Expositiones Mathematicae最新文献

Let $h$ be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair $(X,H)$, consisting of a connected space $X$ and an $h$-perfect normal subgroup $H$ of the fundamental group ${\pi }_1\left(X\right)$, an $h$-acyclic map $X\to X_H^{+h}$ inducing the quotient by $H$ on the fundamental group. We show that this map is terminal among the $h$-acyclic maps that kill a subgroup of $H$. When $h$ is an ordinary homology theory with coefficients in a commutative ring with unit $R$, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen’s plus construction. We also clarify the necessity to use a strongly $R$-perfect group $H$ in characteristic zero.

The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we establish a more general, “unified” nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory.

We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on $Z$, obtained from closed sets of the profinite completion $\hat{Z}$. We also discuss various applications to number theory.

We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\mathrm{PSL}}_n\left(q\right)$ is prime. We present heuristic arguments and computational evidence based on the Bateman–Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\mathrm{PSL}}_n\left(q\right)$, ${\mathrm{PSU}}_n\left(q\right)$ and ${\mathrm{PSp}}_{2n}\left(q\right)$ which arise in the work of Dixon and Zalesskii on linear groups of prime degree.

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A $Hom$-finite category is an additive category $A$ for which there is a commutative unital ring $k$, such that each $Hom$-set in $A$ is a finite length $k$-module. The aim of this note is to provide a proof that a $Hom$-finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.

Fix a unital $C^{\ast }$-algebra $A$, and write $A_{\mathrm{sa}}$ for the set of self-adjoint elements of $A$. Also, if $f:R\to ℂ$ is a continuous function, then write $f_A:A_{\mathrm{sa}}\to A$ for the operator function $a\mapsto f\left(a\right)$ defined via functional calculus. In this paper, we introduce and study a space $N{C}^k\left(R\right)$ of $C^k$ functions $f:R\to ℂ$ such that, no matter the choice of $A$, the operator function $f_A:A_{\mathrm{sa}}\to A$ is $k$-times continuously Fréchet differentiable. In other words, if $f\in N{C}^k\left(R\right)$, then $f$ “lifts” to a $C^k$ map $f_A:A_{\mathrm{sa}}\to A$, for any (possibly noncommutative) unital $C^{\ast }$-algebra $A$. For this reason, we call $N{C}^k\left(R\right)$ the space of noncommutative $C^k$ functions. Our proof that $f_A\in C^k(A_{\mathrm{sa}};A)$, which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estim