analogs of projective planes

In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin numbers satisfying the Hirzebruch signature formula and a set of congruence relations, which turns out to be equivalent to finding solutions to a system of Diophantine equations.

Algebraic models of non-connected spaces and homotopy theory of [formula omitted] algebras

We develop a homotopy theory of L∞ algebras based on the Lawrence–Sullivan construction, a complete differential graded Lie algebra which, as we show, satisfies the necessary properties to become the right cylinder in this category. As a result, we obtain a general procedure to algebraically model the rational homotopy type of non-connected spaces.

Rational homotopy type of the classifying space for fibrewise self-equivalences

Let p : E → B p \colon E \to B be a fibration of simply connected CW complexes with finite base B B and fibre F F . Let a u t 1 ( p ) {\mathrm {aut}}_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p p . Let B a u t 1 ( p ) {\mathrm {Baut}}_1(p) denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for B a u t 1 ( p ) {\mathrm {Baut}}_1(p) , connecting the results of recent work by the authors and others. We use this model to give classification results for the rational homotopy types represented by B a u t 1 ( p ) {\mathrm {Baut}}_1(p) and also to obtain conditions under which the monoid a u t 1 ( p ) {\mathrm {aut}}_1(p) is a double loop-space after rationalization.

$$L_{\infty }$$ rational homotopy of mapping spaces

In this paper we describe explicit \(L_\infty \) algebras modeling the rational homotopy type of any component of the spaces \(\text{ map}(X,Y)\) and \(\text{ map}^*(X,Y)\) 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.

Classification of Rational Homotopy Type for 8-Cohomological Dimension Elliptic Spaces

The different methods used to classify rational homotopy types of manifolds are in general fascinating and various (see [1,7,8]). In this paper we are interested to a particular case, that of simply connected elliptic spaces, denoted X, by discussing its cohomological dimension. Here we will the discuss the case when dimH*( Χ ;Q)=8 and χ(Χ)=0.

Examples of Rational Toral Rank Complex

There is a CW complex 𝒯(𝑋), which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of X associated with rational toral ranks and also presents certain relations in them. We call it the rational toral rank complex of X. It represents a variety of toral actions. In this note, we will give effective 2-dimensional examples of it when X is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.

Rational homotopy type of the moduli of representations with Borel mold

In this note we study the rational homotopy types of the moduli space of representations with Borel mold for free monoid and related varieties. The moduli space has a fiber bundle structure over the configuration space in the affine space. We show that the minimal model of the moduli space with mixed Hodge structure is equivalent to the tensor product of minimal models of the configuration space and of the fiber.