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Let $n\geq 2$. Let $VB_n$ (resp. $VP_n$) denote the virtual braid group (resp. virtual pure braid group), let $WB_n$ (resp. $WP_n$) denote the welded braid group (resp. welded pure braid group) and let $UVB_n$ (resp. $UVP_n$) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for $n\geq 4$, the group $VP_n$ and for $n\geq 3$ the groups $WP_n$ and $UVP_n$ are characteristic subgroups of $VB_n$, $WB_n$ and $UVB_n$, respectively. In the second part of the paper we show that, for $n\geq 2$, the virtual braid group $VB_n$, the unrestricted virtual pure braid group $UVP_n$, and the unrestricted virtual braid group $UVB_n$ have the R$_\infty$-property. As a consequence of the technique used for few strings we also prove that, for $n=2,3,4$, the welded braid group $WB_n$ has the R$_\infty$-property and that for $n=2$ the corresponding pure braid groups have the R$_\infty$-property. On the other hand for $n\geq 3$ it is unknown if the R$_\infty$-property holds or not for the virtual pure braid group $VP_n$ and the welded pure braid group $WP_n$., Comment: 26 pages

Let $G$ be a finite group with order $|G|=\ell$ and $2\leq q\leq \ell$. For a free $G$-space $X$, we introduce a notion of $q$-th index of $(X,G)$, denoted by $\text{ind}_q(X,G)$. Our concept is relevant in the Borsuk-Ulam theory. We draw general estimates for the $q$-th index in terms of the sectional category of the quotient map $X\to X/G$, denoted by $\text{secat}(X\to X/G)$. This property connects a standard problem in Borsuk-Ulam theory to current research trends in sectional category. Under certain hypothesis we observed that $\text{secat}(X\to X/G)=\text{ind}_2(X,G)+1$. As an application of our results, we present new results in Borsuk-Ulam theory and sectional category., Comment: 15 pages. Comments are welcome

In this paper we characterize primitive branched coverings with minimal defect over the projective plane with respect to the properties decomposable and indecomposable. This minimality is achieved when the covering surface is also the projective plane, which corresponds to the last case to be solved. As a consequence, we have extended the family of realisations of branched coverings on the projective plane and established a type of generalisation of results on primitive permutation groups., Comment: 26 pages

Let $M$ and $N$ be fiber bundles over the same base $B$, where $M$ is endowed with a free involution $\tau$ over $B$. A homotopy class $\delta \in [M,N]_{B}$ (over $B$) is said to have the Borsuk-Ulam property with respect to $\tau$ if for every fiber-preserving map $f\colon M \to N$ over $B$ which represents $\delta$ there exists a point $x \in M$ such that $f(\tau(x)) = f(x)$. In the cases that $B$ is a $K(\pi ,1)$-space and the fibers of the projections $M \to B$ and $N \to B$ are $K(\pi,1)$ 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 $\tau$ 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 $\tau$ over $\mathbb{S}^1$.

For finite connected graphs $\Gamma$ and $G$, with $\Gamma$ admitting a free involution $\tau$, we characterize the based homotopy classes $\alpha\in[\Gamma,G]$ for which the Borsuk-Ulam property holds in the sense of Gon\c{c}alves, Guaschi and Casteluber-Laass, i.e., the homotopy classes $\alpha$ so that each of its representatives $f\in\alpha$ satisfies $f(x) = f(\tau\cdot x)$ for some $x\in\Gamma$. This is attained through a graph-braid-group perspective aided by the use of discrete Morse theory., Comment: 11 pages, 2 figures

Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $\tau \colon\thinspace G \times M \to M$ be a proper free action of $G$. In this paper, we define a Borsuk-Ulam-type property for homotopy classes of maps from $M$ to $N$ with respect to the pair $(G,\tau)$ that generalises the classical antipodal Borsuk-Ulam theorem of maps from the $n$-sphere $\mathbb{S}^n$ to $\mathbb{R}^n$. In the cases where $M$ is a finite pathwise-connected CW-complex, $G$ is a finite, non-trivial Abelian group, $\tau$ is a proper free cellular action, and $N$ is either $\mathbb{R}^2$ or a compact surface without boundary different of $\mathbb{S}^2$ and $\mathbb{RP}^2$, we give an algebraic criterion involving braid groups to decide whether a free homotopy class $\beta \in [M,N]$ has the Borsuk-Ulam property. As an application of this criterion, we consider the case where $M$ is a compact surface without boundary equipped with a free action $\tau$ of the finite cyclic group $\mathbb{Z}_n$. In terms of the orientability of the orbit space $M_\tau$ of $M$ by the action $\tau$, the value of $n$ modulo $4$ and a certain algebraic condition involving the first homology group of $M_\tau$, we are able to determine if the single homotopy class of maps from $M$ to $\mathbb{R}^2$ possesses the Borsuk-Ulam property with respect to $(\mathbb{Z}_n,\tau)$. Finally, we give some examples of surfaces on which the symmetric group acts, and for these cases, we obtain some partial results regarding the Borsuk-Ulam property for maps whose target is $\mathbb{R}^2$., Comment: 18 pages, 2 figures

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $\tau$ if for every representative map $f: M \to N$ of $\beta$, there exists a point $x \in M$ such that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy class of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to any free involution of $T^2$ for which the orbit space is $K^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$. This completes the analysis of the Borsuk-Ulam problem for the case $M=T^2$ and $N=K^2$, and for any free involution $\tau$ of $T^2$.

Let $M$ be a compact surface without boundary, and $n\geq 2$. We analyse the quotient group $B_n(M)/\Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $\Gamma_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is different from the $2$-sphere $\mathbb{S}^2$, we prove that $B_n(M)/\Gamma_2(P_n(M))$ is isomorphic rho $P_n(M)/\Gamma_2(P_n(M)) \rtimes_{\varphi} S_n$, and that $B_n(M)/\Gamma_2(P_n(M))$ is a crystallographic group if and only if $M$ is orientable. If $M$ is orientable, we prove a number of results regarding the structure of $B_n(M)/\Gamma_2(P_n(M))$. We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of $B_n(M)/\Gamma_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_n(M)/\Gamma_2(P_n(M))\to S_n$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups $\tilde{G}_{n,g}$ of $B_n(M)/\Gamma_2(P_n(M))$ of dimension $2ng$ and whose holonomy group is the finite cyclic group of order $n$, and if $\mathcal{X}_{n,g}$ is a flat manifold whose fundamental group is $\tilde{G}_{n,g}$, we prove that it is an orientable K\"ahler manifold that admits Anosov diffeomorphisms.

In this paper we prove that all pure Artin braid groups $P_n$ ($n\geq 3$) have the $R_\infty$ property. In order to obtain this result, we analyse the naturally induced morphism $\operatorname{\text{Aut}}(P_n) \to \operatorname{\text{Aut}}(\Gamma_2 (P_n)/\Gamma_3(P_n))$ which turns out to factor through a representation $\rho\colon S_{n+1} \to \operatorname{\text{Aut}}(\Gamma_2 (P_n)/\Gamma_3(P_n))$. We can then use representation theory of the symmetric groups to show that any automorphism $\alpha$ of $P_n$ acts on the free abelian group $\Gamma_2 (P_n)/\Gamma_3(P_n)$ via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number $R(\alpha)$ of $\alpha$ is $\infty$., Comment: 18 pages

For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \rangle$, with $k\geq1$ odd, we describe representatives for all free and based homotopy classes of maps from $K$ into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all $k\geq1$ odd, $H^2(K;mathbb{Z})=0$ and, for $k\geq3$ odd, there exist maps from $K$ into the real projective plane which are strongly surjective. For $k=1$, there is no such a strongly surjective map. 55N25, 57M20., Comment: The revised version has a more suitable title and some improvements on the language

Let $M$ be a topological space that admits a free involution $\tau$, and let $N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to have {\it the Borsuk-Ulam property with respect to $\tau$} if for every representative map $f: M \to N$ of $\beta$, there exists a point $x \in M$ such that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy classes of maps from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam property with respect to a free involution $\tau_1$ of $T^2$ for which the orbit space is $T^2$. Our results are given in terms of a certain family of homomorphisms involving the fundamental groups of $T^2$ and $K^2$.

Let $E$ be the fundamental group of a closed Sol 3-manifold. We describe the groups $Aut(E)$ and $Out(E)$. We first consider the case where $E$ is the fundamental group of a torus bundle, and then the case where $E$ is the fundamental group of a closed Sol 3-manifold which is not a torus bundle. The groups are described in terms of some iterated semi-direct products of well known groups, where one exception is for $Out(E)$ for some $E$'s, in which case $Out(E)$ is described as an extension and a presentation is given.

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

We investigate the exponents of the total Cohen groups $[\Omega(\mathbb S^{r+1}), \Omega(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[\Omega(\mathbb S^{r+1}), \Omega(\mathbb S^{2n+1})]$ and $[\Omega(\mathbb S^{r+1}), \Omega(\mathbb S^{2n})]$ 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 $\Sigma(n)/G$ for a homotopy $n$-sphere $\Sigma(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space $\mathbb{H}P^n$ for $1\le n\le \infty$., Comment: 20 pages

Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{\Gamma_l(P_n)\}_{l\in \mathbb{N}}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous results show that a necessary condition for such an embedding to exist is that $|G|$ is odd (resp. is relatively prime with $6$) if $k=2$ (resp. $k=3$), where $|G|$ denotes the order of $G$. We show that any finite group $G$ of odd order (resp. of order relatively prime with $6$) embeds in $B_{|G|}/\Gamma_2(P_{|G|})$ (resp. in $B_{|G|}/\Gamma_3(P_{|G|})$). The result in the case of $B_{|G|}/\Gamma_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $n < |G|$ and $k \in \{2, 3\}$. If $G$ is of the form $\mathbb{Z}_{p^r} \rtimes_{\theta} \mathbb{Z}_d$, where the action $\theta$ is injective, $p$ is an odd prime (resp. $p \geq 5$ is prime) $d$ is odd (resp. $d$ is relatively prime with $6$) and divides $p-1$, we show that $G$ embeds in $B_{p^r}/\Gamma_2(P_{p^r})$ (resp. in $B_{p^r}/\Gamma_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/\Gamma_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/\Gamma_2(P_9)$ of the two non-Abelian groups of order $27$, namely the semi-direct product $\mathbb{Z}_9 \rtimes \mathbb{Z}_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.

Let $n, k \geq 3$. In this paper, we analyse the quotient group $B\_n/\Gamma\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup $\Gamma\_k(P\_n)$ 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 $\tau \in N$ is such that $gcd(\tau, 6) = 1$, we show that $B\_n/\Gamma\_3 (P\_n)$ possesses torsion $\tau$ 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 $\tau$ in $B\_n/\Gamma\_3 (P\_n)$ with those of elements of order $\tau$ in the symmetric group $S\_n$. We also exhibit a presentation for the almost-crystallographic group $B\_n/\Gamma\_3 (P\_n)$. Finally, we obtain some $4$-dimensional almost-Bieberbach subgroups of $B\_3/\Gamma\_3 (P\_3)$, we explain how to obtain almost-Bieberbach subgroups of $B\_4/\Gamma\_3(P\_4)$ and $B\_3/\Gamma\_4(P\_3)$, and we exhibit explicit elements of order $5$ in $B\_5/\Gamma\_3 (P\_5)$.

Let $n\geq 1$, and let $\iota\_{n}\colon\thinspace F\_{n}(M) \longrightarrow \prod\_{1}^{n} M$ 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 $\iota\_{n}$, its homotopy fibre $I\_{n}$, and the induced homomorphisms $(\iota\_{n})\_{#k}$ 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 $(\iota\_{n})\_{#k}$ 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 $K(R\_{n-1},1) \times \Omega(\prod\_{1}^{n-1} \mathbb{S}^{2})$, 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 $I\_{n} \longrightarrow F\_n(M) \stackrel{\iota\_{n}}{\longrightarrow} \prod\_{1}^{n} M$ in geometric terms, and notably the boundary homomorphism $\pi\_{k+1}(\prod\_{1}^{n} M)\longrightarrow \pi\_{k}(I\_{n})$. From this, if $M=\mathbb{R}P^{2}$ and $n\geq 2$, we show that $\ker{(\iota\_{n})\_{#1}}$ 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.

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., Comment: 1 figure

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 $\phi\colon\thinspace X \multimap X$ 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 $\phi\circ q\colon\thinspace \widehat{X} \multimap X$., Comment: To appear in Science China Math

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.

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 Gon\c{c}alves 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).

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., Comment: 30 pages

Let M and N be topological spaces such that M admits a free involution $\\tau$. A homotopy class $\beta$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\\tau$ if for every representative map f : M $\rightarrow$ N of $\beta$, there exists a point x $\in$ M such that f ($\\tau$ (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 $\\tau$. 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 $\\tau$ : 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 $\\tau$. Secondly, we prove that up to a certain equivalence relation, there is only one class of free involutions $\\tau$ : 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 $\\tau$ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $\\tau$ : 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 $\\tau$ if and only if the given homotopy class lifts to the torus.

Let G be a virtually cyclic of the form (Z_a x Z_b) x Z or [Z_a x (Z b x Q_{2^i})] x Z. We compute the integral cohomology ring of G, and then obtain the periodicity of the Farell cohomology of these groups., Comment: 22 pages

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{} (\pi\_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.

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 $G/ \gamma_{c+1}(G)$ has the $R_{\infty}$ property, where $\gamma_r(G)$ 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 $2$.

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.

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 $A$., Comment: 14 pages

For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, \tau; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $\tau$ is a free involution, we show if $(M, \tau; R^n)$ 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 $3$., Comment: 17 pages

We construct finite free resolutions of Z over ZG, where G is the fundamental group of a surface distinct from S^2 and RP^2, and define diagonal approximations for these resolutions. We then procceed to give some possible applications that come with the knowledge of those maps., Comment: 12 pages

Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $\Sigma(2n)$ 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 $\Sigma(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $\Sigma(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,\Sigma(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ which act as above on $\Sigma(2n)$. 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 $\Sigma(2n)$ 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 $\Sigma(2n)$., Comment: 19 pages, submitted

Let $F$ be either a free nilpotent group of a given class and of finite rank or a free solvable group of a certain derived length and of finite rank. We show precisely which ones have the $R_{\infty}$ property. Finally, we also show that the free group of infinite rank does not have the $R_{\infty}$ property.

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., Comment: 18 pages

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., Comment: 96 pages, 11 figures

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., Comment: 35 pages

In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.

Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this paper, we formulate this property in terms of a relation in the 2-string braid group B_2(S) of S. If X is a compact, connected surface without boundary, we use this criterion to classify all triples (X, t, S) for which the Borsuk-Ulam property holds. We also consider various cases where X is not necessarily a surface without boundary, but has the property that \pi_1(X/t) is isomorphic to the fundamental group of such a surface. If S is different from the 2-sphere S^2 and the real projective plane RP^2, then we show that the Borsuk-Ulam property does not hold for (X, t, S) unless either \pi_1(X/t) is isomorphic to \pi_1(RP^2), or \pi_1(X/t) is isomorphic to the fundamental group of a compact, connected non-orientable surface of genus 2 or 3 and S is orientable. In the latter case, the veracity of the Borsuk-Ulam property depends further on the choice of involution t; we give a necessary and sufficient condition for it to hold in terms of the surjective homomorphism \pi_1(X/t)-->Z_2 induced by the double covering X-->X/t. The cases S=S^2,RP^2 are treated separately., Comment: 31 pages

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plane due to Bujalance, Cirre and Gamboa. Secondly, using the linearity of B_{n} due to Bigelow and Krammer, we show that the braid groups of compact, connected surfaces of low genus are linear., Comment: 17 pages

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., Comment: This is the version published by Algebraic & Geometric Topology on 4 October 2006

Let M be a compact, connected non-orientable surface without boundary and of genus g greater than or equal to 3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 --> P_m(M {x_1,...,x_n}) --> P_{n+m}(M) --> P_n(M) --> 1, where m,n are positive integers, and the homomorphism p*:P_{n+m}(M) --> P_n(M) corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p:F_{n+m}(M)} --> F_n(M) of configuration spaces, defined by p((x_1,...,x_n,..., x_{n+m}))= (x_1, ..., x_n). We show that p and p* admit a section if and only if n=1. Together with previous results, this completes the resolution of the splitting problem for surfaces pure braid groups., Comment: 17 pages

Let n\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in B_n(S^2), and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi's classification of the torsion elements of B_n(S^2), and explain how the finite subgroups of B_n(S^2) are related to this classification, as well as to the lower central and derived series of B_n(S^2)., Comment: 23 pages