Daniel Tanré
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View article: Cone length and Lusternik-Schnirelmann category in rational homotopy
Cone length and Lusternik-Schnirelmann category in rational homotopy Open
Lusternik-Schnirelmann category (LS-category) of a topological space is the least integer $n$ such that there is a covering of $X$ by $n+1$ open sets, each of them being contractible in $X$. The cone length is the minimum number of cofibat…
View article: Intersection homotopy, refinements and coarsenings
Intersection homotopy, refinements and coarsenings Open
In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity and a filtered space. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to loca…
View article: Relation between intersection homology and homotopy groups
Relation between intersection homology and homotopy groups Open
As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which Gajer associates to a…
View article: Realization of Lie algebras and classifying spaces of crossed modules
Realization of Lie algebras and classifying spaces of crossed modules Open
The category of complete differential graded Lie algebras provides nice\nalgebraic models for the rational homotopy types of non-simply connected\nspaces. In particular, there is a realization functor, $\\langle -\\rangle$, of\nany complet…
View article: Homotopy truncations of homotopically stratified spaces
Homotopy truncations of homotopically stratified spaces Open
Intersection homology of Goresky and MacPherson can be defined from the\nDeligne sheaf, obtained from truncations of complexes of sheaves. As\nintersection homology is not the homology of a particular space, the search for\na family of spa…
View article: Homotopy truncations of homotopically stratified spaces
Homotopy truncations of homotopically stratified spaces Open
Intersection homology of Goresky and MacPherson can be defined from the Deligne sheaf, obtained from truncations of complexes of sheaves. As intersection homology is not the homology of a particular space, the search for a family of spaces…
View article: Perverse homotopy groups
Perverse homotopy groups Open
As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which P. Gajer associates t…
View article: Relation between intersection homology and homotopy groups
Relation between intersection homology and homotopy groups Open
As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which P. Gajer associates t…
View article: SIMPLICIAL INTERSECTION HOMOLOGY REVISITED
SIMPLICIAL INTERSECTION HOMOLOGY REVISITED Open
Intersection homology is defined for simplicial, singular and PL chains. In the case of a filtered simplicial complex, it is well known that the three versions are isomorphic. This isomorphism is established by using the PL case as an inte…
View article: Simplicial intersection homology revisited
Simplicial intersection homology revisited Open
Intersection homology is defined for simplicial, singular and PL chains and it is well known that the three versions are isomorphic for a full filtered simplicial complex. In the literature, the isomorphism, between the singular and the si…
View article: Realization of a Lie algebra and classifying space of crossed modules
Realization of a Lie algebra and classifying space of crossed modules Open
Complete differential graded Lie algebras appear to be a wonderful tool for giving nice algebraic models for the rational homotopy type of non-simply connected spaces. In particular, there is a realization functor of any Lie algebra as a s…
View article: Realization of Lie algebras and classifying spaces of crossed modules
Realization of Lie algebras and classifying spaces of crossed modules Open
The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $\langle -\rangle$, of any complete dif…
View article: POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY
POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY Open
Using a cap product, we construct an explicit Poincaré duality isomorphism between the blown-up intersection cohomology and the Borel–Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseu…
View article: Natural operations in Intersection Cohomology
Natural operations in Intersection Cohomology Open
Eilenberg-MacLane spaces, that classify the singular cohomology groups of topological spaces, admit natural constructions in the framework of simplicial sets. The existence of similar spaces for the intersection cohomology groups of a stra…
View article: Spatial realization of a Lie algebra and Bar construction of a group
Spatial realization of a Lie algebra and Bar construction of a group Open
We prove that the spatial realization of a rational complete Lie algebra $L$, concentrated in degree 0, is isomorphic to the simplicial bar construction on the group, obtained from the Baker-Campbell-Hausdorff product on $L$.
View article: Relative singular value decomposition and applications to LS-category
Relative singular value decomposition and applications to LS-category Open
Let $Sp(n)$ be the symplectic group of quaternionic $(n\times n)$-matrices. For any $1\leq k\leq n$, an element $A$ of $Sp(n)$ can be decomposed in $A= \begin{bmatrix} α&T\cr β&P \end{bmatrix}$ with $P$ a $(k\times k)$-matrix. In this work…
View article: Symmetric Lie models of a triangle
Symmetric Lie models of a triangle Open
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 geometric…
View article: Homotopy theory of complete Lie algebras and Lie models of simplicial sets
Homotopy theory of complete Lie algebras and Lie models of simplicial sets Open
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 grad…
View article: Blown-up intersection cochains and Deligne's sheaves
Blown-up intersection cochains and Deligne's sheaves Open
In a series of papers the authors introduced the so-called blown-up intersection cochains. These cochains are suitable to study products and cohomology operations of intersection cohomology of stratified spaces. The aim of this paper is to…
View article: Blown-up intersection cohomology
Blown-up intersection cohomology Open
In previous works, we have introduced the blown-up intersection cohomology\nand used it to extend Sullivan's minimal models theory to the framework of\npseudomanifolds, and to give a positive answer to a conjecture of M. Goresky\nand W. Pa…
View article: The infinity Quillen functor, Maurer-Cartan elements and DGL realizations
The infinity Quillen functor, Maurer-Cartan elements and DGL realizations Open
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 a…
View article: Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes
Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes Open
In a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to…
View article: Cayley Transform on Stiefel manifolds
Cayley Transform on Stiefel manifolds Open
We define a Cayley transform on Stiefel manifolds. Applications to the Lusternik-Schnirelmann category and optimisation problems are presented.
View article: Singular decompositions of a cap product
Singular decompositions of a cap product Open
In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that thi…