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Daciberg L. Gonçalves

Publications and source records attributed to Daciberg L. Gonçalves.

10 recordsLinked to original sources
arXiv2024

Sectional Category and the parametrized Borsuk-Ulam property

In this paper, for fibrations $p:E\to B$, $p':E'\to B$ over $B$ and a free involution $τ:E\to E$ which satisfy the equality $p\circτ=p$, we discover a connection between the sectional category of the double covers $q:E\to E/τ$ and $q^Y:F_{p'}(E',2)\to F_{p'}(E',2)/\mathbb{Z}_2$ from the $2$-ordered fibre-wise configuration space $F_{p'}(E',2)$ to its unordered quotient $ F_{p'}(E',2)/\mathbb{Z}_2$, and the parametrized Borsuk-Ulam property (PBUP) for the triple $\left((p,τ);p'\right)$. Explicitly, we demonstrate that the triple $\left((p,τ);p'\right)$ satisfies the PBUP if the sectional category of $q$ is bigger than the sectional category of $q^{p'}$. This property connects a standard problem in parametrized Borsuk-Ulam theory to current research trends in sectional category. As applictions of our results, we explore the PBUP for $E, E'$ one of the following fibrations: trivial fibration, Hopf fibration and the Fadell-Neuwirth fibration.

math.AT↗
arXiv2023

Borsuk-Ulam property and Sectional Category

For a Hausdorff space $X$, a free involution $τ:X\to X$ and a Hausdorff space $Y$, we discover a connection between the sectional category of the double covers $q:X\to X/τ$ and $q^Y:F(Y,2)\to D(Y,2)$ from the ordered configuration space $F(Y,2)$ to its unordered quotient $D(Y,2)=F(Y,2)/Σ_2$, and the Borsuk-Ulam property (BUP) for the triple $\left((X,τ);Y\right)$. Explicitly, we demonstrate that the triple $\left((X,τ);Y\right)$ satisfies the BUP if the sectional category of $q$ is bigger than the sectional category of $q^Y$. This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. As an application of our results, we show that the index of $(X,τ)$ coincides with the sectional category of the quotient map $q:X\to X/τ$ minus 1 for any paracompact space $X$. In addition, we present some new results relating Borsuk-Ulam theory and sectional category.

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arXiv2021

Strong surjections from two-complexes with odd order top-cohomology onto the projective plane

Given a finite and connected two-dimensional $CW$-complex $K$ with fundamental group $Π$ and second integer cohomology group $H^2(K;\mathbb{Z})$ finite of odd order, we prove that: (1) for each local integer coefficient system $α:Π\to{\rm Aut}(\mathbb{Z})$ over $K$, the corresponding twisted cohomology group $H^2(K;_α\!\mathbb{Z})$ is finite of odd order, we say order $\mathbb{C}^{\ast}(α)$, and there exists a natural function -- which resemble that one defined by the twisted degree -- from the set $[K;\mathbb{R}P^2]_α^{\ast}$ of the based homotopy classes of based maps inducing $α$ on $π_1$ into $H^2(K;_α\!\mathbb{Z})$, which is a bijection; (2) the set $[K;\mathbb{R}P^2]_α$ of the (free) homotopy classes of based maps inducing $α$ on $π_1$ is finite of order $\mathbb{C}(α)=(\mathbb{C}^{\ast}(α)+1)/2$; (3) all but one of the homotopy classes $[f]\in[K;\mathbb{R}P^2]_α$ are strongly surjective, and they are characterized by the non-nullity of the induced homomorphism $f^{\ast}:H^2(\mathbb{R}P^2;_{\varrho}\!\mathbb{Z})\to H^2(K;_α\!\mathbb{Z})$, where $\varrho$ is the nontrivial local integer coefficient system over the projective plane. Also some calculations of the groups $H^2(K;_α\!\mathbb{Z})$ are provided for several two-complexes $K$ and actions $α$, allowing to compare $H^2(K;\mathbb{Z})$ and $H^2(K;_α\!\mathbb{Z})$ for nontrivial $α$.

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arXiv2018

Deficient and multiple points of maps into a manifold

For a map $f:X \to M$ into a manifold $M$, we study the sets of deficient and multiple points of $f$. In case of the set of deficient points, we estimate its dimension. For multiple points, we study its density in $X$, and we also provide examples where its complement is dense.

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arXiv2015

Groups of PL-homeomorphisms admitting non-trivial invariant characters

We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.

math.GR↗
arXiv2013

Twisted conjugacy in Richard Thompson's group T

Let $f$ be an automorphism of a group $G$. Two elements $x, y$ in $G$ are said to be in the same $f$-twisted conjugacy class if there exists an element $z$ in $G$ such that $y=z x f(z^{-1})$. This is an equivalence relation known as $f$-twisted conjugacy. If the number $R(f)$ of $f$-twisted conjugacy classes is infinite for every automorphism $f$ of $G$ one says that $G$ has the $R_\infty$-property. We show that the Richard Thompson group $T$ has the $R_\infty$-property.

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arXiv2013

Nielsen coincidence theory of fibre-preserving maps and Dold's fixed point index

Let M to B, N to B be fibrations and f1,f2 :M to N be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair f1,f2 over B to a coincidence free pair of maps.In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our ω-invariant turns out to equal Dold's fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over B are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between S^1-bundles over S^1 as well as their Nielsen and Reidemeister numbers.

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arXiv2007

Twisted conjugacy classes in Symplectic groups, Mapping class groups and Braid groups(including an Appendix written with Francois Dahmani)

We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $ϕ$ of $G$, where $G$ is one of the above groups, the number of twisted $ϕ$-conjugacy classes is infinite.

math.GR↗
arXiv2007

Twisted conjugacy classes in R. Thompson's group F

In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers.

math.GR↗