Sullivan Minimal Model Research Articles

Rational homotopy type of projectivization of the tangent bundle of certain spaces

PurposeThe paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces.Design/methodology/approachThe paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of P(E) and establish relationships with the base space.FindingsThe paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model of P(E). We also provide the homogeneous space of P(E) when n = 2. Finally, we prove the formality of P(E) over a homogeneous space of equal rank.Research limitations/implicationsLimitations may include the complexity of computing minimal models for higher-dimensional bundles.Practical implicationsUnderstanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling.Social implicationsWhile the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits.Originality/valueThe paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles.

Evaluation Subgroups of a map between Rational Finite H-Spaces

We use the theory of Sullivan minimal models and derivation to compute the evaluation subgroups and moreover the relative evaluation subgroups of a map f : X → Y between rational finite H-spaces. As a consequence, we show that the G-sequence is exact if f induces a zero map on rational homotopy groups.

Mysterious Triality and Rational Homotopy Theory

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?

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 a DGL-map between derivations of Sullivan minimal models

For a map f:Xrightarrow Y, there is the relative model M(Y)=(Lambda V,d)rightarrow (Lambda Votimes Lambda W,D)simeq M(X) by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let mathrm{Baut}_1X be the Dold–Lashof classifying space of orientable fibrations with fiber X (Dold and Lashof, Ill J Math 3:285–305, 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations mathrm{Der}M(X) of the Sullivan minimal model M(X) of X. Then we consider the condition that the restriction b_f:mathrm{Der} (Lambda Votimes Lambda W,D)rightarrow mathrm{Der}(Lambda V,d) is a DGL-map and the related topics.

A Fundamental Class for Intersection Spaces of Depth One Witt Spaces

By a theorem of Banagl–Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincaré duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincaré duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl–Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincaré duality space to the Goresky–MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata.

Formality criteria in terms of higher Whitehead brackets

We provide two criteria for establishing the non-formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogues of the Massey products of a differential graded associative algebra. We also show that formality of a differential graded Lie algebra is not equivalent to the collapse of its associated Quillen spectral sequence. Finally, we use L∞ algebras and Quillen's formulation of rational homotopy theory to recover and improve a classical theorem for detecting higher Whitehead products in Sullivan minimal models, and give some applications.

Sullivan minimal models of operad algebras

We prove the existence of Sullivan minimal models of operad algebras for a quite wide family of operads in the category of complexes of vector spacesover a field of characteristic zero. Our construction is an adaptation of Sullivan’s original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass, and Lie, as well as over their minimal models Com∞, Ass∞, and Lie∞. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.

Rational cohomologies of classifying spaces for homogeneous spaces of small rank

Let a simply-connected homogeneous space $${X}$$ satisfy the condition of $${{\rm dim} \pi_{\rm even}(X)\otimes {\mathbb{Q}}=2}$$ and $${{\rm dim} \pi_{\rm odd}(X)\otimes {\mathbb{Q}}=3}$$ (then, we say it is of (2, 3) type), which is the smallest rank in non-formal pure spaces. Then, we compute the Sullivan minimal model of the Dold–Lashof classifying space $${{\rm Baut}_1 X}$$ according to Nishinobu and Yamaguchi (Topol Appl 196:290–307, 2015) and observe whether or not its rational cohomology is a polynomial algebra, which is a necessary condition for the Serre spectral sequence of any fibration over a sphere with fibre $${X}$$ to degenerate at term $${E_2}$$ .

Sullivan minimal models of classifying spaces for non-formal spaces of small rank

We observe certain rational homotopical conditions of a simply connected two stage CW complex X such that the rational cohomology of the classifying space Baut1X for fibrations with fiber X is (not) free and we illustrate their examples. First, we consider when is Baut1X a rational factor of Baut1(X×Sn) for a non-formal elliptic space X of rank 3 and an odd-integer n. Second, we compute the Sullivan minimal models of Baut1X when X are certain non-formal pure spaces of rank 5.

THE LEFSCHETZ CONDITION ON PROJECTIVIZATIONS OF COMPLEX VECTOR BUNDLES

We consider a condition under which the projectivization <TEX>$P(E^k)$</TEX> of a complex k-bundle <TEX>$E^k{\rightarrow}M$</TEX> over an even-dimensional manifold M can have the hard Lefschetz property, affected by [10]. It depends strongly on the rank k of the bundle <TEX>$E^k$</TEX>. Our approach is purely algebraic by using rational Sullivan minimal models [5]. We will give some examples.

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.

The de Rham homotopy theory and differential graded category

This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations and closed dg-categories with subsidiary data.

Rationalized evaluation subgroups of a map I: Sullivan models, derivations and [formula omitted]-sequences

We identify the homomorphism induced on rational homotopy groups by the evaluation map ω : map ( X , Y ; f ) → Y in terms of a map of complexes of derivations constructed directly from the Sullivan minimal model of f . This allows us to characterize the rationalized n th evaluation subgroup of f and, more generally, the rationalization of the so-called G -sequence of the map f . We use these results to study the G -sequence in the context of rational homotopy theory.

Relative topology of real algebraic varieties in their complexifications

We investigate, for a given smooth closed manifoldM, the existence of an algebraic model X forM (i.e., a nonsingular real algebraic variety di.eomorphic to M) such that some non-singular projective complexi.cation i : X ?? XC of X admits a retraction r : XC ?? X. If such an X exists, we show that M must be formal in the sense of Sullivan's minimal models, and that all rational Massey products on M are trivial. We also study the homomorphism on cohomology induced by I for algebraic models X of M. Using ?etale cohomology, we see that mod p Steenrod powers give an obstruction for the induced map on cohomology, i. : Hk(XC,Zp) ?? Hk(X, Zp), to be onto, if we require that X is de.ned over rational numbers.

Free torus actions and two-stage spaces

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

Explicit formulae for the rational L–S category of some homogeneous spaces

Explicit formulae for rational Lusternik–Schnirelman (L–S) category (cat 0) are rare, but some are available for a class of spaces which includes homogeneous spaces G/ H when H is a product of at most 3 rank 1 groups, and rank G− rank H⩽1 . We extend the applicability of these formulae to the case when rank G=5 and H is a 4-torus or ( SU 2) 4. With a Sullivan minimal model as data, implementing the formula requires the selection of a regular subsequence of length 4 from a sequence f 1,…, f 5 of homogeneous polynomials in 4 variables satisfying dim Q [x 1,…,x 4]/(f 1,…,f 5)<∞. Such subsequences are readily obtainable, and the ease of computation is in contrast to most available methods for determining rational L–S category, which usually involve both upper and lower bounds and a good measure of luck. The proof of the formula is a pretty application of ideal class groups in algebraic topology. We also present some examples to illustrate our result.

Model categories in algebraic topology

This survey of model categories and their applications in algebraic topology is intended as an introduction for non homotopy theorists, in particular category theorists and categorical topologists. We begin by defining model categories and the homotopy-like equivalence relation on their morphisms. We then explore the question of compatibility between monoidal and model structures on a category. We conclude with a presentation of the Sullivan minimal model of rational homotopy theory, including its application to the study of Lusternik–Schnirelmann category.

Minimal models for non-free circle actions

Let $\Phi \colon \sbat \times M \to M$ be a smooth action of the unit circle $ \sbat$ on a manifold $M$. In this work, we compute the minimal model of $M$ in terms of the orbit space $B$ and the fixed point set $F\subset B$, as a dg-module over the Sullivan's minimal model of $B$.

Algèbres de Lie de dérivations de certaines algèbres pures

Let E ( X ) be the H -space of homotopy self-equivalences which are homotopic to the identity of a homogeneous Kähler manifold with maximal rank. The Lie algebra of derivations of a pure differential graded algebra with zero homotopy Euler characteristic allows us to use Sullivan's minimal model to study the rational homotopy theory of E ( X ). As an application we compute dim (π ∗ (E(X)) ⊗ Q ) in the case when X is a certain flag manifold.