Let $M$ be a 1-connected closed manifold of dimension $m$ and $LM$ be the space of free loops on $M$. M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_{*}(LM;k)$. When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology $H{H}^{*}(C^{*}\left(M\right);C^{*}\left(M\right))$ which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $H{H}^{*}(C^{*}\left(M\right);C^{*}\left(M\right))$ and the shifted homology $H_{*+m}(LM;k)$. We also prove...

We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, $D_{k}Spⁿ$ and $D_k\Lambda ⁿ$, in terms of the first terms in the Taylor towers of $S{p}^t$ and ${\Lambda }^t$ for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $S{p}^t$ and ${\Lambda }^t$. We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k}Spⁿ$ and $D_k\Lambda ⁿ$.

We introduce two entire functions $f_{{\mathrm{A}}_{\frac{1}{2}\infty }}$ and $f_{{\mathrm{D}}_{\frac{1}{2}\infty }}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps ${\mathbf{C}}^2\to \mathbf{C}$ define topologically locally trivial fibrations over $\mathbf{C}\setminus \{0,1\}$. All critical points in the singular fibers over $0$ and $1$ are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed infinite quivers of type ${\mathrm{A}}_{\frac{1}{2}\infty }$ and ${\mathrm{D}}_{\frac{1}{2}\infty }$, respectively. Coxeter elements...

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f:ℂ{\mathbb{P}}^k\to ℂ{\mathbb{P}}^k$, when k >1. Classical theory describes U(f) as the complement in $ℂ{\mathbb{P}}^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ{\mathbb{P}}^k$, we give a definition of linking number between closed loops in $ℂ{\mathbb{P}}^k\setminus suppS$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁\left(ℂ{\mathbb{P}}^k\setminus suppS\right)$. As an application, we use...

Given an algebraic number field $k$ and a finite group $\Gamma$, we write $ℛ\left(O_k\left[\Gamma \right]\right)$ for the subset of the locally free classgroup $\mathrm{Cl}\left(O_k\left[\Gamma \right]\right)$ consisting of the classes of rings of integers $O_N$ in tame Galois extensions $N/k$ with $\mathrm{Gal}(N/k)\cong \Gamma$. We determine $ℛ\left(O_k\left[\Gamma \right]\right)$, and show it is a subgroup of $\mathrm{Cl}\left(O_k\left[\Gamma \right]\right)$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $\Gamma =V⋊C$, where $V$ is an elementary abelian group of order $p^r$ and $C$ is a cyclic group of order $m\&gt;1$ acting faithfully...

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum${\left(k\right)}_F$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum${\left(k\right)}_F$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined...

Given a smooth proper dg algebra $A$, a perfect dg $A$-module $M$ and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of the algebra $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ${\sum }_{n=1}^{\infty }x\left(n\right)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of ${\sum }_{n=1}^{\infty }b\left(n\right)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....

Let $\Sigma$ be a compact connected oriented surface with one boundary component, and let $\pi$ be the fundamental group of $\Sigma$. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $\Sigma$, whose $k$-th term consists of the self-homeomorphisms of $\Sigma$ that act trivially at the level of the $k$-th nilpotent quotient of $\pi$. Morita defined a homomorphism from the $k$-th term of the Johnson filtration to the third homology group of the $k$-th nilpotent quotient of $\pi$. In this paper, we...

A subset of a product of topological spaces is called $n$-thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_3$ space $X$ without isolated points such that $X^n$ contains an $n$-thin dense subset, but $X^{n+1}$ does not contain any $n$-thin dense subset. We also observe that part of the construction can be carried out under MA.

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We study the compositum $k^{\left[d\right]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{\left[d\right]}$ contains all subextensions of degree less than $d$ if and only if $d\le 4$. We prove that for $d\&gt;2$ there is no bound $c=c\left(d\right)$ on the degree of elements required to generate finite subextensions of $k^{\left[d\right]}/k$. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$, but that one can take $c=d$ when $d$ is prime. This question was inspired by work of...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...