arXiv subjects

Weslem Liberato Silva

Publications and source records attributed to Weslem Liberato Silva.

5 recordsLinked to original sources

arXiv2026

Homotopy minimal periods for fiber maps on circle bundles over circle

Given a fiber bundle $Y\rightarrow M\xrightarrow{p}B$ and a fiber map $f: M\rightarrow M$ over $B$, we introduce the notion of fiberwise homotopy minimal periods, denoted by $H_BPer(f).$ This invariant records the periods that occur among representatives of the fiberwise homotopy class of $f.$ We investigate the case in which both the base and the fiber are circles. Up to isomorphism, the corresponding total spaces are the torus and the Klein bottle. Using Nielsen theory for fiber maps and Nielsen-type periodic numbers, we obtain a complete classification of $H_{S^1}Per(f)$ for fiber maps of the torus and the Klein bottle over $S^1$. In particular, we identify the exceptional cases in which some periods can be removed by fiberwise homotopy, including the case $H_{S^1}Per(f)=\mathbb{N}\setminus{2}.$

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arXiv2026

A Lefschetz type homomorphism for coincidence of several maps

Given $p$-maps $f_1, \cdots, f_p : X \to M,$ $p \geq 2,$ from an arbitrary topological space to an orientable closed connected $m$-manifold, in this paper we define a graded homomorphism $Λ_{f_1 \cdots f_p}: H(X) \to H(M^{p-1})$ of degree $-m(p-1)$ called by Lefschetz homomorphism. If the Lefschetz homomorphism is nontrivial then there is a point $x \in X$ such that $f_1(x) = \cdots = f_p(x).$ The Lefschetz homomorphism $Λ_{f_1 \cdots f_p}$ can be represented as a Knill-like trace.

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arXiv2024

A Nielsen type periodic number for maps over $B$

Let $Y \to E \stackrel{p}{\to} B$ be a fibration and let $f: E \to E$ be a fiber map over $B$. In this work, we study the geometric and algebraic Reidemeister classes of the iterates of $f$ and introduce a Nielsen-type periodic number over $B$, denoted by $N_B P_n(f)$. When $B$ is a point, then $N_B P_n(f)$ coincides with the classical Nielsen periodic number.

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arXiv2023

Computing the one-parameter Nielsen number for homotopies on the n-torus

Let $F: T^{n} \times I \to T^{n}$ be a homotopy on a n-dimensional torus. The main purpose of this paper is to present a formula for the one-parameter Nielsen number $N(F)$ of $F$ in terms of its induced homomorphism. If $L(F)$ is the one-parameter Lefschetz class of $F$ then $L(F)$ is given by $L(F) = \ N(F)α,$ for some $α\in H_{1}(π_{1}(T^{n}),\mathbb{Z}).$

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arXiv2023

The Borsuk-Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

Let $M$ and $N$ be fiber bundles over the same base $B$, where $M$ is endowed with a free involution $τ$ over $B$. A homotopy class $δ\in [M,N]_{B}$ (over $B$) is said to have the Borsuk-Ulam property with respect to $τ$ if for every fiber-preserving map $f\colon M \to N$ over $B$ which represents $δ$ there exists a point $x \in M$ such that $f(τ(x)) = f(x)$. In the cases that $B$ is a $K(π,1)$-space and the fibers of the projections $M \to B$ and $N \to B$ are $K(π,1)$ closed surfaces $S_M$ and $S_N$, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map $f\colon M \to N$ over $B$ has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of $M$, the orbit space of $M$ by $τ$ and a type of generalized braid groups of $N$ that we call parametrized braid groups. As an application, we determine the homotopy classes of self fiber-preserving maps of some 2-torus bundles over $\mathbb{S}^1$ that satisfy the Borsuk-Ulam property with respect to certain involutions $τ$ over $\mathbb{S}^1$.

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