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Urtzi Buijs

Publications and source records attributed to Urtzi Buijs.

At least 19 recordsLinked to original sources
arXiv2026

Higher dimensional visual proofs, Nicomachus' 4D Theorem and the mysterious irreducible factor $(3n^2+3n-1)$ in the sum of fourth powers

Sums of powers $S_p(n)=\sum_{k=1}^n k^p$ can be described by Faulhaber's formula in terms of the Bernoulli numbers. The first cases of this formula admit visual proofs of various kinds, which lead to factorized Faulhaber polynomials. In this article we present a technique that yields higher-dimensional visual proofs for these factorized formulas, providing a geometric interpretation of the roots that appear. In particular, we prove Nicomachus's Theorem in four dimensions, and we visually explain the appearance, in dimension five, of the irreducible factor $(3n^2 +3n-1)$ in the polynomial ring over the rational numbers.

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arXiv2024

Sectional category à la Quillen

In this note we give a characterization of the sectional category of a map between rational spaces in terms of its Koszul-Quillen model.

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arXiv2024

Explicit Quillen models for Cartesian products of $2$-cones

We give an explicit minimal Quillen model for the Cartesian product $X\times Y$ of rational $2$-cones in terms of derivations and a binary operation $\star \colon \mathbb{M}(V)\otimes \mathbb{L}(W)\to \mathbb{L}(V\oplus W\oplus s(V\otimes W))$, where $(\mathbb{L}(V), \partial)$ and $(\mathbb{L}(W), \partial)$ are Quillen minimal models for $X$ and $Y$ respectively and $\mathbb{M}$ denotes the free magma on $W$. The model presented also allows us to explicitly describe a model for the diagonal map $Δ\colon X\to X\times X$.

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arXiv2019

A-infinity structures and Massey products

We show how and when it is possible to detect and recover higher Massey products on the cohomology $H$ of a differential graded algebra $A$ with higher multiplications on quasi-isomorphic $A_\infty$ structures on $H$.

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arXiv2019

Formality criteria in terms of higher Whitehead brackets

We provide two criteria for discarding the formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogue 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 the Quillen spectral sequence. Finally, we use $L_\infty$ 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.

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arXiv2018

Symmetric Lie models of a triangle

R. Lawrence and D. Sullivan have constructed a Lie model for an interval from the geometrical idea of flat connections and flows of gauge transformations. Their model supports an action of the symmetric group $Σ_2$ reflecting the geometrical symmetry of the interval. In this work, we present a Lie model of the triangle with an action of the symmetric group $Σ_3$ compatible with the geometrical symmetries of the triangle. We also prove that the model of a graph consisting of a circuit with $k$ vertices admits a Maurer-Cartan element stable by the automorphisms of the graph.

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arXiv2018

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(\mathbb R^m,k), S^n)$ and $ map^*(F(\mathbb R^m,k), S^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.

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arXiv2018

Homotopy theory of complete Lie algebras and Lie models of simplicial sets

In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow up of this work. We show that when X is a finite connected simplicial set, then the realization of the model of X is the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on the category of complete differential graded algebras making the two previous functors a Quillen pair, and we construct an explicit cylinder. In particular, these functors preserve homotopies and weak equivalences and therefore, this gives the basis for developing a Lie rational homotopy theory for all spaces.

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arXiv2018

The infinity Quillen functor, Maurer-Cartan elements and DGL realizations

We show an alternative construction of the cosimplicial free complete diferential graded Lie algebra $\mathfrak{L}_\bullet=\widehat{\mathbb{L}}(s^{-1}Δ^\bullet)$ based on a new Lie bracket formulae for Lie polynomials on a general tensor algebra. Based on it,we prove that for any complete differential graded Lie algebra $L$, its geometrical realization $\langle L\rangle=\text{Hom}_{\text{cdgl}}(\mathfrak{L}_\bullet,L)$ is isomorphic to its nerve $γ_\bullet(L)$, a deformation retract of the Getzler-Hinich realization $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.

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arXiv2017

Weight decompositions of Thom spaces of vector bundles in rational homotopy theory

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. Lastly, we link the theory of weight decompositions with mixed Hodge theory and apply our results to motivic Thom spaces.

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arXiv2017

Lie models of simplicial sets and representability of the Quillen functor

Extending the model of the interval, we explicitly define for each $n\ge 0$ a free complete differential graded Lie algebra $\mathfrak{L}_n$ generated by the simplices of $Δ^n$, with desuspended degrees, in which the vertices are Maurer-Cartan elements and the differential extends the simplicial chain complex of the standard $n$-simplex. The family $\{\mathfrak{L}_\bullet\}_{n\ge 0}$ is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct a pair of adjoint functors between the categories of simplicial sets and complete differential graded Lie algebras given by $\langle L\rangle_\bullet=\text{ DGL} (\mathfrak{L}_\bullet,L)$ and $ \mathfrak{L}(K)=\varinjlim_K\mathfrak{L}_{\bullet} $. This new tools let us extend Quillen rational homotopy theory approach to any simplicial set $K$ whose path components are non necessarily simply connected. We prove that $\mathfrak{L} (K)$ contains a model of each component of $K$. When $K$ is a $1$-connected finite simplicial complex, the Quillen model of $K$ can be extracted from $\mathfrak{L} (K)$. When $K$ is connected then, for a perturbed differential $\partial_a$, $H_0(\mathfrak{L} (K),\partial_a)$ is the Malcev Lie completion of $π_1(K)$. Analogous results are obtained for the realization $\langle L\rangle$ of any complete $\text{DGL}$.

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arXiv2016

Maurer-Cartan elements in the Lie models of finite simplicial complexes

In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

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arXiv2016

The gauge action, DG Lie algebra and identities for Bernoulli numbers

In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers $(a,b,c)$ with $a+b+c=n-1$, $n\ge 4$. These identities are deduced while translating into homotopical terms the gauge action on the Maurer Cartan Set which can be seen an abstraction of the behaviour of gauge infinitesimal transformations in classical gauge theory. We show that Euler and Miki's identities, well known and apparently non related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.

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arXiv2015

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 ${\rm Hom}(H_*(X; \mathbb{Q}), π_*(ΩY)\otimes\mathbb{Q})$, 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 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.

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arXiv2014

The homotopy fixed point set of Lie group actions on elliptic spaces

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, that for $G$ a torus, $π_*(k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.

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arXiv2013

Algebraic models of non-connected spaces and homotopy theory of $L_\infty$ algebras

We develop a homotopy theory of $L_\infty$ 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.

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