arXiv subjects

Carolina de Miranda e Pereiro

Publications and source records attributed to Carolina de Miranda e Pereiro.

3 recordsLinked to original sources

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 rôle in our analysis.

math.GT↗

arXiv2023

The BNS invariants of the braid groups and pure braid groups of some surfaces

We compute and explicitly describe the Bieri-Neumann-Strebel invariants $Σ^1$ for the full and pure braid groups of the sphere $\mathbb{S}^2$, the real projective plane $\mathbb{R}P^2$ and specially the torus $\mathbb{T}$ and the Klein bottle $\mathbb{K}$. In order to do this for $M=\mathbb T$ or $M=\mathbb K$, and $n \geq 2$, we use the $n^{th}$-configuration space of $M$ to show that the action by homeomorphisms of the group $Out(P_n(M))$ on the character sphere $S(P_n(M))$ contains certain permutation of coordinates, under which $Σ^1(P_n(\mathbb T))^c$ and $Σ^1(P_n(\mathbb K))^c$ are invariant. Furthermore, $Σ^1(P_n(\mathbb T))^c$ and $Σ^1(P_n(\mathbb{S}^2))^c$ (the latter with $n \geq 5$) are finite unions of pairwise disjoint circles, and $Σ^1(P_n(\mathbb K))^c$ is finite. This last fact implies that there is a normal finite index subgroup $H \leq Aut(P_n(\mathbb K))$ such that the Reidemeister number $R(φ)$ is infinite for every $φ\in H$.

math.AT↗

arXiv2023

The Borsuk-Ulam theorem for n-valued maps between surfaces

In this work we analysed the validity of a type of Borsuk-Ulam theorem for multimaps between surfaces. We developed an algebraic technique involving braid groups to study this problem for $n$-valued maps. As a first application we described when the Borsuk-Ulam theorem holds for splits and non-splits multimaps $ϕ\colon X \multimap Y$ in the following two cases: $(i)$ $X$ is the $2$-sphere eqquiped with the antipodal involution and $Y$ is either a closed surface or the Euclidean plane; $(ii)$ $X$ is a closed surface different of the $2$-sphere eqquiped with a free involution $τ$ and $Y$ is the Euclidean plane. The results are exhaustive and in the case $(ii)$ are described in terms of an algebraic condition involving the first integral homology group of the orbit space $X / τ$.

math.AT↗