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Daciberg Lima Gonçalves

Publications and source records attributed to Daciberg Lima Gonçalves.

At least 37 records · Page 2Linked to original sources
arXiv2018

Exponents of $[Ω(\mathbb S^{r+1}), Ω(Y)]$

We investigate the exponents of the total Cohen groups $[Ω(\mathbb S^{r+1}), Ω(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n+1})]$ and $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $Σ(n)/G$ for a homotopy $n$-sphere $Σ(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space $\mathbb{H}P^n$ for $1\le n\le \infty$.

math.AT↗
arXiv2018

Almost-crystallographic groups as quotients of Artin braid groups

Let $n, k \geq 3$. In this paper, we analyse the quotient group $B\_n/Γ\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup $Γ\_k(P\_n)$ belonging to the lower central series of the Artin pure braid group $P\_n$. We prove that it is an almost-crystallographic group. We then focus more specifically on the case $k=3$. If $n \geq 5$, and if $τ\in N$ is such that $gcd(τ, 6) = 1$, we show that $B\_n/Γ\_3 (P\_n)$ possesses torsion $τ$ if and only if $S\_n$ does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order $τ$ in $B\_n/Γ\_3 (P\_n)$ with those of elements of order $τ$ in the symmetric group $S\_n$. We also exhibit a presentation for the almost-crystallographic group $B\_n/Γ\_3 (P\_n)$. Finally, we obtain some $4$-dimensional almost-Bieberbach subgroups of $B\_3/Γ\_3 (P\_3)$, we explain how to obtain almost-Bieberbach subgroups of $B\_4/Γ\_3(P\_4)$ and $B\_3/Γ\_4(P\_3)$, and we exhibit explicit elements of order $5$ in $B\_5/Γ\_3 (P\_5)$.

math.GT↗
arXiv2017

The homotopy fibre of the inclusion $F\_n(M) \lhook\joinrel\longrightarrow \prod\_{1}^{n} M$ for $M$ either $\mathbb{S}^2$ or$\mathbb{R}P^2$ and orbit configuration spaces

Let $n\geq 1$, and let $ι\_{n}\colon\thinspace F\_{n}(M) \longrightarrow \prod\_{1}^{n} M$ be the natural inclusion of the $n$th configuration space of $M$ in the $n$-fold Cartesian product of $M$ with itself. In this paper, we study the map $ι\_{n}$, its homotopy fibre $I\_{n}$, and the induced homomorphisms $(ι\_{n})\_{#k}$ on the $k$th homotopy groups of $F\_{n}(M)$ and $\prod\_{1}^{n} M$ for $k\geq 1$ in the cases where $M$ is the $2$-sphere $\mathbb{S}^{2}$ or the real projective plane $\mathbb{R}P^{2}$. If $k\geq 2$, we show that the homomorphism $(ι\_{n})\_{#k}$ is injective and diagonal, with the exception of the case $n=k=2$ and $M=\mathbb{S}^{2}$, where it is anti-diagonal. We then show that $I\_{n}$ has the homotopy type of $K(R\_{n-1},1) \times Ω(\prod\_{1}^{n-1} \mathbb{S}^{2})$, where $R\_{n-1}$ is the $(n-1)$th Artin pure braid group if $M=\mathbb{S}^{2}$, and is the fundamental group $G\_{n-1}$ of the $(n-1)$th orbit configuration space of the open cylinder $\mathbb{S}^{2}\setminus \{\widetilde{z}\_{0}, -\widetilde{z}\_{0}\}$ with respect to the action of the antipodal map of $\mathbb{S}^{2}$ if $M=\mathbb{R}P^{2}$, where $\widetilde{z}\_{0}\in \mathbb{S}^{2}$. This enables us to describe the long exact sequence in homotopy of the homotopy fibration $I\_{n} \longrightarrow F\_n(M) \stackrel{ι\_{n}}{\longrightarrow} \prod\_{1}^{n} M$ in geometric terms, and notably the boundary homomorphism $π\_{k+1}(\prod\_{1}^{n} M)\longrightarrow π\_{k}(I\_{n})$. From this, if $M=\mathbb{R}P^{2}$ and $n\geq 2$, we show that $\ker{(ι\_{n})\_{#1}}$ is isomorphic to the quotient of $G\_{n-1}$ by its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order $2$ generated by the centre of $P\_{n}(\mathbb{R}P^{2})$ that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in a previous paper.

math.GT↗
arXiv2017

Fixed points of n-valued maps on surfaces and the Wecken property -- a configuration space approach

In this paper, we explore the fixed point theory of $n$-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp.\ the $2$-sphere ${\mathbb S}^{2}$) has the Wecken property for $n$-valued maps for all $n\in {\mathbb N}$ (resp.\ all $n\geq 3$). In the case $n=2$ and ${\mathbb S}^{2}$, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split $n$-valued map $ϕ\colon\thinspace X \multimap X$ of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering $q\colon\thinspace \widehat{X} \to X$ with a subset of the coordinate maps of a lift of the $n$-valued split map $ϕ\circ q\colon\thinspace \widehat{X} \multimap X$.

math.GT↗
arXiv2017

Fixed points of n-valued maps, the fixed point property and the case of surfaces -- a braid approach

We study the fixed point theory of n-valued maps of a space X using the fixed point theory of maps between X and its configuration spaces. We give some general results to decide whether an n-valued map can be deformed to a fixed point free n-valued map. In the case of surfaces, we provide an algebraic criterion in terms of the braid groups of X to study this problem. If X is either the k-dimensional ball or an even-dimensional real or complex projective space, we show that the fixed point property holds for n-valued maps for all n $\ge$ 1, and we prove the same result for even-dimensional spheres for all n $\ge$ 2. If X is the 2-torus, we classify the homotopy classes of 2-valued maps in terms of the braid groups of X. We do not currently have a complete characterisation of the homotopy classes of split 2-valued maps of the 2-torus that contain a fixed point free representative, but we give an infinite family of such homotopy classes.

math.GT↗
arXiv2017

Indecomposable branched coverings over the projective plane by surfaces $M$ with $χ(M) \leq 0$

In this work we study the decomposability property of branched coverings of degree $d$ odd, over the projective plane, where the covering surface has Euler characteristic $\leq 0$. The latter condition is equivalent to say that the defect of the covering is greater than $d$. We show that, given a datum $\mathscr{D}=\{D_{1},\dots,D_{s}\}$ with an even defect greater than $d$, it is realizable by an indecomposable branched covering over the projective plane. The case when $d$ is even is known.

math.GT↗
arXiv2016

Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces

In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman and Chillingworth and of Gonçalves and Guaschi to show that the mapping class group MCG(N ; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all compact, connected aspherical surfaces without boundary. Finally, if the genus of N is greater than or equal to 2, we give upper bounds for the virtual cohomological dimension of MCG(N ; k).

math.GT↗
arXiv2016

The cohomology ring of the sapphires that admit the Sol geometry

Let $G$ be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of $\mathbb{Z}$ over $\mathbb{Z}G$ and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings $H^*(G; A)$ for $A = \mathbb{Z}$ and $A = \mathbb{Z}/p$ for an odd prime $p$, and indicate how to compute the groups $H^*(G; A)$ and the multiplicative structure given by the cup product for any system of coefficients $A$.

math.AT↗
arXiv2016

On the homotopy fibre of the inclusion map F\_n(X) $\rightarrow$ $\prod$\_1^n X for some orbit spaces X

Under certain conditions, we describe the homotopy type of the homo-topy fibre of the inclusion map F\_n(X) $\rightarrow$ $\prod$\_1^n X for the n-th configuration space F\_n(X) of a topological manifold X without boundary such that dim(X) $\ge$ 3. We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space S^k/G of a tame, free action of a Lie group G on the k-sphere S^k. If the group G is finite and k is odd, we give a full description of the long exact sequence in homotopy of the homotopy fibration of the inclusion map F\_n(S^k/G) $\rightarrow$ $\prod$\_1^n S^k/G.

math.GT↗
arXiv2016

The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero

Let M and N be topological spaces such that M admits a free involution $\τ$. A homotopy class $β$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\τ$ if for every representative map f : M $\rightarrow$ N of $β$, there exists a point x $\in$ M such that f ($\τ$ (x)) = f (x). In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N , and of the fundamental groups of M and the orbit space of M with respect to the action of $\τ$. If M = N is either the 2-torus T^2 or the Klein bottle K^2 , we then solve the problem of deciding which homotopy classes of [M, M ] have the Borsuk-Ulam property. First, if $\τ$ : T^2 $\rightarrow$ T^2 is a free involution that preserves orientation, we show that no homotopy class of [T^2 , T^2 ] has the Borsuk-Ulam property with respect to $\τ$. Secondly, we prove that up to a certain equivalence relation, there is only one class of free involutions $\τ$ : T^2 $\rightarrow$ T^2 that reverse orientation, and for such involutions, we classify the homotopy classes in [T^2 , T^2 ] that have the Borsuk-Ulam property with respect to $\τ$ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $\τ$ : K^2 $\rightarrow$ K^2 is a free involution, then a homotopy class of [K^2 , K^2 ] has the Borsuk-Ulam property with respect to $\τ$ if and only if the given homotopy class lifts to the torus.

math.GT↗
arXiv2015

The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (π\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was later proved by C.Goldberg for compact surfaces without boundary different from the 2-sphere S^2 and the projective plane RP^2. In this paper, we prove the conjecture for S^2 and RP^2. In the case of RP^2, we prove that Ker(i\_\#) is equal to the commutator subgroup of P\_n(RP^2), we show that it may be decomposed in a manner similar to that of P\_n(S^2) as a direct sum of a torsion-free subgroup L\_n and the finite cyclic group generated by the full twist braid, and we prove that L\_n may be written as an iterated semi-direct product of free groups. Finally, we show that the groups B\_n(S^2) and P\_n(S^2) (resp. B\_n(RP^2) and P\_n(RP^2)) have finite virtual cohomological dimension equal to n-3 (resp. n-2), where B\_n(M) denotes the full n-string braid group of M. This allows us to determine the virtual cohomological dimension of the mapping class groups of the mapping class groups of S^2 and RP^2 with marked points, which in the case of S^2, reproves a result due to J.Harer.

math.GT↗
arXiv2015

Quotients of the Artin braid groups and crystallographic groups

Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class of any Abelian subgroup of odd order of S\_n is realised by a subgroup of B\_n/[P n,P\_n]. Finally, we discuss the realisation of non-Abelian subgroups of S\_n of odd order as subgroups of B\_n/[P n,P\_n], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in B\_n/[P n,P\_n] for all n greater than or equal to 7.

math.GR↗
arXiv2014

Involutions on sapphire Sol 3-manifolds and the Borsuk-Ulam theorem for maps into $R^n$

For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, τ; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $τ$ is a free involution, we show if $(M, τ; R^n)$ has the Borsuk-Ulam property or not. It is known that for $n>3$ the Borsuk-Ulam property does not hold independent of the involution, so we provide a classification when $n=2$ and $3$.

math.AT↗
arXiv2012

Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane

We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the 2-sphere S^2 or the real projective plane RP^2. We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction on X, and secondly when X consists of elements of finite order. This improves on results of Berrick and Matthey in the case of S^2, and extends them in the case of RP^2. We begin by recalling the situation for the Artin braid groups. As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for M=S^2 or RP^2, the induced action of B_n(M) on H_3 of the universal covering of the n th configuration space of M is trivial.

math.GT↗
arXiv2011

The classification of the virtually cyclic subgroups of the sphere braid groups

We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete classification. For small even values of n, the classification is complete up to an explicit finite number of open cases. In order to prove our main theorem, we obtain a number of other results of independent interest, notably the characterisation of the centralisers and normalisers of the finite cyclic and dicyclic subgroups of B_n(S^2), a result concerning conjugate powers of finite order elements, an analysis of the isomorphism classes of the amalgamated products that occur as subgroups of B_n(S^2), as well as an alternative proof of the fact that the universal covering space of the n-th configuration space of S^2 has the homotopy type of S^3 if n is greater than or equal to three.

math.GT↗
arXiv2010

The lower central and derived series of the braid groups of the projective plane

We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and derived series are known. If n>2 (resp. n>4), we show that the lower central (resp. derived) series of B_n(RP^2) is constant from the commutator subgroup onwards, and we exhibit a presentation of the commutator subgroup. In the exceptional cases n=3,4, we determine explicitly the complete derived series of B_3(RP^2), we calculate the derived series of B_4(RP^2) up to and including its fifth term, and we obtain many of the derived series quotients in these two cases.

math.GT↗