In this paper, using Sullivan's approach to rational homotopy theory of simply connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X; \mathbb{Q})}(\mathbb{Q}, C^{\ast}(X; \mathbb{Q}))$ with a graded commutative algebra structure. Thus, we introduce new algebraic invariants referred to as the $Ext$-versions of the ordinary higher, module, and homology Topological Complexities of $X_0$, the rationalization of $X$. For Gorenstein spaces, we establish, under additional hypotheses, that the new homology topological complexity, denoted $HTC^{\mathcal{E}xt}_n(X,\mathbb{Q})$, lowers the ordinary $HTC_n(X)$ and, in case of equality, we extend Carasquel's characterization for $HTC_n(X)$ to some class of Gorenstein spaces (Theorem 1.2). We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristic especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.

This paper explores recent developments in applying fuzzy mathematics to rational homotopy theory. Fuzzy mathematical concepts allow for impreciseness and vagueness to be incorporated into mathematical models. This has enabled new techniques for analyzing topological spaces and homotopy groups. After reviewing foundational concepts in fuzzy mathematics and rational homotopy theory, this paper examines three novel approaches for integrating fuzzy methods into rational homotopy theory: fuzzy homotopy groups, fuzzy topological spaces, and fuzzy homological algebra. Challenges and opportunities for further research are also discussed.

Abstract The space of degree d smooth projective hypersurfaces of $\mathbb{C}P^n$ admits a scanning map to a certain space of sections. We compute a rational homotopy model of the action by conjugation of the group $U(n+1)$ on this space of sections, from which we deduce that the scanning map induces a monomorphism on cohomology when d > 2. Our main technique is the rational homotopy theory of Sullivan and, more specifically, a Sullivan model for the action by conjugation of a connected topological group on a space of sections.

This is the continuation of the study of differential graded (dg) vertex algebras defined in our previous paper [Caradot et al., “Differential graded vertex operator algebras and their Poisson algebras,” J. Math. Phys. 64, 121702 (2023)]. The goal of this paper is to construct a functor from the category of dg vertex Lie algebras to the category of dg vertex algebras which is left adjoint to the forgetful functor. This functor not only provides an abundant number of examples of dg vertex algebras, but it is also an important step in constructing a homotopy theory [see Avramov and Halperin, “Through the looking glass: A dictionary between rational homotopy theory and local algebra,” in Algebra, Algebraic Topology and their Interactions, Lecture Notes in Mathematics, edited by J. E. Roos (Springer, Berlin, Heidelberg, 1986), Vol. 1183, pp. 1–27 and D. G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics (Springer, Berlin, Heidelberg, 1967), Vol. 43] in the category of vertex algebras. Vertex Lie algebras were introduced as analogues of vertex algebras, but in which we only consider the singular part of the vertex operator map and the equalities it satisfies. In this paper, we extend the definition of vertex Lie algebras to the dg setting. We construct a pair of adjoint functors between the categories of dg vertex algebras and dg vertex Lie algebras, which leads to the explicit construction of dg vertex (operator) algebras. We will give examples based on the Virasoro algebra, the Neveu–Schwarz algebra, and dg Lie algebras.

ABSTRACT We study the homotopy groups of open books in terms of those of their pages and bindings. Under homotopy theoretic conditions on the monodromy we prove an integral decomposition result for the based loop space on an open book, and under more relaxed conditions we prove a rational loop space decomposition. The latter case allows for a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that for Milnor’s open book decomposition of an odd sphere with monodromy of finite order the induced action of the monodromy on the homology groups of its page cannot be nilpotent.

Abstract We exhibit the Hodge degeneration from nonabelian Hodge theory as a$2$-fold delooping of the filtered loop space$E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree$1$recovers the filtered circle$S^1_{fil}$of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an$E_2$-cogroupoid object in the$\infty $-category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on$S^1$, as well as the Todd class of the Lie algebroid$\mathbb {T}_{X}$; this is an invariant of a smooth and proper schemeXthat arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.

We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1 -formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3 -dimensional manifolds.

We describe two major string topology operations, the Chas–Sullivan product and the Goresky–Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom–Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3 -dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate–Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply-connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically.

Rational homotopy theory originated in the late 1960s and the early 1970s with the simultaneous but distinct approaches of Quillen (1969), Sullivan (1977) and Bousfield–Kan (1972). Each approach associated to a path connected space X an “algebraic object” A which is then used to construct a rational completion of X , X \to X_{\mathbb Q} . These constructions are homotopy equivalent for simply connected CW complexes of finite type, in which case H_*(X_{\mathbb Q})\cong H_*(X)\otimes \mathbb Q and \pi_*(X_{\mathbb Q}) \cong \pi_*(X)\otimes \mathbb Q . Otherwise, they may be different; in fact, Quillen’s construction is only available for simply connected spaces. In this review, discussion is limited to Sullivan’s completions, and the notation X\to X_{\mathbb Q} is reserved for these. We briefly review the construction, and follow that with a review of developments and examples over the subsequent decades, but often without the proofs. Since the explicit form of Sullivan’s completion has lent itself to a wide variety of applications in a range of fields, this survey will necessarily be modest in scope.

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa (Adv Theor Math Phys 5:769–808, 2002) as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series $$E_k$$ . It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics that gives rise to the same $$E_k$$ symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere $$S^4$$ , and then forming its iterated cyclic loop spaces $$\mathscr {L}_c^k S^4$$ , within which we discover the $$E_k$$ symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its $$E_k$$ symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space $$\mathscr {L}_c^k S^4$$ is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of $$(11-k)$$ -dimensional supergravity as the defining equations of the Sullivan minimal model of $$\mathscr {L}_c^k S^4$$ . This gives an explicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa’s Mysterious Duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: Is there an explicit relation between the del Pezzo surfaces $$\mathbb {B}_k$$ and iterated cyclic loop spaces of $$S^4$$ which would explain the common $$E_k$$ symmetry pattern?

The Toral Rank Conjecture and variants of equivariant formality

Absorbing homogeneous polynomials arising from rational homotopy theory and graph theory

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, At(G) of finite injective dimension, a homology theory π⁎At(⋅) taking values in At(G) based on the homology of the Borel construction, and a finite Adams spectral sequenceExtAt(G)⁎,⁎(π⁎At(X),π⁎At(Y))⇒[X,Y]⁎G for rational G-spectra X and Y.

Vector bundles of non-negative curvature over cohomogeneity one manifolds

Let G $G$ be an affine algebraic group defined over a field k $k$ of characteristic 0. We study the derived moduli space of G $G$ -local systems on a pointed connected CW complex X $X$ trivialized at the basepoint of X $X$ . This derived moduli space is represented by an affine DG scheme R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ : we call the (co)homology of the structure sheaf of R Loc G ( X , * ) $ \bm{\mathrm{R}}\mathrm{Loc}_G(X,\ast )$ the representation homology of X $X$ in G $G$ and denote it by HR * ( X , G ) $ \mathrm{HR}_\ast (X,G)$ . The 0-dimensional homology, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ , is isomorphic to the coordinate ring of the G $G$ -representation variety Rep G [ π 1 ( X ) ] $ {\rm {Rep}}_G[\pi _1(X)]$ of the fundamental group of X $X$ — a well-known algebro-geometric invariant that plays a role in many areas of topology. The higher representation homology is much less studied. In particular, when X $X$ is simply connected, HR 0 ( X , G ) $ \mathrm{HR}_0(X,G)$ is trivial but HR ∗ ( X , G ) $ \mathrm{HR}_*(X,G)$ is still an interesting rational invariant of X $X$ that depends on the Lie algebra of G $G$ . In this paper, we use Quillen's rational homotopy theory to compute the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Lie and Sullivan algebraic models. When G $G$ is reductive, we also compute HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ , the G $G$ -invariant part of representation homology, and study the question when HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is free of locally finite type as a graded commutative algebra. This question turns out to be related to the so-called Strong Macdonald Conjecture, a celebrated result in representation theory proposed (as a conjecture) by Feigin and Hanlon in the 1980s and proved by Fishel, Grojnowski and Teleman in 2008. Reformulating the Strong Macdonald Conjecture in topological terms, we give a simple characterization of spaces X $X$ for which HR * ( X , G ) G $ \mathrm{HR}_\ast (X,G)^G$ is a graded symmetric algebra for any complex reductive group G $G$ .

We show that the infinite symmetric product of a connected graded-commutative algebra over ℚ is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over ℚ is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold–Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded-commutative algebras are free; this allows us to determine minimal Sullivan models for finite symmetric products of complex projective spaces.

Abstract We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.

In this paper we study families of representations of the outer automorphism groups indexed on a collection of finite groups \mathcal{U} . We encode this large amount of data into a convenient abelian category which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of T. Church et al. [Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)], we investigate for which choices of \mathcal{U} the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibit representation stability.

The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a -graded transform whose homogeneous components are the Hori-Fourier transforms in twisted cohomology associated with integral multiples of a basic pair of T-dual closed 3-forms. We show how in the rational homotopy theory approximation of T-duality, such a map is naturally realized as a pull-iso-push transform, where the isomorphism part corresponds to the canonical equivalence between the left and the right gerbes associated with a T-duality configuration.

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. This proposes a new A_infty model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.