For a real central arrangement $𝒜$, Salvetti introduced a construction of a finite complex Sal$\left(𝒜\right)$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement ${𝒜}_{k-1}$, the Salvetti complex Sal$\left({𝒜}_{k-1}\right)$ serves as a good combinatorial model for the homotopy type of the configuration space $F(ℂ,k)$ of $k$ points in $C$, which is homotopy equivalent to the space $C_2\left(k\right)$ of k little $2$-cubes. Motivated by the importance of little cubes in homotopy theory, especially in...

The class of loop spaces of which the mod $p$ cohomology is Noetherian is much larger than the class of $p$-compact groups (for which the mod $p$ cohomology is required to be finite). It contains Eilenberg–Mac Lane spaces such as $ℂP^{\infty }$ and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space $BX$ of such an object and prove it is as small as expected, that is, comparable to that of $BℂP^{\infty }$. We also show that $B$X differs basically from the classifying space of a $p$-compact...

The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles...

Let $d$ be a square free integer and $L_d:=\mathbb{Q}({\zeta }_8,\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, ${\mathrm{Cl}}_2\left(L_d\right)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

Let $F$ be an imaginary quadratic field and $𝒪$ its ring of integers. Let $𝔫\subset 𝒪$ be a non-zero ideal and let $p\>5$ be a rational inert prime in $F$ and coprime with $𝔫$. Let $V$ be an irreducible finite dimensional representation of ${\overline{𝔽}}_p\left[{\mathrm{GL}}_2\left({𝔽}_{p^2}\right)\right]$. We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $V$ already lives in the cohomology with coefficients in ${\overline{𝔽}}_p\otimes de{t}^e$ for some $e\ge 0$; except possibly in some few cases.

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,4}$ to compute the generators of the ${\mathbb{Z}}_2$–cohomology groups $H^j\left({\tilde{G}}_{n,4}\right)$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n,3}$ we conjecture some predictions.

We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical $\mathrm{Ext}$-computations as well as new results. In particular, we get a cohomological version of the “fundamental theorems” from classical invariant theory for $G{L}_n$ for $n$ big enough (and we give a conjecture for smaller values of $n$). We also study the “twisting spectral sequence” $E^{s,t}(F,G,r)$ converging to the extension groups...

Let $(R,𝔪)$ be a commutative Noetherian local ring, $𝔞$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $𝔞M\ne M$ and $\mathrm{cd}(𝔞,M)-\mathrm{grade}(𝔞,M)\le 1$, where $\mathrm{cd}(𝔞,M)$ is the cohomological dimension of $M$ with respect to $𝔞$ and $\mathrm{grade}(𝔞,M)$ is the $M$-grade of $𝔞$. Let $D(-):={\mathrm{Hom}}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/𝔪)$ is the injective hull of the residue field $R/𝔪$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0⟶& H_{𝔞}^{n-2}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)⟶D\left(M\right)\\ \hfill ⟶& H_{𝔞}^{n-1}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^{n+1}\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)\\ \hfill ⟶& H_{𝔞}^n\left(D\left(H_{(x_1,...,x_{n-1})}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{(}^{n-1}M\right)\right))⟶...,\end{array}$$ where $n:=\mathrm{cd}(𝔞,M)$ is a non-negative integer, $x_1,...,x_{n-1}$ is a regular sequence in $𝔞$ on $M$ and, for an $R$-module $L$, $H_{𝔞}^i\left(L\right)$ is the $i$th local cohomology module...

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

Let $M$ be any compact simply-connected oriented $d$-dimensional smooth manifold and let $𝔽$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $H{H}^{*}(S^{*}\left(M\right),S^{*}\left(M\right))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjectured to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}\left(LM\right)$ introduced by Chas and Sullivan. We also show that the negative cyclic...

We establish in this paper a lower bound for the volume of a unit vector field $\overrightarrow{v}$ defined on ${𝐒}^n\setminus \{\pm x\}$, $n=2,3$. This lower bound is related to the sum of the absolute values of the indices of $\overrightarrow{v}$ at $x$ and $-x$.

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$-theory, Chow groups, Grothendieck’s $K_0$, etc.) with formal group law $F$. We construct a ring from $F$ and the characters of $T$, that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁(G/B)$ where $G/B$ is the variety of Borel subgroups of $G$. Our main result says that when the...

General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}\left({\mathrm{F}}_p^2\left(K\right)\right|K)$, that is the Galois group of the second Hilbert $p$-class field ${\mathrm{F}}_p^2\left(K\right)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group ${\mathrm{Cl}}_p\left(K\right)$, the position of $G$ is restricted to certain admissible branches...

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of ${ℳ}_{1,n}$ and ${\overline{ℳ}}_{1,n}$. In the second part, we study the orbifold intersection theory of ${\overline{ℳ}}_{1,n}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

For $\varphi$ a normalized positive definite function on a locally compact abelian group $G$, let ${\pi }_{\varphi }$ be the unitary representation associated to $\varphi$ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,{\pi }_{\varphi })$ and reduced 1-cohomology ${\overline{H}}^1(G,{\pi }_{\varphi })$. For example, ${\overline{H}}^1(G,{\pi }_{\varphi })=0$ if and only if either $\text{Hom}(G,ℂ)=0$ or ${\mu }_{\varphi }\left(1_G\right)=0$, where $1_G$ is the trivial character of $G$ and ${\mu }_{\varphi }$ is the probability measure on the Pontryagin dual $\widehat{G}$ associated to $\varphi$ by Bochner’s Theorem. This streamlines an argument of Guichardet...

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This...