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Daniel Tanré

Publications and source records attributed to Daniel Tanré.

33 records · Page 2Linked to original sources
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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arXiv2018

Lie models for nilpotent spaces

Let $(L,d)$ be a differential graded Lie algebra, where $L=L(V)$ is free as graded Lie algebra and $V=V_{\geq 0}$ is a finite type graded vector space. We prove that the injection of $(L,d)$ into its completion $(\widehat{L},d)$ is a quasi-isomorphism if and only if $H(L,d)$ is a finite type pronilpotent graded Lie algebra. As a consequence, we obtain an equivalence between graded Lie models for nilpotent spaces in rational homotopy theory.

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arXiv2017

Poincaré duality with cap products in intersection homology

For having a Poincaré duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure need a coefficient field or an additional hypothesis on the torsion. In this work, by using the classical geometric process of blowing-up, adapted to a simplicial setting, we build a cochain complex which gives a Poincaré duality via a cap product with intersection homology, for any commutative ring of coefficients. We prove also the topological invariance of the blown-up intersection cohomology with compact supports in the case of a paracompact pseudomanifold with no codimension one strata. This work is written with general perversities, defined on each stratum and not only in function of the codimension of strata. It contains also a tame intersection homology, suitable for large perversities.

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arXiv2017

Blown-up intersection cohomology

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivan's minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon on Steenrod squares in intersection homology. In this paper, we establish the main properties of this cohomology. One of its major feature is the existence of cap and cup products for any filtered space and any commutative ring of coefficients, at the cochain level. Moreover, we show that each stratified map induces an homomorphism between the blown-up intersection cohomologies, compatible with the cup and cap products. We prove also its topological invariance in the case of a pseudomanifold with no codimension one strata. Finally, we compare it with the intersection cohomology studied by G. Friedman and J.E. McClure. A great part of our results involves general perversities, defined independently on each stratum, and a tame intersection homology adapted to large perversities.

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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

Singular decompositions of a cap product

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 this classical cap product is compatible with a cap product in intersection (co)-homology, that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincaré duality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities.

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arXiv2014

Steenrod squares on Intersection cohomology and a conjecture of M. Goresky and W. Pardon

We prove a conjecture raised by M. Goresky and W. Pardon, concerning the range of validity of the perverse degree of Steenrod squares in intersection cohomology. This answer turns out of importance for the definition of characteristic classes in the framework of intersection cohomology. For this purpose, we present a construction of ${\rm cup}_{i}$-products on the cochain complex, built on the blow-up of some singular simplices and introduced in a previous work. We extend to this setting the classical properties of the associated Steenrod squares, including Adem and Cartan relations, for any generalized perversities. In the case of a pseudomanifold, we prove that our definition coincides with M. Goresky's definition. Several examples of concrete computation of perverse Steenrod squares are given, including the case of isolated singularities and, more especially, we describe the Steenrod squares on the Thom space of a vector bundle, in function of the Steenrod squares of the basis and the Stiefel-Whitney classes. We detail also an example of a non trivial square, $\sq^2\colon H_{\ bar{p}}\to H_{\ bar{p}+2}$, whose information is lost if we consider it as values in $H_{2\ bar{p}}$, showing the interest of the Goresky and Pardon's conjecture.

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arXiv2014

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, we use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology. For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (cdga's) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse cdga's, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PL-pseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in CP(4), is intersection-formal.

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arXiv2010

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category $\Tops$ of stratified spaces, that are topological spaces $X$ endowed with a partition $\cF$ and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element $(X,\cF)$ of $\Tops$ together with a class $\cA$ of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If $(X,\cF)\in\Tops$, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any $S\in \cF$. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with $\cA$ the class of transverse subsets, coincide with the tangential category and are homotopical invariants.

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arXiv2007

Simplicial resolutions and Ganea fibrations

In this work, we compare the two approximations of a path-connected space $X$, by the Ganea spaces $G_n(X)$ and by the realizations $\|Λ_\bullet X\|_{n}$ of the truncated simplicial resolutions emerging from the loop-suspension cotriple $ΣΩ$. For a simply connected space $X$, we construct maps $\|Λ_\bullet X\|_{n-1}\to G_n(X)\to \|Λ_\bullet X\|_{n}$ over $X$, up to homotopy. In the case $n=2$, we prove the existence of a map $G_2(X)\to\|Λ_\bullet X\|_{1}$ over $X$ (up to homotopy) and conjecture that this map exists for any $n$.

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arXiv2003

The cohomology algebra of unordered configuration spaces

Given an $N$-dimensional compact manifold $M$ and a field $\bk$, F. Cohen and L. Taylor have constructed a spectral sequence, $\cE(M,n,\bk)$, converging to the cohomology of the space of ordered configurations of $n$ points in $M$. The symmetric group $Σ_n$ acts on this spectral sequence giving a spectral sequence of $Σ_n$ differential graded commutative algebras. Here, we provide an explicit description of the invariants algebra $(E_1,d_1)^{Σ_n}$ of the first term of $\cE(M,n,\Q)$. We apply this determination in two directions: -- in the case of a complex projective manifold or of an odd dimensional manifold $M$, we obtain the cohomology algebra $H^*(C_n(M);\Q)$ of the space of unordered configurations of $n$ points in $M$ (the concrete example of $P^2(\C)$ is detailed), -- we prove the degeneration of the spectral sequence formed of the $Σ_n$-invariants $\cE(M,n,\Q)^{Σ_n}$ at level 2, for any manifold $M$. These results use a transfer map and are also true with coefficients in a finite field $\F_p$ with $p>n$.

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