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Daciberg Gonçalves

Publications and source records attributed to Daciberg Gonçalves.

12 recordsLinked to original sources
arXiv2026

Fixed point free homeomorphisms and the $R_{\infty}$-property

Let $f:M\to M$ be a diffeomorphism on a compact connected smooth manifold of dimension at least $5$. It is known that the Nielsen number $N(f)$ of $f$ vanishes if and only if $f$ is isotopic to a fixed point free map. For any $n\ge 5$, there exists a compact $n$-dimensional nilmanifold $M$ such that {\it every} self homeomorphism on $M$ is isotopic to a fixed point free map. The proof depends on the fact that nilmanifolds are of {\it Jiang-type} and the fundamental group $π_1(M)$ has the property $R_{\infty}$ for certain nilmanifolds $M$. In this paper we construct the first known examples (an infinite family) of non-aspherical closed manifolds whose fundamental groups have property $R_{\infty}$ and that are also of Jiang-type. In particular, every self homeomorphism on such a manifold is isotopic to be fixed point free. The main objective of this work is to construct non-aspherical manifolds whose fundamental groups have property $R_{\infty}$ and that are also of Jiang-type.

math.AT↗
arXiv2018

Coincidence Wecken property for nilmanifolds

Let $f,g:X\to Y$ be maps from a compact infra-nilmanifold $X$ to a compact nilmanifold $Y$ with $\dim X\ge \dim Y$. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number $N(f,g)$ vanishes then $f$ and $g$ are deformable to be coincidence free. We also show that if $X$ is a connected finite complex $X$ and the Reidemeister coincidence number $R(f,g)=\infty$ then $f\sim f'$ so that $C(f',g)=\{x\in X \mid f'(x)=g(x)\}$ is empty.

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arXiv2017

Mapping degrees between spherical $3$-manifolds

Let $D(M,N)$ be the set of integers that can be realized as the degree of a map between two closed connected orientable manifolds $M$ and $N$ of the same dimension. For closed $3$-manifolds with $S^3$-geometry $M$ and $N$, every such degree $deg f\equiv \overline{deg}ψ$ $(|π_1(N)|)$ where $0\le \overline{deg}ψ<|π_1(N)|$ and $\overline{deg}ψ$ only depends on the induced homomorphism $ψ=f_π$ on the fundamental group. In this paper, we calculate explicitly the set $\{\overline{deg}ψ\}$ when $ψ$ is surjective and then we show how to determine $\overline{deg}(ψ)$ for arbitrary homomorphisms. This leads to the determination of the set $D(M,N)$.

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arXiv2017

On groups where the twisted conjugacy class of the unit element is a subgroup

We study groups $G$ where the $φ$-conjugacy class $[e]_φ=\{g^{-1}φ(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $φ$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.

math.GR↗
arXiv2014

Sigma theory and twisted conjugacy-II: Houghton groups and pure symmetric automorphism groups

We say that $x,y\in Γ$ are in the same $ϕ$-twisted conjugacy class and write $x\sim_ϕy$ if there exists an element $γ\in Γ$ such that $y=γxϕ(γ^{-1})$. This is an equivalence relation on $Γ$ called the $ϕ$-twisted conjugacy. Let $R(ϕ)$ denote the number of $ϕ$-twisted conjugacy classes in $Γ$. If $R(ϕ)$ is infinite for all $ϕ\in Aut(Γ)$, we say that $Γ$ has the $R_\infty$-property. The purpose of this note is to show that the symmetric group $S_\infty$, the Houghton groups and the pure symmetric automorphism groups have the $R_\infty$-property. We show, also, that the Richard Thompson group $T$ has the $R_\infty$-property. We obtain a general result establishing the $R_\infty$-property of finite direct product of finitely generated groups.

math.GR↗
arXiv2014

On the group structure of $[Ω\mathbb S^2, ΩY]$

Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),ΩY]\cong \lim\limits_{\leftarrow} [J_n(X),ΩY]$ for any space $Y$. When $X=\mathbb S^1$, the circle, $J(\mathbb S^1)=ΩΣ\mathbb S^1=Ω\mathbb S^2$. Furthermore, there is a bijection between $[J(\mathbb S^1),ΩY]$ and the product $\prod_{i=2}^\infty π_i(Y)$, as sets. In this paper, we describe the group structure of $[J_n(\mathbb S^1),ΩY]$ by determining the co-multiplication structure on the suspension $ΣJ_n(\mathbb S^1)$.

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arXiv2014

The $R_{\infty} property for abelian groups

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them. In fact we construct an uncountable number of infinite countable abelian groups which do have the $R_{\infty}$ property. We also construct an abelian group such that the cardinality of the Reidemeister classes is uncountable for any automorphism of that group. 8 pages, no figures

math.GR↗
arXiv2011

Twisted conjugacy classes for polyfree groups

Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.

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arXiv2008

Twisted conjugacy classes in nilpotent groups

A group is said to have the $R_\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\infty$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_\infty$ property. The $R_{\infty}$ property for virtually abelian and for $\mathcal C$-nilpotent groups are also discussed.

math.GR↗
arXiv2007

On Fox spaces and Jacobi identities

In 1945, R. Fox introduced the so-called Fox torus homotopy groups in which the usual homotopy groups are embedded and their Whitehead products are expressed as commutators. A modern treatment of Fox torus homotopy groups and their generalization has been given and studied. In this note, we further explore these groups and their properties. We discuss co-multiplications on Fox spaces and a Jacobi identity for the generalized Whitehead products and the $Γ$-Whitehead products.

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