arXiv subjects

John Guaschi

Publications and source records attributed to John Guaschi.

At least 19 recordsLinked to original sources

arXiv2025

Braid groups of the projective plane, mapping class groups of non-orientable surfaces and algebraic K-theory of their group rings

We describe the lower algebraic $K$-theory of the integral group ring of both the pure and full braid groups of the real projective plane $\mathbb{R}P^2$ with $3$ strings, as well as that of the integral group ring of the mapping class group of $\mathbb{R}P^2$ with $3$ marked points. In addition, we give a general formula for the algebraic $K$-theory groups of the group ring of the mapping class group of non-orientable surfaces with k marked points, where $k \geq 3$.

math.GT↗

arXiv2025

The Splitting of Generalisations of the Fadell-Neuwirth short exact sequence

We study some generalisations to mixed braid groups of the Fadell-Neuwirth short exact sequence and the possible splitting of this sequence. In certain cases, we determine conditions under which the projection from the mixed braid group $B_{n_{1},\ldots,n_{k}}(M)$ to $B_{n_{1},\ldots, n_{k-q}}(M)$ admits a section, where $M$ is either the torus or the Klein bottle, $n_{1}, \ldots, n_{k},q \in \mathbb{N}$, and $1\leq q \leq k-1$. For $k\geq 2$ and $q=k-1$, we show that this projection admits a section if and only if $n_{1}$ divides $n_{i}$ for all $i=2,\ldots, k$. We present some partial conclusions in the case $k\geq 3$ and $q=1$. To obtain our results, we compute and make use of suitable mixed braid groups of $M$, as well as certain key quotients that play a central r\^{o}le in our analysis.

math.GT↗

arXiv2022

Free cyclic actions on surfaces and the Borsuk-Ulam theorem

Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $τ\colon\thinspace G \times M \to M$ 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 $(G,τ)$ 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, $τ$ 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 $β\in [M,N]$ 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 $τ$ of the finite cyclic group $\mathbb{Z}_n$. In terms of the orientability of the orbit space $M_τ$ of $M$ by the action $τ$, the value of $n$ modulo $4$ and a certain algebraic condition involving the first homology group of $M_τ$, 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 $(\mathbb{Z}_n,τ)$. 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 $\mathbb{R}^2$.

math.GT↗

arXiv2022

Unrestricted virtual braids and crystallographic braid groups

We show that the crystallographic braid group $B_n/[P_n,P_n]$ embeds naturally in the group of unrestricted virtual braids $UVB_n$, we give new proofs of known results about the torsion elements of $B_n/[P_n,P_n]$, and we characterise the torsion elements of $UVB_n$.

math.GR↗

arXiv2021

The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle -- part 2

Let $M$ be a topological space that admits a free involution $τ$, and let $N$ be a topological space. A homotopy class $β\in [ M,N ]$ is said to have the Borsuk-Ulam property with respect to $τ$ if for every representative map $f: M \to N$ of $β$, there exists a point $x \in M$ such that $f(τ(x))= f(x)$. 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 $τ$ of $T^2$.

math.GT↗

arXiv2021

Crystallographic groups and flat manifolds from surface braid groups

Let $M$ be a compact surface without boundary, and $n\geq 2$. We analyse the quotient group $B_n(M)/Γ_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $Γ_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is different from the $2$-sphere $\mathbb{S}^2$, we prove that $B_n(M)/Γ_2(P_n(M))$ is isomorphic rho $P_n(M)/Γ_2(P_n(M)) \rtimes_φ S_n$, and that $B_n(M)/Γ_2(P_n(M))$ 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 $B_n(M)/Γ_2(P_n(M))$. 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 $B_n(M)/Γ_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_n(M)/Γ_2(P_n(M))\to S_n$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups $\tilde{G}_{n,g}$ of $B_n(M)/Γ_2(P_n(M))$ 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 Kähler manifold that admits Anosov diffeomorphisms.

math.GR↗

arXiv2019

The Borsuk-Ulam property for homotopy classes of maps between the torus and the Klein bottle

Let $M$ be a topological space that admits a free involution $τ$, and let $N$ be a topological space. A homotopy class $β\in [ M,N ]$ is said to have {\it the Borsuk-Ulam property with respect to $τ$} if for every representative map $f: M \to N$ of $β$, there exists a point $x \in M$ such that $f(τ(x))= f(x)$. 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 $τ_1$ 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 $K^2$.

math.GT↗

arXiv2018

Embeddings of finite groups in $B_n/Γ_k(P_n)$ for $k=2, 3$

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 $B_n/Γ_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{Γ_l(P_n)\}_{l\in \mathbb{N}}$ 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 $B_{|G|}/Γ_2(P_{|G|})$ (resp. in $B_{|G|}/Γ_3(P_{|G|})$). The result in the case of $B_{|G|}/Γ_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/Γ_k(P_n)$, where $n < |G|$ and $k \in \{2, 3\}$. If $G$ is of the form $\mathbb{Z}_{p^r} \rtimes_θ \mathbb{Z}_d$, where the action $θ$ 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 $B_{p^r}/Γ_2(P_{p^r})$ (resp. in $B_{p^r}/Γ_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/Γ_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/Γ_2(P_9)$ 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 $3$.

math.GT↗

arXiv2018

Lower central series, surface braid groups, surjections and permutations

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.

math.GT↗

arXiv2018

The lower algebraic $K$-theory of virtually cyclic subgroups of the braid groups of the sphere and of $\mathbb{Z}[B\_4(\mathbb{S}^2)]$

We study $K$-theoretical aspects of the braid groups $B\_n(\mathbb{S}^{2})$ on $n$ strings of the $2$-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture~\cite{JM}. In light of this, in order to determine the algebraic $K$-theory of the group ring $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$, one should first compute that of its virtually cyclic subgroups, which were classified by D.~L.~Gon{\c c}alves and the first author. We calculate the Whitehead and $K\_{-1}$-groups of the group rings of the finite subgroups (dicyclic and binary polyhedral) of $B\_n(\mathbb{S}^{2})$ for all $4\leq n\leq 11$. Some new phenomena occur, such as the appearance of torsion for the $K\_{-1}$-groups. We then go on to study the case $n=4$ in detail, which is the smallest value of $n$ for which $B\_n(\mathbb{S}^{2})$ is infinite. We show that $B\_n(\mathbb{S}^{2})$ is an amalgamated product of two finite groups, from which we are able to determine a universal space for proper actions of the group $B\_n(\mathbb{S}^{2})$. We also calculate the algebraic $K$-theory of the infinite virtually cyclic subgroups of $B\_n(\mathbb{S}^{2})$, including the Nil groups of the quaternion group of order $8$. This enables us to determine the lower algebraic $K$-theory of $\mathbb{Z}[B\_n(\mathbb{S}^{2})]$.

math.KT↗

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↗

arXiv2018

The lower central and derived series of the braid groups of compact surfaces

Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B\_n(M) and P\_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P\_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P\_n(K). Finally, if M is a non-orientable compact surface without boundary, we determine the values of n for which B\_n(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

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↗

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

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↗