Finite-dimensional spaces in resolving classes

Jeffrey Strom

Displaying similar documents to “Finite-dimensional spaces in resolving classes”

Homotopy invariance of higher signatures and $3$ -manifold groups

Michel Matthey, Hervé Oyono-Oyono, Wolfgang Pitsch (2008)

Bulletin de la Société Mathématique de France

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For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable $3$ -manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable $3$ -manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients...

On the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun (2019)

Czechoslovak Mathematical Journal

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We show that the GVC (generalized vanishing conjecture) holds for the differential operator $Λ = (\partial_{x} - Φ (\partial_{y})) \partial_{y}$ and all polynomials $P (x, y)$ , where $Φ (t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

G-functors, G-posets and homotopy decompositions of G-spaces

Stefan Jackowski, Jolanta Słomińska (2001)

Fundamenta Mathematicae

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We describe a unifying approach to a variety of homotopy decompositions of classifying spaces, mainly of finite groups. For a group G acting on a poset W and an isotropy presheaf d:W → (G) we construct a natural G-map $h o c o l i m_{_{d}} G / d (-) \to | W |$ which is a (non-equivariant) homotopy equivalence, hence $h o c o l i m_{_{d}} E G \times_{G} F_{d} \to E G \times_{G} | W |$ is a homotopy equivalence. Different choices of G-posets and isotropy presheaves on them lead to homotopy decompositions of classifying spaces. We analyze higher limits over the categories associated to isotropy...

Minimal component numbers of fixed point sets

Xuezhi Zhao (2003)

Fundamenta Mathematicae

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Let f: (X,A) → (X,A) be a relative map of a pair of compact polyhedra. We introduce a new relative homotopy invariant $N^{C} (f; X, A)$ , which is a lower bound for the component numbers of fixed point sets of the self-maps in the relative homotopy class of f. Some properties of $N^{C} (f; X, A)$ are given, which are very similar to those of the relative Nielsen number N(f;X,A).

On weak $i$ -homotopy equivalences of modules

Zheng-Xu He (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si definisce il gruppo di $i$ —omotopia di un singolo modulo e si introduce la nozione di equivalenza $i$ -omotopica debole. Sotto determinate condizioni per l'anello di base $\Lambda$ oppure per i moduli considerati, le equivalenze $i$ -omotopiche deboli coincidono con le equivalenze $i$ -omotopiche (forti).

The strength of the projective Martin conjecture

C. T. Chong, Wei Wang, Liang Yu (2010)

Fundamenta Mathematicae

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We show that Martin’s conjecture on Π¹₁ functions uniformly $\leq_{T}$ -order preserving on a cone implies Π¹₁ Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant $Π ¹_{2 n + 1}$ functions is equivalent over ZFC to $Σ ¹_{2 n + 2}$ -Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π¹₁ functions implies the consistency of the existence of a Woodin cardinal.

A geometric construction for spectrally arbitrary sign pattern matrices and the $2 n$ -conjecture

Dipak Jadhav, Rajendra Deore (2023)

Czechoslovak Mathematical Journal

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We develop a geometric method for studying the spectral arbitrariness of a given sign pattern matrix. The method also provides a computational way of computing matrix realizations for a given characteristic polynomial. We also provide a partial answer to $2 n$ -conjecture. We determine that the $2 n$ -conjecture holds for the class of spectrally arbitrary patterns that have a column or row with at least $n - 1$ nonzero entries.

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

A decomposition of $E^{3}$ into straight arcs and singletons

S. Armentrout

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CONTENTS1. Introduction............................................................................................................................................. 52. Notation and. terminology................................................................................................................... 63. Description of G.................................................................................................................................... 64. Preliminaries to the...

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

Greenberg’s conjecture for $ℤ_{p}^{d}$ -extensions

Andrea Bandini (2003)

Acta Arithmetica

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On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta Słomińska (2001)

Fundamenta Mathematicae

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Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $(B_{p} (G)) / G$ of the classifying space of the category associated to the G-poset $_{p} (G)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{E ₙ} / (N E ₀ \cap . . . \cap N E ₙ)$ defined over the poset $(s d_{p} (G)) / G$ , where sd is the barycentric subdivision....

On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

Recent progress on the Jacobian Conjecture

Michiel de Bondt, Arno van den Essen (2005)

Annales Polonici Mathematici

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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x + {(A x)}^{* 3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f \in k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

On the connectivity of finite subset spaces

Jacob Mostovoy, Rustam Sadykov (2012)

Fundamenta Mathematicae

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We prove that the space $e x p_{k} ⋁ S^{m + 1}$ of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space $e x p_{k} X$ is (m+k-2)-connected for any m-connected cell complex X.

Order of the smallest counterexample to Gallai's conjecture

Fuyuan Chen (2018)

Czechoslovak Mathematical Journal

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In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai’s conjecture is a graph on 12 vertices. We prove that Gallai’s conjecture is true for every connected graph $G$ with $α^{'} (G) \leq 5$ , which implies that Zamfirescu’s conjecture is true.

On the homotopy transfer of $A_{\infty}$ structures

Jakub Kopřiva (2017)

Archivum Mathematicum

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The present article is devoted to the study of transfers for $A_{\infty}$ structures, their maps and homotopies, as developed in [7]. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.

Strong density for higher order Sobolev spaces into compact manifolds

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2015)

Journal of the European Mathematical Society

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Given a compact manifold $N^{n}$ , an integer $k \in ℕ_{*}$ and an exponent $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ of smooth maps on the cube with values into $N^{n}$ is dense with respect to the strong topology in the Sobolev space $W^{k, p} (Q^{m}; N^{n})$ when the homotopy group $π_{⌊ k p ⌋} (N^{n})$ of order $⌊ k p ⌋$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m - ⌊ k p ⌋ - 1$ , without any restriction on the homotopy group of $N^{n}$ .

Derived dimension via $τ$ -tilting theory

Yingying Zhang (2021)

Czechoslovak Mathematical Journal

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Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support $τ$ -tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given $τ$ -tilting module.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $q \geq 2$ and denote by $s_{q}$ the sum-of-digits function in base $q$ . For $j = 0, 1, \dots, q - 1$ consider $# {0 \leq n < N : s_{q} (2 n) \equiv j (mod q)} .$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N / q$ and, respectively, less than $N / q$ for infinitely many $N$ , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

Unstable homotopy invariance for finite fields

Kevin P. Knudson (2002)

Fundamenta Mathematicae

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We prove that if k is a finite field with $p^{d}$ elements, then the natural map $H_{i} (G L ₙ (k), ℤ) \to H_{i} (G L ₙ (k [t]), ℤ)$ is an isomorphism for 0 ≤ i < d(p-1) and for all n.

Non-Wieferich primes in number fields and $a b c$ -conjecture

Srinivas Kotyada, Subramani Muthukrishnan (2018)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be an algebraic number field of class number one and let $𝒪_{K}$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $𝒪_{K}$ under the assumption of the $a b c$ -conjecture for number fields.

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

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Given a set A ⊂ ℕ let $σ_{A} (n)$ denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, $σ_{A} (n)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and $σ_{A} (n ̅) \leq 5120$ for all n̅ ∈ ℤₘ.

Troesch complexes and extensions of strict polynomial functors

Antoine Touzé (2012)

Annales scientifiques de l'École Normale Supérieure

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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 $Ext$ -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...