arXiv subjects

Alberto Arabia

Publications and source records attributed to Alberto Arabia.

3 recordsLinked to original sources

arXiv2018

Espaces de configuration généralisés. Espaces topologiques $i$-acycliques. Suites spectrales "basiques"

The generalized (ordered) configuration spaces associated to a topological space $X$ are the spaces $Δ_{\leq\ell}X^{m}:=\{(x_1,\ldots,x_{m})\in X^{m}\mid\#\{x_1,\ldots,x_{m}\}\leq \ell\}$ and $Δ_{\ell}X^{m}:=Δ_{\leq\ell}X^{m}\setminus Δ_{\leq\ell-1}$. They are equipped with the action of the symmetric group $S_m$ permuting coordinates. When $X$ has no interior cohomology (i.e. is $i$-acyclic) we are able to compute explicitly the character formula of $S_m$ acting on the cohomology of these spaces, and if $X$ is furthermore a connected and oriented pseudomanifold of dimension $\geq2$ we generalize Church's representation stability theorem to the case of the families $\{Δ_{\leq m-a}X^m\}_m$ and $\{Δ_{\ell-a}X^m\}_m$. We show that, for fixed $a,i\in\mathbb N$, the families of representations $\{ S_m: H ^{i}(Δ_{?m-a}X^{m})\}_{m}$ are monotone and stationary for $m\geq4i+4a$, if $d_{X}=2$, and for $m\geq2i+4a$, if $d_{X}\geq3$. The corresponding families of characters and Betti numbers are (hence) polynomial and the families of integers $\{\mathop{\rm Betti}_{i}({Δ_{?m-a}X^{m} / S_m})\}_{m}$ are constant within the same range of integers $m$. We further show that the family $\{\mathop{\rm Betti}_{i}({Δ_{m}X^{m}/ S_m})\}_{m}$ is constant for $m\geq 2i$, if $d_{X}=2$, and for $m\geq i$, if $d_{X}\geq3$. In particular, complex algebraic varieties whether they are smooth on not verify these generalizations of Church's stability theorems.

math.AT↗

arXiv2018

On the equivalence of two stability conditions of FB-modules

We give a proof of the fact that for an FB-module the properties of being "representation stable" (RS) and "having a polynomial character" (PC) are equivalent. We obtain optimal estimates for the gap between the ranks of the polynomiality and of representation stability. As a by-product, we show that the degree of the polynomial character and the weight at infinity of an FB-module coincide, which we apply to determine the weight at infinity of a tensor product of FB-modules.

math.RT↗

arXiv2017

On Equivariant Poincaré Duality, Gysin Morphisms and Euler Classes

The aim of these notes, originally intended as an appendix to a book on the foundations of equivariant cohomology, is to set up the formalism of the $G$-equivariant Poincaré duality for oriented $G$-manifolds, for any connected compact Lie group $G$, following the work of J.-L. Brylinski leading to the spectral sequence $$\mathop{\rm Extgr}\nolimits_{H_G}(H_{G,\rm c} (M),H_G)\Rightarrow H_{G}(M)[d_{M}]\,.$$ The equivariant Gysin functor $(\_)_!:=Ω_{G}(\_)\in\mathcal D^{+}(\mathord{\rm DGM}(H_{G}))$ (resp. $(\_)_{*}:=Ω_{G,\rm c}(\_)$) is then defined in the category of oriented $G$-manifolds and proper maps (resp. unrestricted maps) with values in the derived category of the category of differential graded modules over $H_{G}$, as the composition of the Cartan complex of equivariant differential forms functor $Ω_{G,\rm c}(\_)$ (resp. $Ω_{G}(\_)$) with the duality functor $I\mkern-4.5muR\,{\rm Hom}_{H_{G}}^{\bullet}(\_,H_{G})$ and the equivariant Poincaré adjunction $I\mkern-4.5muD_{G} (M):Ω_{G} (M)[d_{M}]\to I\mkern-4.5muR\,{\rm Hom}_{H_{G}}^{\bullet}(Ω_{G,\rm c} (M),H_{G} )$ (resp. $I\mkern-4.5muD_{G}' (M):Ω_{G,\rm c} (M)[d_{M}]\to I\mkern-4.5muR\,{\rm Hom}_{H_{G}}^{\bullet}(Ω_{G} (M),H_{G} )$). Equivariant Euler classes are next introduced for any closed embedding $i:N\subseteq M$ as ${\rm Eu}_{G}(N,M):=i^{*}i_{!}(1)$ where $i^{*}i_{!}:H_{G}(N)\to H_{G}(N)$ is the push-pull operator. Some localization and fixed point theorems finish the notes. The idea of introducing Gysin morphisms through an equivariant Poincaré duality formalism à la Grothendieck-Verdier has many theoretical advantages and is somewhat uncommon in the equivariant setting, warranting publication of these notes.

math.AT↗