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Daciberg Lima Goncalves

Publications and source records attributed to Daciberg Lima Goncalves.

8 recordsLinked to original sources
arXiv2025

At most n-valued maps

This paper concerns various models of ``at-most-$n$-valued maps''. That is, multivalued maps $f:X\multimap Y$ for which $f(x)$ has cardinality at most $n$ for each $x$. We consider 4 classes of such maps which have appeared in the literature: $\mathcal U$, the set of exactly $n$-valued maps, or unions of such; $\mathcal F$, the set of $n$-fold maps defined by Crabb; $\mathcal S$, the set of symmetric product maps; and $\mathcal W$, the set of weighted maps with weights in $\mathbb N$. Our main result is roughly that these classes satisfy the following containments: \[ \mathcal U \subsetneq \mathcal F \subsetneq \mathcal S = \mathcal W \] Furthermore we define the general class $\mathcal C$ of all at-most-$n$-valued maps, and show that there are maps in $\mathcal C$ which are outside of any of the other classes above. We also describe a configuration-space point of view for the class $\mathcal C$, defining a configuration space $C_n(Y)$ such that any at-most-$n$-valued map $f:X\multimap Y$ corresponds naturally to a single-valued map $f:X\to C_n(Y)$. We give a full calculation of the fundamental group and homology groups of $C_n(S^1)$.

math.GN↗
arXiv2015

The $R_\infty$ property for nilpotent quotients of surface groups

It is well known that when $G$ is the fundamental group of a closed surface of negative Euler characteristic, it has the $R_{\infty}$ property. In this work we compute the least integer $c$, {\it called the $R_{\infty}$-nilpotency degree of $G$}, such that the group $G/ γ_{c+1}(G)$ has the $R_{\infty}$ property, where $γ_r(G)$ is the $r$-th term of the lower central series of $G$. We show that $c=4$ for $G$ the fundamental group of any orientable closed surface $S_g$ of genus $g>1$. For the fundamental group of the non-orientable surface $N_g$ (the connected sum of $g$ projective planes) this number is $2(g-1)$ (when $g>2$). A similar concept is introduced using the derived series $G^{(r)}$ of a group $G$. Namely {\it the $R_{\infty}$-solvability degree of $G$}, which is the least integer $c$ such that the group $G/G^{(c)}$ has the $R_{\infty}$ property. We show that the fundamental group of an orientable closed surface $S_g$ has $R_{\infty}$-solvability degree $2$.

math.GR↗
arXiv2014

Free and properly discontinuous actions of groups on homotopy $2n$-spheres

Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $Σ(2n)$ which has the homotopy type of the $2n$- sphere $\mathbb{S}^{2n}$. Then, this action induces an action of the group $G$ on the top cohomology of $Σ(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $Σ(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,Σ(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ which act as above on $Σ(2n)$. It turns out that they consist of free groups and certain semi-direct products $F\rtimes \mathbb{Z}_2$ with $F$ a free group. For those groups $G$ and a given action of $G$ on $\mbox{Aut}(\mathbb{Z})$, we present an algebraic criterion equivalent to the realizability of an action $G$ on $Σ(2n)$ which induces the given action on its top cohomology. Then, we obtain a classification of those groups together with actions on the top cohomology of $Σ(2n)$.

math.AT↗
arXiv2010

Decomposability problem on branched coverings

Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This shows that decomposable and indecomposable realizations may coexist. Moreover, we characterize the branch data of a decomposable primitive branched covering.

math.GT↗
arXiv2009

Cohomology of preimages with local coefficients

Let M,N and B\subset N be compact smooth manifolds of dimensions n+k,n and \ell, respectively. Given a map f from M to N, we give homological conditions under which g^{-1}(B) has nontrivial cohomology (with local coefficients) for any map g homotopic to f. We also show that a certain cohomology class in H^j(N,N-B) is Poincare dual (with local coefficients) under f^* to the image of a corresponding class in H_{n+k-j}(f^{-1}(B)) when f is transverse to B. This generalizes a similar formula of D Gottlieb in the case of simple coefficients.

math.GT↗
arXiv2006

Homotopy types of orbit spaces and their self-equivalences for the periodic groups Z/a \rtimes (Z/b x T^\star_n) and Z/a \rtimes (Z/b x O^\star_n)

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW-complex with the homotopy type of an n-sphere. We study the automorphism group Aut(G) to compute the number of distinct homotopy types of orbit spaces X(2dn-1)/μwith respect to free and cellular G-actions μon all CW-complexes X(2dn-1). At the end, the groups E(X(2dn-1)/μ) of self homotopy equivalences of orbit spaces X(2dn-1)/μassociated with free and cellular G-actions μon X(2dn-1) are determined.

math.AT↗
arXiv2004

The braid groups of the projective plane

Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP^2. In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the `full twist' braid. Our main results may be summarised as follows: first, the pure braid group short exact sequence 1 --> P_{m-n}(RP^2 - {x_1,...,x_n}) --> P_m(RP^2) --> P_n(RP^2) --> 1 does not split if m > 3 and n=2,3. Now let n > 1. Then in B_n(RP^2), there is a k-torsion element if and only if k divides either 4n or 4(n-1). Finally, the full twist braid has a k-th root if and only if k divides either 2n or 2(n-1).

math.AT↗