# The Borsuk-Ulam Theorem
for $n$-valued maps between surfaces

Vinicius Casteluber Laass  VINICIUS CASTELUBER LAAASS
Universidade Federal da Bahia, IME, Departamento de Matemática
Av. Milton Santos, S/N Ondina CEP: 40170-110, Salvador-BA, Brazil.
<EMAIL_ADDRESS>and  Carolina de Miranda e Pereiro  CAROLINA DE MIRANDA
E PEREIRO
Universidade Federal do Espírito Santo, UFES, Departamento de Matemática
29075-910, Vitória, Espírito Santo, Brazil<EMAIL_ADDRESS>
(Date: 27th August 2024)

###### Abstract.

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 $\phi\colon X\multimap Y$ 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 $\tau$ 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 $X/\tau$.

††footnotetext: _2020 Mathematics Subject Classification_. Primary: 55M20,
57M07; Secondary: 20F36.
_Key Words and phrases_. Borsuk-Ulam theorem, surfaces, multifunction, braid
groups

##  1\. Introduction

The celebrated Borsuk-Ulam theorem states that for any continuous map
$f\colon\mathbb{S}^{m}\rightarrow\mathbb{R}^{m}$ there exists a point
$x\in\mathbb{S}^{m}$ such that $f(x)=f(-x)$ [6, Satz II]. More information
about the history, some generalisations and applications of these result may
be found for example in the book of Matoušek [29], as well as in the paper
[35] of Steinlein published in 1985, which was the most complete survey of
this subject at that time.

In 1983, H. Schirmer proved a version of the Borsuk-Ulam theorem for a family
of $n$-valued maps $\phi\colon X\multimap\mathbb{R}^{m}$, where $X$ is a
compact metric space, simply connected and has the homology of
$\mathbb{S}^{m}$ (see [33, Theorem 2 and Corollary]). Others authors were also
interested in discuss antipodes problems for multivalued maps and there are
some results in this direction as we can see in [23] and [26] for example. On
the other hand, the fixed point problem for multivalued maps between surfaces
were studied in a recent work of D. L. Gonçalves and J. Guaschi using a braid
approach (see [21]).

Motivated by the previous works, the main goal of this paper is to establish a
version of the Borsuk-Ulam theorem for $n$-valued maps between certain
surfaces and we develop a new perspective using configuration spaces and braid
groups as tools. Before to enunciate more precisely the results, we recall
some definitions and properties of multivalued maps.

A multivalued function (or a multifunction) $\phi\colon X\multimap Y$ is a
correspondence that associates to each $x\in X$ a non-empty subset $\phi(x)$
of $Y$. Following [21] a multifunction $\phi$ is _upper semi-continous_ if for
all $x\in X$, $\phi(x)$ is closed, and given an open set $V$ in $Y$, the set
$\left\\{x\in X\,:\,\phi(x)\subset V\right\\}$ is open in $X$, and $\phi$ is
_lower semi-continuous_ if the set $\left\\{x\in X\,:\,\phi(x)\cap
V\neq\emptyset\right\\}$ is open in $X$, and is _continous_ if it is upper
semi-continuous and lower semi-continuous. An _$n$ -valued map_ (or multimap)
$\phi:X\multimap Y$ is a continuous multifuction that to each $x\in X$ assigns
a unordered subset of $Y$ of cardinal exactly $n$. Schirmer [34] defined that
a $n$-valued map $\phi:X\multimap Y$ is _split_ if there exists single-valued
maps $f_{1},\ldots,f_{n}:X\rightarrow Y$ such that
$\phi(x)=\left\\{f_{1}(x),\ldots,f_{n}(x)\right\\}$ for all $x\in X$. In
particular, all split $n$-valued maps are continuous if $Y$ is an Hausdorff
topological space [21, Propositon 42].

A convenient way of dealing with $n$-valued function is making use of
configuration spaces, as explored by R. F. Brown and D. L. Gonçalves [7] and
also by the latter author and J. Guaschi [21], and we describe in the
following. Given a topological space $Y$, we recall that

$F_{n}(Y)=\left\\{(y_{1},\ldots,y_{n})\in Y^{n}\,:\,y_{i}\neq
y_{j}\,\,\mbox{if}\,\,i\neq j\right\\}$

is the _$n^{th}$ ordered configuration space_ of $Y$. The symmetric group
$S_{n}$ on $n$ elements acts freely on $F_{n}(Y)$ by permuting the
coordinates. The corresponding quotient space is denoted by
$D_{n}(Y)=F_{n}(Y)/S_{n}$ and it is called the _$n^{th}$ unordered
configuration space_ of $Y$.

There is a natural correspondence between a $n$-valued map $\phi\colon
X\multimap Y$ and a single map $\Phi\colon X\rightarrow D_{n}(Y)$, if we
consider $\Phi(x)=\phi(x)$. In the particular case where $X$ and $Y$ are
metric spaces, it is helpful to use this correspondence because the multimap
$\phi$ is continuous if, and only if, the single-valued map $\Phi$ is also
continuous by [21, Theorem 8].

Now, we are able to define the Borsuk-Ulam property for multivalued maps. Let
$X$ and $Y$ be topological spaces such that $X$ admits a free involution
$\tau$. Given a multivalued map $\phi\colon X\multimap Y$, we define a
_Borsuk-Ulam coincidence of $\phi$ with respect to $\tau$_ as a pair
$\left\\{x,\tau(x)\right\\}$ of points of $X$ such that
$\phi(x)\cap\phi(\tau(x))\neq\emptyset$. The behavior of the $n$-valued maps
is divided between the distinct classes splits and non-splits and since we are
interested in classify these maps in relation to Borsuk-Ulam coincidences, we
define that the triple $(X,\tau;Y)$ has the _$n$ -split Borsuk-Ulam property_
(resp.  _$n$ -non-split Borsuk-Ulam property_) if every split (resp. non-
split) $n$-valued map from $X$ to $Y$ has a Borsuk-Ulam coincidence with
respect to $\tau$. Finally we say that the triple $(X,\tau;Y)$ has the _$n$
-Borsuk-Ulam property_ if it has the $n$-split and $n$-non-split Borsuk-Ulam
property. In the remaining of the text, we abbreviate Borsuk-Ulam property by
BUP.

Note that all $1$-valued maps are split and the above definition coincides
with the BUP for single-valued maps in the same sense that were used in the
works [9, 30], for example. In the particular cases where $X$ is a closed
surface and $Y$ is either a closed surface or the Euclidean plane, the
complete classification of the triples $(X,\tau;Y)$ that possess the $1$-BUP
can be found in [16, 20].

Some results about Borsuk-Ulam coincidences for $n$-valued maps are
straightforward consequence from the single-valued case. For example, if the
triple $(X,\tau;Y)$ has the $1$-BUP, then any split $n$-valued map
$\phi=\left\\{f_{1},\ldots,f_{n}\right\\}\colon X\multimap Y$ has Borsuk-Ulam
coincidence, for all $n$. This follows with the same proof given by H.
Schirmer [33, Theorem 2]. Nevertheless, we can not conclude the same for non-
split $n$-valued maps, as we shall see in Theorem 2, where we have a triple
$(X,\tau;Y)$ that satisfies the $1$-BUP but does not satifies the $2$-BUP, for
example. On the other hand, if $\tau$ is the antipodal map on the $2$-sphere,
then the triple $(\mathbb{S}^{2},\tau;\mathbb{S}^{2})$ is an example that does
not satisfies the $1$-BUP although it satisfies de $n$-BUP for all $n\geq 2$,
which is one of the consequences of the following main theorem.

######  Theorem 1.

Let $n$ be a positive integer, let
$\tau\colon\mathbb{S}^{2}\longrightarrow\mathbb{S}^{2}$ be the antipodal map
and let $Y$ be $\mathbb{R}^{2}$ or a closed surface. The triple
$(\mathbb{S}^{2},\tau;Y)$ has the $1$-BUP if, and only if,
$Y\neq\mathbb{S}^{2}$ and it has the $n$-BUP for all $n\geq 2$.

The above result is a complete classification due the fact that the antipodal
map is essentially the only free involution on $\mathbb{S}^{2}$ (see for
example [4, Theorem 1]).

Besides the theorem above, we also give a full classification of all triples
$(X,\tau,\mathbb{R}^{2})$ in relation to BUP, where $X$ is a closed surface.
Before to state the other main result of this paper let us describe some facts
and notation. Suppose that $X$ is a pathwise connected $CW$-complex equipped
with a free cellular involution $\tau$. We say that two points $x,y\in X$ are
equivalent if $y\in\\{x,\tau(x)\\}$. We denote the correspondent quotient
space by $X_{\tau}$, which we also call the orbit space, and the natural
projection by $p_{\tau}\colon X\longrightarrow X_{\tau}$. Note that $X_{\tau}$
is also a pathwise $CW$-complex and $p_{\tau}$ is a double covering map.
Therefore we have the following short exact sequence:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(p_{\tau})_{\\#}}$$\textstyle{\pi_{1}(X_{\tau})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\tau}}$$\textstyle{\dfrac{\pi_{1}(X_{\tau})}{(p_{\tau})_{\\#}\left(\pi_{1}(X)\right)}\cong\mathbb{Z}_{2}=\\{\overline{0},\overline{1}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$
(1.1)

where $\theta_{\tau}$ is the natural projection. The homomorphism
$\theta_{\tau}$ has a factorization to the abelianization of
$\pi_{1}(X_{\tau})$ which is the $1$-homology group of $X_{\tau}$. We denote
this factorization by $\hat{\theta_{\tau}}\colon
H_{1}(X_{\tau})\longrightarrow\mathbb{Z}_{2}$. If $X_{\tau}$ is a closed non-
orientable surface, we denote by $\hat{\delta}$ the only non-trivial element
of order two of $H_{1}(X_{\tau})$. We obtain the next result that extends the
$1$-BUP case due to D. L. Gonçalves [16].

######  Theorem 2.

Let $n\geq 1$ and $X$ be a closed surface different of $\mathbb{S}^{2}$ which
admits a free involution $\tau\colon X\rightarrow X$. The following conditions
are true:

  1. (1)

If $X_{\tau}$ is a closed orientable surface or $X_{\tau}$ is a closed non-
orientable surface and $\hat{\theta_{\tau}}(\hat{\delta})=\overline{0}$, then
the triple $(X,\tau;\mathbb{R}^{2})$ does not have the $n$-split BUP and it
does not have the $n$-non-split BUP.

  2. (2)

If $X_{\tau}$ is a closed non-orientable surface,
$\hat{\theta_{\tau}}(\hat{\delta})=\overline{1}$ and $n$ is even, then the
triple $(X,\tau;\mathbb{R}^{2})$ has the $n$-split BUP, but it does not have
the $n$-non-split BUP.

  3. (3)

If $X_{\tau}$ is a closed non-orientable surface,
$\hat{\theta_{\tau}}(\hat{\delta})=\overline{1}$ and $n$ is odd, then the
triple $(X,\tau;\mathbb{R}^{2})$ has the $n$-BUP.

We emphasize to the reader that the above result is complete in the sense that
it is possible to know when a triple $(X,\tau;\mathbb{R}^{2})$ has (or not)
the $n$-BUP. Also, we analyzed the behavior of $n$-split and $n$-non-split
multimaps in relation to BUP separately.

We also obtain partial results about the classfication of the triples
$(X,\tau;Y)$ in relation to the $n$-BUP, where $X$ and $Y$ are both closed
surfaces, $X\neq\mathbb{S}^{2}$. These results are rather straightforward and
it is a consequence of Theorem 2 and the classification of the triples in
relation to $1$-BUP given in [20]. We enunciate and prove these examples in
Section 6. The study to obtain a full classification of the above mentioned
triples in relation to $n$-split BUP and $n$-non-split BUP is a work in
progress.

Besides the introduction and Section 6, the manuscript is organised as
follows. In Section 2 we give a brief resume of the braid groups of surfaces.
We define some important subgroups which is usefull to obtain Theorem 3. This
result gives a necessary and suficient algebraic condition to study the BUP
for $n$-valued maps whose domain is a CW-complex eqquiped with a free cellular
involution and the target is a closed surface. In Section 3 we derive some
properties for the subgroups mentioned above when the surface is the Euclidean
plane which is usefull to prove Theorem 2. Section 4 of the paper is devoted
to proving Theorem 1 and the proof will follow from Propositions 9–11. We
prove Theorem 2 at Section 5 and the proof will follow from Propositions
14–16.

##  2\. Preliminaries and generalities

Let $n$ be a positive integer and let $Y$ be either a closed surface or
$\mathbb{R}^{2}$. The braid groups and the pure braid groups of the plane were
defined by Artin [2, 3] and generalized for surfaces by Fox and Neuwirth [12]
using configuration spaces. The fundamental group $\pi_{1}(D_{n}(Y))$ (resp.
$\pi_{1}(F_{n}(Y))$) is the $n$-string braid group (resp. $n$-string pure
braid group) of $Y$ and it is denoted by $B_{n}(Y)$ (resp. $P_{n}(Y)$). This
give rise to the following short exact sequence induced by the
$\left(n!\right)$-fold covering map $F_{n}(Y)\longrightarrow D_{n}(Y)$:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$
(2.1)

The notion of _mixed braid groups_ was defined in [17, 19] taking an
intermediary quotient of the configuration space $F_{m+n}(Y)$ by the free
action of the subgroup $S_{m}\times S_{n}\subset S_{m+n}$. The orbit space
$F_{m+n}(Y)/{S_{m}\times S_{n}}$ is denoted by $D_{m,n}(Y)$ and the mixed
braid groups is defined by $B_{m,n}(Y)=\pi_{1}(D_{m,n}(Y))$. Therefore, we
have the following short exact sequence induced by the
$\left(\left(m!\right)\times\left(n!\right)\right)$-fold covering map
$F_{m,n}(Y)\longrightarrow D_{m,n}(Y)$:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{m,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{m,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{m}\times
S_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$
(2.2)

An arbitrary element of $D_{m,n}(Y)$ can be seen as a pair
$(\bar{x},\bar{y})$, where $\bar{x}$ and $\bar{y}$ are disjoints subsets of
$Y$ of cardinality $m$ and $n$, respectively. If $m=n$, let $D^{2}_{n,n}(Y)$
be the orbit space of the free involution $\hat{\tau}:D_{n,n}(Y)\rightarrow
D_{n,n}(Y)$ defined by $\hat{\tau}(\bar{x},\bar{y})=(\bar{y},\bar{x})$.
Therefore, we obtain another important subgroup of $B_{2n}(Y)$ defined by
$B^{2}_{n,n}(Y)=\pi_{1}(D^{2}_{n,n}(Y))$, which are related by the following
short exact sequence induced by the double covering map
$D_{n,n}(Y)\longrightarrow D_{n,n}^{2}(Y)$:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{n,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{2}_{n,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1.}$
(2.3)

Theses subgroups of $B_{2n}(Y)$ are important for this work, and they will
appear in the following theorem. The result below is a generalization of [20,
Proposition 13] and it will play a key rôle in the rest of the paper.

######  Theorem 3.

Let $n$ be a positive integer, let $X$ be a pathwise connected and locally
compact $CW$-complex eqquiped with a cellular free involution $\tau$ and let
$Y$ be either $\mathbb{R}^{2}$ or a closed surface. Suppose that the $n$-split
BUP (resp. n-non-split BUP) does not hold for the triple $(X,\tau;Y)$. Then
there exists a homomorphism $\Psi\colon\pi_{1}(X_{\tau})\rightarrow
B^{2}_{n,n}(Y)$ such that
$\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\subset P_{2n}(Y)$
(resp. $\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\not\subset
P_{2n}(Y)$) and the following diagram is commutative:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.89778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}(X_{\tau})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.7442pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\Psi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 82.58669pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.92479pt\raise-25.80498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0pt\hbox{$\scriptstyle{\theta_{\tau}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.89778pt\raise-32.8053pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
46.74223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 82.58669pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B^{2}_{n,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
73.39122pt\raise-24.45082pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
58.58669pt\raise-32.71892pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
139.27907pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 169.27907pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-39.88776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.89778pt\raise-39.88776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Z}_{2}.}$}}}}}}}{\hbox{\kern
95.93288pt\raise-39.88776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
139.27907pt\raise-39.88776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
169.27907pt\raise-39.88776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(2.4)

Conversely, if such a factorization $\Psi$ of the diagram (2.4) exists and it
satisfies $\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\subset
P_{2n}(Y)$ (resp.
$\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\not\subset
P_{2n}(Y)$), then the $n$-split BUP (resp. $n$-non-split BUP) does not hold
for the triple $(X,\tau;Y)$ in the following cases:

  1. (a)

The dimension of the CW-complex $X$ is less than or equal to two and
$Y=\mathbb{S}^{2}$.

  2. (b)

The dimension of the CW-complex $X$ is less than or equal to three and
$Y=\mathbb{RP}^{2}$.

  3. (c)

$Y$ is different from $\mathbb{S}^{2}$ and $\mathbb{RP}^{2}$.

######  Remark 4.

The Theorem 3 gives us a necessary and sufficient algebraic condition to
investigate the validity of the BUP for $n$-valued maps from $X$ to $Y$ in the
cases where $X$ is a closed surface and $Y$ is a closed surface or the
Euclidean plane.

###### Proof of Theorem 3.

First of all, the hypothesis over $X$ implies that $X$ is a metric space by
[25, Chapter IV, Proposition 5.8 and Theorem 5.14]. Since $Y$ is the Euclidean
plane or a closed surface, then $Y$ is also a metric space. Thus, we can
consider $n$-valued maps from $X$ to $Y$ as single maps from $X$ to $D_{n}(Y)$
by [21, Theorem 8].

Suppose first that the triple $(X,\tau;Y)$ does not have the $n$-BUP which
means that there exists a map $\Phi\colon X\longrightarrow D_{n}(Y)$ without
Borsuk-Ulam coincidence. Then, the map $\hat{\Phi}\colon X\longrightarrow
D_{n,n}(Y)$ given by $\hat{\Phi}(x)=(\Phi(x),\Phi(\tau(x)))$ is well defined.
Note that $\hat{\Phi}$ is $(\tau,\hat{\tau})$-equivariant, where $\hat{\tau}$
is the free involution on $D_{n,n}(Y)$ defined by
$\hat{\tau}(\overline{x},\overline{y})=(\overline{y},\overline{x})$. Using
standard covering spaces arguments analougous that was used in the first part
of the proof of [22, Lemma 5] there exists an homomorphism
$\Psi:\pi_{1}(X_{\tau})\rightarrow B^{2}_{n,n}(Y)$ such that the following
diagram is commutative:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
23.38164pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.35243pt\raise 6.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\hat{\Psi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 82.58669pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.45834pt\hbox{$\scriptstyle{(p_{\tau})_{\\#}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.06888pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
46.74223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 82.58669pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B_{n,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
101.67197pt\raise-28.93112pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}(X_{\tau})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
45.38954pt\raise-35.17722pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\Psi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
85.32578pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.91402pt\raise-66.40884pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0pt\hbox{$\scriptstyle{\theta_{\tau}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.89778pt\raise-73.4269pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
46.74223pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
85.32578pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B^{2}_{n,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
74.55591pt\raise-65.05469pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
58.58669pt\raise-73.72055pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
144.75725pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
174.75725pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-80.52663pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
40.89778pt\raise-80.52663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Z}_{2},}$}}}}}}}{\hbox{\kern
98.67197pt\raise-80.52663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
144.75725pt\raise-80.52663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
174.75725pt\raise-80.52663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(2.5)

where $\hat{\Psi}=\hat{\Phi}_{\\#}$. If the triple $(X,\tau;Y)$ does not have
the $n$-split BUP, then we may suppose without loss of generality that the
multimap $\Phi$ is split. This occurs if, and only if, there exists maps
$f_{1},\ldots f_{n}\colon X\longrightarrow Y$ such that
$\hat{\Phi}(x)=\left(\\{f_{1}(x),\ldots,f_{n}(x)\\},\\{f_{1}(\tau(x)),\ldots,f_{n}(\tau(x))\\}\right)$
for all $x\in X$. The above equality is equivalent that the map $\hat{\Phi}$
has a lift with respect to the covering $F_{2n}(Y)\longrightarrow D_{n,n}(Y)$
which is algebraically equivalent to $P_{2n}(Y)\supset{\rm
Im}\left(\hat{\Psi}\right)=\Psi(\left({\rm
Im}(p_{\tau})_{\\#}\right)=\Psi\left(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right)\right)$.
Analogous preceding arguments show that
$\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\not\subset
P_{2n}(Y)$ if the triple $(X,\tau;Y)$ does not have the $n$-non-split BUP. The
diagram (2.4) is the bottom of diagram (2.5) so we proved the first part of
the theorem.

Now, we will prove the converse part of the theorem. We assert that there
exists a map $\overline{\Phi}\colon\thinspace X_{\tau}\longrightarrow
D^{2}_{n,n}(Y)$ that induces $\Psi$. We treat the three cases of the statement
in turn. Before to do this, note that the dimension of $X_{\tau}$ as a CW-
complex is the same as $X$ because $\tau$ is free and cellular.

  1. (a)

The existence of the map $\overline{\Phi}$ follows using classical arguments
of obstruction theory in the same manner that were used in [20, Proof of
Proposition 13, item (a)].

  2. (b)

The universal covering of $D^{2}_{n,n}(Y)$ is the same of
$F_{2n}(\mathbb{RP}^{2})$ which is $\mathbb{S}^{3}$ by [18, Proposition 6].
Since $X_{\tau}$ has dimension less or equal to three, once more using
classical obstruction theory, we may construct the map $\overline{\Phi}$.

  3. (c)

The space $F_{2n}(Y)$ is a ${\rm K}(\pi,1)$ if
$Y\neq\mathbb{S}^{2},\mathbb{RP}^{2}$ by [10, Chapter 4, Theorem 1.1] and [11,
Corollary 2.2], and then $D^{2}_{n,n}(Y)$ is also a ${\rm K}(\pi,1)$. The
existence of the map $\overline{\Phi}$ is assured by [22, Theorem 4].

Let $\hat{\Phi}\colon\thinspace X\longrightarrow D_{n,n}(Y)$ be the lift of
the map $\overline{\Phi}\circ p_{\tau}$ with respect to the covering
$D_{n,n}(Y)\longrightarrow D^{2}_{n,n}(Y)$. So we have the following
commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.2705pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.36894pt\raise 6.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\hat{\Phi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.75874pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{\tau}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.73557pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.75874pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{D_{n,n}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
52.0791pt\raise-28.93112pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{\tau}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.93384pt\raise-46.40776pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{\Phi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
34.10895pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.10895pt\raise-40.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{D^{2}_{n,n}(Y).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(2.6)

Let $\hat{\Psi}$ be the induced homomorphism by $\hat{\Phi}$ on the level of
fundamental group. So, from the diagrams (2.4) and (2.6) we obtain the
algebraic commutative diagram (2.5). Using similar arguments that were used in
[22, Proofs of Lemma 5 and Theorem 7], we conclude that $\hat{\Phi}$ is a
$(\tau,\hat{\tau})$-equivariant map and there exists a map
$\Phi\colon\thinspace X\longrightarrow D_{n}(Y)$ that satisfies
$\hat{\Phi}(x)=(\Phi(x),\Phi(\tau(x)))$ for all $x\in X$. Therefore the map
$\Phi$ does not have Borsuk-Ulam coincidence. Finally, using similar arguments
that were used in the first part of the proof, we conclude that $\Phi$ is
split (resp. non-split) if
$\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\subset P_{2n}(Y)$
(resp. $\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\not\subset
P_{2n}(Y)$) and the conclusion of the second part of the theorem follows. ∎

The next result will be useful to prove Theorem 2 in Section 5.

######  Theorem 5.

Given integers numbers $n,k\geq 1$, there exists an homomorphism
$\tilde{\Psi}\colon B^{2}_{n,n}(\mathbb{R}^{2})\rightarrow
B^{2}_{nk,nk}(\mathbb{R}^{2})$ for which the following diagram is commutative:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.41286pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B^{2}_{n,n}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.47952pt\raise 6.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\tilde{\Psi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.10176pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.3427pt\raise-24.47667pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
41.41286pt\raise-33.15059pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
47.25731pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 83.10176pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B^{2}_{nk,nk}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
74.846pt\raise-24.73901pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\pi_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
59.10176pt\raise-33.14195pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
41.41286pt\raise-40.22441pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Z}_{2},}$}}}}}}}{\hbox{\kern
97.51462pt\raise-40.22441pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(2.7)

where $\pi_{1}$ and $\pi_{2}$ are as defined in (2.3).

###### Proof.

Given a point $z=(x_{1},x_{2},\ldots,x_{n},y_{1},y_{2},\ldots,y_{n})\in
F_{2n}(\mathbb{R}^{2})$, consider the real number

$\delta(z)=\dfrac{1}{2k}\cdot\displaystyle\min_{1\leq i<j\leq
n}\\{||x_{i}-x_{j}||,||y_{i}-y_{j}||,||x_{i}-y_{i}||,||x_{i}-y_{j}||,||x_{j}-y_{i}||\\}.$

Note that $\delta(z)\neq 0$. Now, consider
$\varepsilon(z)=(\delta(z),0)\in\mathbb{R}^{2}$. We define

$\displaystyle G(z)=($ $\displaystyle
x_{1}+0\cdot\varepsilon(z),x_{1}+1\cdot\varepsilon(z),\ldots,x_{1}+(k-1)\cdot\varepsilon(z),$
$\displaystyle
x_{2}+0\cdot\varepsilon(z),x_{2}+1\cdot\varepsilon(z),\ldots,x_{2}+(k-1)\cdot\varepsilon(z),$
$\displaystyle\qquad\qquad\qquad\vdots$ $\displaystyle
x_{n}+0\cdot\varepsilon(z),x_{n}+1\cdot\varepsilon(z),\ldots,x_{n}+(k-1)\cdot\varepsilon(z),$
$\displaystyle
y_{1}+0\cdot\varepsilon(z),y_{1}+1\cdot\varepsilon(z),\ldots,y_{1}+(k-1)\cdot\varepsilon(z),$
$\displaystyle
y_{2}+0\cdot\varepsilon(z),y_{2}+1\cdot\varepsilon(z),\ldots,y_{2}+(k-1)\cdot\varepsilon(z),$
$\displaystyle\qquad\qquad\qquad\vdots$ $\displaystyle
y_{n}+0\cdot\varepsilon(z),y_{n}+1\cdot\varepsilon(z),\ldots,y_{n}+(k-1)\cdot\varepsilon(z)).$

Note that $G\colon F_{2n}(\mathbb{R}^{2})\longrightarrow
F_{2nk}(\mathbb{R}^{2})$ is a well defined continuous map. Using standard
arguments of quotient topology, there exists maps $\overline{G}\colon
D_{n,n}(\mathbb{R}^{2})\longrightarrow D_{nk,nk}(\mathbb{R}^{2})$ and
$\tilde{G}\colon D^{2}_{n,n}(\mathbb{R}^{2})\longrightarrow
D^{2}_{nk,nk}(\mathbb{R}^{2})$ such that the following diagram is commutative:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.38702pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F_{2n}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.76666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.40808pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{G}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 47.36003pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.36003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{F_{2nk}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.8657pt\raise-29.76666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{D_{n,n}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-71.43774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
26.88702pt\raise-35.56554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{G}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
44.38702pt\raise-41.40442pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
44.38702pt\raise-41.40442pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{D_{nk,nk}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.8657pt\raise-71.43774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{D^{2}_{n,n}(\mathbb{R}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.46062pt\raise-76.46439pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\tilde{G}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
48.82889pt\raise-83.0755pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
48.82889pt\raise-83.0755pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{D^{2}_{nk,nk}(\mathbb{R}^{2}).}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(2.8)

Using covering spaces properties, similar that were used in [22, Proof of
Lemma 5], we may take $\tilde{\Psi}=\tilde{G}_{\\#}\colon
B^{2}_{n,n}(\mathbb{R}^{2})=\pi_{1}(D^{2}_{n,n}(\mathbb{R}^{2}))\longrightarrow
B^{2}_{nk,nk}(\mathbb{R}^{2})=\pi_{1}(D^{2}_{nk,nk}(\mathbb{R}^{2}))$ to
obtain the diagram (2.7). ∎

##  3\. Presentations

In this section we present some general properties and results about the braid
groups of surfaces, notably the braid group of the plane, that will be useful
in this paper. As defined in the previous section, $B_{n}(\mathbb{R}^{2})$ is
the fundamental group of the $n^{th}$ unordered configuration space
$D_{n}(\mathbb{R}^{2})$, and it is usually denoted by $B_{n}$ and called the
_$n^{th}$ Artin braid group_. From now on, we will denote all the subgroups of
$B_{n}$ without mention the surface $Y$ in the case that $Y=\mathbb{R}^{2}$.

For $n\geq 2$, the group $B_{n}$ is generated by the elements
$\sigma_{1},\ldots,\sigma_{n-1}$, subject to the following relations, called
_Artin relations_ [2]:

$\left\\{\begin{aligned}
&\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}&\text{for
all $1\leq i\leq n-2$;}\\\ &\sigma_{j}\sigma_{i}=\sigma_{i}\sigma_{j}&\text{if
$|i-j|\geq 2$ and $1\leq i,j\leq n-1$.}\end{aligned}\right.$

The elements of $B_{n}$, called braids, may be considered as a geometric
representation of strings and braids of hair, and the generators of the Artin
braid group (also called as _Artin generators_) can be visualized in Figure
(1), where each path is called string. Also, since each braid is defined in
the interior of a disc, the inclusion $j:\mathbb{D}^{2}\rightarrow int(Y)$
allows us to see each braid in $B_{n}$ as a braid in $B_{n}(Y)$, where $Y$ is
any surface.

$\cdots$$\cdots$$1$$i-1$$i$$i+1$$i+2$$n$$\sigma_{i}$ Figure 1. The braid
$\sigma_{i}$

There is a natural correspondence that associates to each braid a permutation
of $S_{n}$, considering the projection $\pi:B_{n}\rightarrow S_{n}$,
$\sigma_{i}\longmapsto(i,i+1)$. The _$n^{th}$ Artin pure braid group_
$P_{n}=\pi_{1}(F_{n}(\mathbb{R}^{2}))$ is isomorphic to the kernel of $\pi$.
Thus, the pure braids are the elements that has the trivial permutation. The
subgroup $P_{n}$ is generated by the set $\left\\{A_{i,j}\,:\,1\leq i<j\leq
n\right\\}$ [24, Chapter 1, Lemma 4.2], where

$A_{i,j}=\sigma_{j-1}\cdots\sigma_{i+1}\sigma^{2}_{i}\sigma^{-1}_{i+1}\cdots\sigma^{-1}_{j-1}.$
(3.1)

The _Garside element_ (or ‘half-twist’) $\Delta_{n}$ of $B_{n}$ is defined by

$\Delta_{n}=(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})(\sigma_{1}\sigma_{2}\cdots\sigma_{n-2})\cdots(\sigma_{1}\sigma_{2})(\sigma_{1}),$
(3.2)

and can be visualized in Figure 2 for $n=6$.

Figure 2. The Garside element $\Delta_{6}$ of $B_{6}$

This element has an important rôle in the theory of braid group, notably in
the resolution of the conjugacy problem in $B_{n}$ [5, 13]. Using the Artin
relations, it is easy to proof that

$\Delta_{n}\sigma_{i}\Delta^{-1}_{n}=\sigma_{n-i}\,\,\mbox{for\,all}\,\,1\leq
i\leq n-1.$ (3.3)

The _‘full twist’_ is the braid defined as the square of the half-twist and it
can be written in terms of generators of $P_{n}$ as

$\Delta^{2}_{n}=A_{1,2}(A_{1,3}A_{2,3})\cdots(A_{1,n}A_{2,n}\cdots
A_{n-1,n}),$ (3.4)

and this pure braid generates the centre of $B_{n}(Y)$ for
$Y=\mathbb{R}^{2},\,\mathbb{S}^{2},\,\mathbb{R}P^{2}$ [8, 14, 32]. Those
elements will be important in this paper, as we will see in the next section.

The others intermediary subgroups $B_{n,n}$ and $B^{2}_{n,n}$ of the Artin
braid group also have a useful geometric representation. We can visualize the
braids in $B_{n,n}$ as braids with $2n$ strings, where the first $n$ strings
has as final point the first $n$ points and the last $n$ strings has as final
point the last $n$ points, as in Figure 3(a). The group $B^{2}_{n,n}$ contains
$B_{n,n}$ and also all the braids that the first $n$ strings has as final
point the last $n$ points and the last $n$ strings has as final point the
first $n$ points, as in Figure 3(b).

(a) Braid in $B_{2,2}$(b) Braid in $B^{2}_{4,4}$ Figure 3. Examples in
$B_{n,n}$ and $B^{2}_{n,n}$

######  Remark 6.

It is easy to see that the permutation of $\Delta_{2n}$ is
$\prod^{n}_{i=1}(i,\,2n+1-i)$, then, the element $\Delta_{2n}\in
B^{2}_{n,n}\setminus B_{n,n}$. By (2.3), we know that $B_{n,n}$ is a subgroup
of $B^{2}_{n,n}$ of index $2$, therefore all elements in $B^{2}_{n,n}\setminus
B_{n,n}$ can be written as a product of $\Delta_{2n}$ with a braid in
$B_{n,n}$.

If $n=1$, we have $B_{1,1}=P_{2}$. For $n\geq 2$, the following proposition
give a presentation of $B_{n,n}$ that will be used in this paper, and it was
based in the presentation of $B_{n,n}(\mathbb{S}^{2})$ given by [19,
Proposition 10].

######  Proposition 7.

Let $n\geq 2$, the group $B_{n,n}$ admits the following presentation:

Generators:
$\left\\{\sigma_{1},\ldots,\sigma_{n-1},\sigma_{n+1},\ldots,\sigma_{2n-1},A_{i,j}\,:\,1\leq
i<j\leq 2n\right\\}$.

Relations:

  1. (1)

Pure Braid relations:
$A^{-1}_{r,s}A_{i,j}A_{r,s}=\left\\{\begin{array}[]{ll}A_{i,j}\hfill
i<r<s<j\mbox{\,\,\,\,or\,\,}&r<s<i<j;\\\
A_{r,j}A_{i,j}A^{-1}_{r,j}&r<i=s<j;\\\
A_{r,j}A_{s,j}A_{i,j}A^{-1}_{s,j}A^{-1}_{r,j}&i=r<s<j;\\\
A_{r,j}A_{s,j}A^{-1}_{r,j}A^{-1}_{s,j}A_{i,j}A_{s,j}A_{r,j}A^{-1}_{s,j}A^{-1}_{r,j}&r<i<s<j.\end{array}\right.$

  2. (2)

Artin relations:
$\left\\{\begin{array}[]{lr}\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},&\quad\,1\leq
i\leq n-2\quad\mbox{or}\quad n+1\leq i\leq 2n-2;\\\
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},&\,|i-j|\geq 2.\end{array}\right.$

  3. (3)

$\sigma^{2}_{i}=A_{i,i+1},\quad\,1\leq i\leq n-1\quad\mbox{or}\quad n+1\leq
i\leq 2n-1$.

  4. (4)

Conjugates:
$\sigma_{k}A_{i,j}\sigma^{-1}_{k}=\left\\{\begin{array}[]{lr}A_{i,j},&k\neq
i-1,i,j-1,j;\\\ A_{i,j},&k=i,j=k+1;\\\ A_{i,k+1},&k=j;\\\
A_{i,k+1}^{-1}A_{i,k}A_{i,k+1},&j=k+1,i<k;\\\ A_{i+1,j},&i=k<j-1;\\\
A^{-1}_{k+1,j}A_{k,j}A_{k+1,j},&i=k+1.\\\ \end{array}\right.$

###### Proof.

We use the extension method described in [27, Chapter 10, Proposition 1]
applyed in the short exact sequence (2.2) for $m=n$ and $Y=\mathbb{R}^{2}$.
According this method, a set of generators for $B_{n,n}$ is given by the union
of all generators of $P_{2n}$ with a coset representatives of generators of
$S_{n}\times S_{n}$. By [24, Chapter 1, Lemma 4.2], we may choose
$\\{A_{i,j}\,:\,1\leq i<j\leq 2n\\}$ as the set of generators of $P_{2n}$.
Since the transpositions generates the symetric group, we may choose
$\sigma_{i}$, $i=1,\ldots,n-1,n+1,\ldots,2n-1$ as the remain generators of
$B_{n,n}$. There are three types of relations. The first type is obtained by
taking all the relations of the generators of $P_{2n}$ and we use again [24,
Chapter 1, Lemma 4.2] to obtain the relations (1). The second type is given by
rewriting the relations of $S_{n}\times S_{n}$ in terms of the chosen cosets
representatives and expressing the corresponding element as words in terms of
the generators of $P_{2n}$. This step can be view in a simple geometric way,
which gives relations (2) and (3). Finally the third type is obtained by
writing the conjugates of the generators of the kernel by the coset
representatives as words written entirely in terms of the generators of the
kernel. This rewriting process is described in details in [28, Lemma 3.1],
which gives relation (4). ∎

######  Remark 8.

Using (3.1), the elements $A_{i,j}\in B_{n,n}$, $1\leq i<j\leq n$ or $n+1\leq
i<j\leq 2n$, can be written as product of $\sigma_{k}$,
$k=1,\ldots,n-1,n+1,\ldots,2n-1$. If $1\leq i\leq n$ and $n+1\leq j\leq 2n$,
the braid $A_{i,j}\in B_{n,n}$ can be written as

$A_{i,j}=(\sigma_{j-1}\cdots\sigma_{n+1})(\sigma^{-1}_{i}\cdots\sigma^{-1}_{n-1})A_{n,n+1}(\sigma_{i}\cdots\sigma_{n-1})(\sigma^{-1}_{n+1}\cdots\sigma^{-1}_{j-1}).$
(3.5)

##  4\. Proof of Theorem 1

In this section we prove Theorem 1 using a sequence of propositions. Recall
that $\tau$ denote the antipodal map on $\mathbb{S}^{2}$. First, note that the
triple $(\mathbb{S}^{2},\tau;\mathbb{S}^{2})$ does not have $1$-BUP, because
the identity map of $\mathbb{S}^{2}$ does not collapses antipodal points. This
fact with the classical Borsuk-Ulam theorem and [20, Theorem 2 and Proposition
4] is summarized in the following propositon:

######  Proposition 9.

Let $Y$ be $\mathbb{R}^{2}$ or a closed surface. Then the triple
$(\mathbb{S}^{2},\tau;Y)$ has the $1$-BUP if, and only if,
$Y\neq\mathbb{S}^{2}$.

With the exception of the case $Y=\mathbb{S}^{2}$, the next result extends the
previous one and the prove uses the same idea of [33, Proof of Theorem 2].

######  Proposition 10.

Let $Y$ be $\mathbb{R}^{2}$ or a closed surface different of $\mathbb{S}^{2}$.
Then the triple $(\mathbb{S}^{2},\tau;Y)$ has the $n$-BUP if $n\geq 2$.

###### Proof.

Let $\phi\colon\thinspace\mathbb{S}^{2}\multimap Y$ be $n$-valued map. Then
$\phi$ is split by [21, Lemma 13] which means that there exists maps
$f_{1},f_{2},\ldots,f_{n}\colon\thinspace\mathbb{S}^{2}\longrightarrow Y$ such
that $\phi(x)=\\{f_{1}(x),f_{2}(x),\ldots,f_{n}(x)\\}$ for all
$x\in\mathbb{S}^{2}$. By Proposition 9, there exists a point
$x\in\mathbb{S}^{2}$ such that $f_{1}(x)=f_{1}(-x)$ and therefore
$\phi(x)\cap\phi(-x)\neq\emptyset$ which ends the proof. ∎

Note that Theorem 1 is a summary of Proposition 9 and 10 and the next result.

######  Proposition 11.

The triple $(\mathbb{S}^{2},\tau;\mathbb{S}^{2})$ has the $n$-BUP for all
$n\geq 2$.

###### Proof.

We argue by contradiction. Suppose that $(\mathbb{S}^{2},\tau;\mathbb{S}^{2})$
does not have the $n$-BUP for $n\geq 2$. By Theorem 3 and the fact the
quotient of $\mathbb{S}^{2}$ by the antipodal involution is $\mathbb{RP}^{2}$,
we have the following commutative diagram:

$\textstyle{\pi_{1}(\mathbb{RP}^{2})\cong\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\theta_{\tau}\cong{\rm
Id}}$$\textstyle{B^{2}_{n,n}(\mathbb{S}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathbb{Z}_{2}.}$

So, $\Psi$ maps the generator of $\pi_{1}(\mathbb{RP}^{2})$ in a braid $b\in
B_{2n}(\mathbb{S}^{2})$ which has order two and it belongs to
$B^{2}_{n,n}(\mathbb{S}^{2})\setminus B_{n,n}(\mathbb{S}^{2})$. However, this
is an absurd because the full twist $\Delta_{2n}^{2}$ is the only element of
order two of $B_{2n}(\mathbb{S}^{2})$ by [14, Theorem 3.5] and
$\Delta_{2n}^{2}\in P_{2n}(\mathbb{S}^{2})\subset B_{n,n}(\mathbb{S}^{2})$. ∎

######  Remark 12.

We also could prove Proposition 10 in a similar way as Proposition 11, because
$B_{m}(Y)$ is torsion-free if $Y$ is $\mathbb{R}^{2}$ or a closed surface
different of $\mathbb{S}^{2}$ and for $\mathbb{RP}^{2}$ by [11, Theorem 8],
and if $Y=\mathbb{R}P^{2}$, then the full twist $\Delta^{2}_{2n}$ is also the
only element of order two of $B_{2n}(\mathbb{RP}^{2})$ by [18, Proposition
23].

##  5\. Proof of Theorem 2

The main goal of this section is to prove Theorem 2. Its proof will follow
from Propositions 14-16. In all of the enunciates $X$ denotes a closed surface
different of $\mathbb{S}^{2}$ that admits a free involution $\tau$, the orbit
space is denoted by $X_{\tau}$ and
$\theta_{\tau}\colon\pi_{1}(X_{\tau})\longrightarrow\mathbb{Z}_{2}$ is the
homomorphism defined in (1.1). Recall that $X_{\tau}$ is also a closed
surface. In what follows, we will distinguish the following three
possibilities, and in each case, we define an element
$\delta\in\pi_{1}(X_{\tau})$:

  1. (I)

$X_{\tau}$ is a closed orientable surface of genus $m\geq 1$,

$\pi_{1}(X_{\tau})=\left\langle
a_{1},a_{2},\ldots,a_{2m-1},a_{2m}\,:\,[a_{1},a_{2}]\ldots[a_{2m-1},a_{2m}]=1\right\rangle\text{
and }\delta=1.$

  2. (II)

$X_{\tau}$ is a closed non-orientable surface of genus $2m+2$, $m\geq 0$,

$\pi_{1}(X_{\tau})=\left\langle
u,v,a_{1},a_{2},\ldots,a_{2m-1},a_{2m}\,:\,uvuv^{-1}[a_{1},a_{2}]\ldots[a_{2m-1},a_{2m}]=1\right\rangle\text{
and }\delta=u.$

  3. (III)

$X_{\tau}$ is a closed non-orientable surface of genus $2m+1$, $m\geq 1$,

$\pi_{1}(X_{\tau})=\left\langle
c,a_{1},a_{2},\ldots,a_{2m-1},a_{2m}\,:\,c^{2}[a_{1},a_{2}]\ldots[a_{2m-1},a_{2m}]=1\right\rangle\text{
and }\delta=c.$

######  Remark 13.

Recall that $\hat{\theta_{\tau}}\colon\thinspace
H_{1}(X_{\tau})\longrightarrow\mathbb{Z}_{2}$ is the factorization of the
homomorphism $\theta_{\tau}$ to the abelianizition of $\pi_{1}(X_{\tau})$ and
$\hat{\delta}$ is the only-non trivial element of order two of
$H_{1}(X_{\tau})$ if $X_{\tau}$ is non-orientable. So, with the notation of
(II) and (III) above, we have that
$\hat{\theta_{\tau}}(\hat{\delta})=\theta_{\tau}(\delta)$.

Let $n$ be a positive integer. With the notation of (I)-(III), in order to
estabilish Theorem 2, by Remark 13, it suffices to prove the following
propositions:

######  Proposition 14.

If $\theta_{\tau}(\delta)=\overline{0}$, then the triple
$(X,\tau;\mathbb{R}^{2})$ does not have the $n$-split BUP and the $n$-non-
split BUP.

######  Proposition 15.

If $\theta_{\tau}(\delta)=\overline{1}$ and $n$ is even, then the triple
$(X,\tau;\mathbb{R}^{2})$ has the $n$-split BUP, but it does not have the
$n$-non-split BUP.

######  Proposition 16.

If If $\theta_{\tau}(\delta)=\overline{1}$ and $n$ is odd, then the triple
$(X,\tau;\mathbb{R}^{2})$ has the $n$-BUP.

The remaining of this paper is devoted to prove Propositions 14-16. We use
without comments the notations estabilished in Sections 3.

###### Proof of Proposition 14.

The case $n=1$ is a direct consequence of [16, Theorem 2.5]. So suppose that
$n\geq 2$. For each generator $g\in\pi_{1}(X_{\tau})$, let $i(g)\in\\{0,1\\}$
such that $\theta_{\tau}(g)=\overline{i(g)}$. Consider the $2n$-braid
$\omega_{n}=\prod^{n-1}_{j=0}\prod^{2n-1-j}_{i=n-j}\sigma_{i}$ ($\omega_{4}$
is geometrically represented in Figure 3(b)). Note that $w_{n}\in
B^{2}_{n,n}\setminus B_{n,n}$ and therefore $\pi(w_{n})=\overline{1}$. We
define $\Psi\colon\thinspace\pi_{1}(X_{\tau})\longrightarrow B^{2}_{n,n}$ by
$\Psi(g)=\omega_{n}^{i(g)}$. Using the hypothesis
$\theta_{\tau}(\delta)=\overline{0}$, in each case (I)-(III) listed in the
beginning of the section we may check that $\Psi$ is a well-defined
homomorphism such that the diagram (2.4) is commutative. If
$a\in\operatorname{\text{Ker}}\left({\theta_{\tau}}\right)$, then $\Psi(a)$ is
a even power of $\omega_{n}$ which is an element of $P_{2n}$. Thus
$\Psi(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\subset P_{2n}$.
Follows by Theorem 3 item (c) that $(X,\tau;\mathbb{R}^{2})$ does not have the
$n$-split BUP. Now, let us show that $(X,\tau;\mathbb{R}^{2})$ does not have
the $n$-non-split BUP. Once more, we may check that
$\Psi^{\prime}\colon\thinspace\pi_{1}(X_{\tau})\longrightarrow B^{2}_{n,n}$
defined by $\Psi^{\prime}(g)=\left(\sigma_{1}\omega_{n}\right)^{i(g)}$ extends
to a homomorphism such that the diagram (2.4) is commutative. Since
$\theta_{\tau}$ is an epimorphism, there exists an generator $g_{1}$ of
$\pi_{1}(X_{\tau})$ such that $\theta_{\tau}(g_{1})=\overline{1}$. Then
$g_{1}^{2}\in\operatorname{\text{Ker}}\left({\theta_{\tau}}\right)$ and
$\Psi^{\prime}(g_{1}^{2})=(\sigma_{1}\omega_{n})^{2}\notin P_{2n}$. Thus
$\Psi^{\prime}(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right))\not\subset
P_{2n}$ which ends the proof by Theorem 3 item (c). ∎

###### Proof of Proposition 15.

The hypothesis $\theta_{\tau}(\delta)=\overline{1}$ implies that
$(X,\tau;\mathbb{R}^{2})$ has the $1$-BUP by [16, Theorem 2.5]. From this and
using analogous arguments that we used in the proof of Proposition 10 we
conclude that $(X,\tau;\mathbb{R}^{2})$ has the $n$-split BUP for all $n$ (in
particular for $n$ even). Now, let us prove that $(X,\tau;\mathbb{R}^{2})$
does not have $n$-non-split BUP. We claim that for $n=2$ there exists a
homomorphism $\Psi_{1}\colon\thinspace\pi_{1}(X_{\tau})\longrightarrow
B_{2,2}^{2}$ such that the diagram (2.4) is commutative. Assume for the moment
that the claim holds. Let $\tilde{\Psi}\colon\thinspace
B^{2}_{2,2}\longrightarrow B^{2}_{2k,2k}$ be the homomorphism obtaneid by
Theorem 5, where $k=n/2$. Then the homomorphism
$\Psi\colon\thinspace\pi_{1}(X_{\tau})\longrightarrow B^{2}_{2k,2k}$ given by
the composition $\Psi=\tilde{\Psi}\circ\Psi_{1}$ makes the diagram (2.4)
commutative. By Theorem 3 item (c), the triple $(X,\tau;\mathbb{R}^{2})$ does
not have $n$-split BUP (if
$\Psi\left(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right)\right)\subset
P_{2n}$) or the triple does not have $n$-non-split BUP (if
$\Psi\left(\operatorname{\text{Ker}}\left({\theta_{\tau}}\right)\right)\not\subset
P_{2n}$). However, we already showed in the first part of the proof that
$(X,\tau;\mathbb{R}^{2})$ has the $n$-split BUP and therefore we conclude that
the triple does not have the $n$-non-split BUP.

In order to complete the proof, we now prove the claim, _i.e._ , let us define
a homomorfism $\Psi_{1}\colon\thinspace\pi_{1}(X_{\tau})\longrightarrow
B_{2,2}^{2}$ such that the diagram (2.4) is commutative. We will divide the
proof into the mutually disjoint cases (i)-(iii) as below. We will make use
the following elements of $B^{2}_{2,2}\setminus B_{2,2}$: $\Delta_{4}$ (the
half twist as defined in (3.2)) and
$\Omega=\sigma_{2}\sigma_{3}\sigma_{1}^{-1}\sigma_{2}^{-1}$. Using (3.3), we
have

$\Delta_{4}\Omega\Delta^{-1}_{4}=\Delta_{4}(\sigma_{2}\sigma_{3}\sigma^{-1}_{1}\sigma^{-1}_{2})\Delta^{-1}_{4}=\sigma_{2}\sigma_{1}\sigma^{-1}_{3}\sigma^{-1}_{2}=\Omega^{-1}.$
(5.1)

For each $g\in\pi_{1}(X_{\tau})$, let $i(g)\in\\{0,1\\}$ such that
$\theta_{\tau}(g)=\overline{i(g)}$.

  1. (i)

$\pi_{1}(X_{\tau})$ is as in (II). We define

$\Psi_{1}\colon\thinspace\begin{cases}u\longmapsto\Omega;\\\
v\longmapsto\Delta_{4}\Omega^{1-i(v)};\\\ a_{k}\longmapsto\Omega^{i(k)}\text{
for all }k.\end{cases}$

By (5.1), we have

$\displaystyle\Psi_{1}\left(uvuv^{-1}[a_{1},a_{2}]\ldots[a_{2m-1},a_{2m}]\right)$
$\displaystyle=\Omega\Delta_{4}\Omega^{1-i(v)}\Omega(\Delta_{4}\Omega^{1-i(v)})^{-1}[\Omega^{i(a_{1})},\Omega^{i(a_{2})}]\ldots[\Omega^{i(a_{2m-1})},\Omega^{i(a_{2m})}]$
$\displaystyle=\Omega\Delta_{4}\Omega\Delta^{-1}_{4}=1.$

So $\Psi_{1}$ extends to a homomorphism from $\pi_{1}(X_{\tau})$ to
$B^{2}_{2,2}$. By the hypothesis, we have that $\theta_{\tau}(u)=\overline{1}$
and from this we may conclude that the diagram (2.4) is commutative.

  2. (ii)

$\pi_{1}(X_{\tau})$ is as in (III) and $\theta_{\tau}(a_{1})=\overline{1}$ or
$\theta_{\tau}(a_{2})=\overline{1}$. We define

$\Psi_{1}\colon\thinspace\begin{cases}c\longmapsto\Omega;\\\
a_{1}\longmapsto\Delta_{4}^{i(a_{2})}\Omega^{i(a_{1})+i(a_{2})-2};\\\
a_{2}\longmapsto\Delta_{4}^{1-i(a_{2})}\Omega;\\\
a_{k}\longmapsto\Omega^{i(k)}\text{ for }k\neq 1,2.\end{cases}$

Suppose first that $\theta_{\tau}(a_{1})=\overline{1}$ and
$\theta_{\tau}(a_{2})=\overline{0}$. By (5.1) we have

$\displaystyle\Psi_{1}(c^{2}[a_{1},a_{2}][a_{3},a_{4}]\ldots[a_{2m-1},a_{2m}])$
$\displaystyle=\Omega^{2}[\Omega^{-1},\Delta_{4}\Omega][\Omega^{i(a_{3})},\Omega^{i(a_{4})}]\ldots[\Omega^{i(a_{2m-1})},\Omega^{i(a_{2m})}]$
$\displaystyle=\Omega\Delta_{4}\Omega\Delta^{-1}_{4}=1.$

Once more $\Psi_{1}$ extends to a homomorphism from $\pi_{1}(X_{\tau})$ to
$B^{2}_{2,2}$ and the diagram (2.4) is commutative because
$\theta_{\tau}(c)=\overline{1}$ by hypothesis. The proof of the cases
($\theta_{\tau}(a_{1})=\overline{0}$ and $\theta_{\tau}(a_{2})=\overline{1}$)
and ($\theta_{\tau}(a_{1})=\overline{1}$ and
$\theta_{\tau}(a_{2})=\overline{1}$) are completely similar and we omit the
details.

  3. (iii)

$\pi_{1}(X_{\tau})$ is as in (III) and
$\theta_{\tau}(a_{1})=\theta_{\tau}(a_{2})=\overline{0}$. We leave to the
reader to verify that the braids $\sigma_{2}\Omega^{-1}\sigma_{2}^{-1}$ and
$\sigma_{2}\sigma_{3}^{2}\sigma_{2}^{-1}$ belongs to $B_{2,2}$. We define

$\Psi_{1}\colon\thinspace\begin{cases}c\longmapsto\Omega;\\\
a_{1}\longmapsto\sigma_{2}\Omega^{-1}\sigma_{2}^{-1};\\\
a_{2}\longmapsto\sigma_{2}\sigma_{3}^{2}\sigma_{2}^{-1};\\\
a_{k}\longmapsto\Omega^{i(k)}\text{ for }k\neq 1,2.\end{cases}$

From the Artin relations in $B_{4}$ we deduce that
$\sigma_{i}^{\varepsilon}\sigma_{i+1}\sigma_{i}^{-\varepsilon}=\sigma_{i+1}^{-\varepsilon}\sigma_{i}\sigma_{i+1}^{\varepsilon}$,
for $i=1,2$ and $\varepsilon=-1,1$. Using this and
$\sigma_{1}\sigma_{3}=\sigma_{3}\sigma_{1}$ we have

$\displaystyle\Psi_{1}(c^{2}[a_{1},a_{2}][a_{3},a_{4}]\ldots[a_{2m-1},a_{2m}])$
$\displaystyle=\Omega^{2}[\sigma_{2}\Omega^{-1}\sigma_{2}^{-1},\sigma_{2}\sigma_{3}^{2}\sigma_{2}^{-1}][\Omega^{i(a_{3})},\Omega^{i(a_{4})}]\ldots[\Omega^{i(a_{2m-1})},\Omega^{i(a_{2m})}]$
$\displaystyle=\Omega^{2}\sigma^{2}_{2}\sigma_{1}\sigma^{-1}_{3}(\sigma^{-1}_{2}\sigma^{2}_{3}\sigma_{2})\sigma_{3}\sigma^{-1}_{1}\sigma^{-1}_{2}\sigma^{-2}_{3}\sigma^{-1}_{2}$
$\displaystyle=\Omega^{2}\sigma^{2}_{2}\sigma_{1}\sigma^{-1}_{3}(\sigma_{3}\sigma^{2}_{2}\sigma^{-1}_{3})\sigma_{3}\sigma^{-1}_{1}\sigma^{-1}_{2}\sigma^{-2}_{3}\sigma^{-1}_{2}$
$\displaystyle=\Omega^{2}\sigma^{2}_{2}(\sigma_{1}\sigma^{2}_{2}\sigma^{-1}_{1})\sigma^{-1}_{2}\sigma^{-2}_{3}\sigma^{-1}_{2}$
$\displaystyle=\Omega^{2}\sigma^{2}_{2}(\sigma^{-1}_{2}\sigma^{2}_{1}\sigma_{2})\sigma^{-1}_{2}\sigma^{-2}_{3}\sigma^{-1}_{2}$
$\displaystyle=\Omega^{2}\sigma_{2}\sigma^{2}_{1}\sigma^{-2}_{3}\sigma^{-1}_{2}=\Omega^{2}\Omega^{-2}=1.$

As in the previous cases we have that the diagram (2.4) is commutative.

∎

###### Proof of Proposition 16.

If $n=1$, then result follows directly from [16, Theorem 2.5]. So let $n$ be
an odd positive integer different from $1$. In order to simplify the proof, we
write

$\pi_{1}(X_{\tau})=\left\langle\alpha,\beta,a_{1},a_{2},\ldots,a_{2m-1},a_{2m}\,:\,\alpha\beta\alpha\beta^{-1}[a_{1},a_{2}]\ldots[a_{2m-1},a_{2m}]=1\right\rangle,$

where $\begin{cases}\alpha=u\\\ \beta=v\end{cases}$ if $\pi_{1}(X_{\tau})$ is
as in (II) and $\begin{cases}\alpha=c\\\ \beta=1\end{cases}$ if
$\pi_{1}(X_{\tau})$ is as in (III). For each generator
$g\in\pi_{1}(X_{\tau})$, let $i(g)\in\\{0,1\\}$ such that
$\theta_{\tau}(g)=\overline{i(g)}$. By hypothesis, we have
$i(\alpha)=\overline{1}$. We argue by contradiciton. Suppose that the triple
$(X,\tau,\mathbb{R}^{2})$ does not have $n$-BUP. By Theorem 3 there exists a
homomorphism $\Psi\colon\thinspace\pi_{1}(X_{\tau})\longrightarrow
B_{n,n}^{2}$ such that the diagram (2.4) is commutative. From this and Remark
6, there exists braids
$b_{\alpha},b_{\beta},b_{1},b_{2},\ldots,b_{2m-1},b_{2m}\in B_{n,n}$ such that

$\Psi\colon\thinspace\begin{cases}\alpha\longmapsto b_{\alpha}\Delta\\\
\beta\longmapsto b_{\beta}\Delta^{i(\beta)}\\\ a_{k}\longmapsto
b_{k}\Delta^{i(a_{k})},\end{cases}$

where $\Delta=\Delta_{2n}$ is the half twist. Applying $\Psi$ in the relation
of the presentation of $\pi_{1}(X_{\tau})$ and using the fact that $B_{n,n}$
is a normal subgroup of $B^{2}_{n,n}$, we obtain the following equation in
$B_{n,n}$:

$b_{\alpha}\Bigl{(}\Delta
b_{\beta}\Delta^{-1}\Bigr{)}\Delta^{2}\Bigl{(}\Delta^{i(\beta)-1}b_{\alpha}\Delta^{-i(\beta)+1}\Bigr{)}b_{\beta}^{-1}.\Biggl{(}b_{1}\Bigl{(}\Delta^{i(a_{1})}b_{2}\Delta^{-i(a_{1})}\Bigr{)}\Bigl{(}\Delta^{i(a_{2})}b_{1}\Delta^{-i(a_{2})}\Bigr{)}^{-1}b_{2}^{-1}\Biggr{)}\ldots\\\
.\Biggl{(}b_{2m-1}\Bigl{(}\Delta^{i(a_{2m-1})}b_{2m}\Delta^{-i(a_{2m-1})}\Bigr{)}\Bigl{(}\Delta^{i(a_{2m})}b_{2m-1}\Delta^{-i(a_{2m})}\Bigr{)}^{-1}b_{2m}^{-1}\Biggr{)}=1,$
(5.2)

where $\Delta^{2}=\Delta^{2}_{2n}$ is the full twist. Follows by Proposition 7
that it is well defined the homomorphism $\varepsilon\colon\thinspace
B_{n,n}\longrightarrow\mathbb{Z}_{2}$ given in the generators by

$\varepsilon\colon\thinspace\begin{cases}\sigma_{k}\longmapsto\overline{0},\text{
if }1\leq k\leq n-1\text{ or }n+1\leq k\leq 2n-1,\\\
A_{i,j}\longmapsto\overline{0},\text{ if }1\leq i<j\leq n\text{ or }n+1\leq
i<j\leq 2n,\\\ A_{i,j}\longmapsto\overline{1},\text{ if }1\leq i\leq n\text{
and }n+1\leq j\leq 2n.\end{cases}$

By (3.3) we have that

$\varepsilon(\Delta\sigma_{k}\Delta^{-1})=\varepsilon(\sigma_{2n-k})=\overline{0}=\varepsilon(\sigma_{k})$

for all $k$. Using the equality above and Remark 8, we conclude

$\varepsilon(\Delta A_{i,j}\Delta^{-1})=\varepsilon(A_{i,j})$

for all $i$ and $j$. Since the target of $\varepsilon$ is $\mathbb{Z}_{2}$, we
conclude that

$\varepsilon(b)+\varepsilon(\Delta b\Delta^{-1})=\overline{0}$ for all $b\in
B_{n,n}$.

Applying $\varepsilon$ in both sides of equation (5.2), using the property
above and (3.4), we obtain:

$\overline{0}=\varepsilon(\Delta^{2})=\displaystyle\sum_{1\leq i<j\leq
2n}\varepsilon(A_{i,j})=\sum_{j=n+1}^{2n}\sum_{i=1}^{n}\overline{1}=\overline{n^{2}}.$

The last equality is an absurd because $n$ is odd. Thus
$(X,\tau;\mathbb{R}^{2})$ has the $n$-BUP. ∎

##  6\. Some additional results about the triples $(X,\tau;Y)$

In this section, we obtain some partial results about the classification of
the triples $(X,\tau;Y)$ in relation to the $n$-BUP, where $X$ and $Y$ are
closed surfaces, $X\neq\mathbb{S}^{2}$. We stated these results in the
following two propositons. The first one show us that there exists triples
that do not have the $n$-BUP and in the second one we prove that the opposite
situation is also true. We start with the following lemma:

######  Lemma 17.

Let $X$ and $Y$ be closed surfaces. If a triple $(X,\tau;\mathbb{R}^{2})$ does
not have the $n$-split-BUP (resp. $n$-non-split BUP), then $(X,\tau;Y)$ also
does not have the $n$-split-BUP (resp. $n$-non-split BUP).

###### Proof.

Let $(X,\tau;\mathbb{R}^{2})$ be a triple that does not have the $n$-split BUP
(resp. $n$-non-split). As used in the proof of Theorem 3, there exists a
continuous map $\Phi_{1}\colon\thinspace X\longrightarrow F_{n}(Y)$ (resp.
$\Phi_{2}\colon\thinspace X\longrightarrow D_{n}(Y)$) such that
$\Phi_{i}(\tau(x))\neq\Phi_{i}(x)$ for all $x\in X$ and $i\in\\{1,2\\}$. Since
$Y$ is a closed surface, there exists an embedding
$\iota\colon\thinspace\mathbb{R}^{2}\longrightarrow Y$ which induces the
embeddings $\iota_{1}\colon\thinspace F_{n}(\mathbb{R}^{2})\longrightarrow
F_{n}(Y)$ and $\iota_{2}\colon\thinspace D_{n}(\mathbb{R}^{2})\longrightarrow
D_{n}(Y)$. Then the maps $\Phi^{\prime}_{i}=\iota_{i}\circ\Phi_{i}$ satisfies
$\Phi^{\prime}_{i}(\tau(x))\neq\Phi^{\prime}_{i}(x)$ for all $x\in X$ and
$i\in\\{1,2\\}$. Therefore the triple $(X,\tau;Y)$ does not have the $n$-split
BUP (resp. $n$-non-split BUP). ∎

Recall that $X_{\tau}$ denotes the orbit space of $X$ by the involution $\tau$
and $\hat{\delta}$ is defined after the sequence (1.1). The following
proposition follows imediately from Theorem 2 and Lemma 17

######  Proposition 18.

Let $n\geq 1$ and $X\neq\mathbb{S}^{2}$ and $Y$ be closed surfaces, where $X$
is equipped with a free involution $\tau$. The following conditions are true:

  1. (1)

If $X_{\tau}$ is a closed orientable surface or $X_{\tau}$ is a closed non-
orientable surface and $\hat{\theta_{\tau}}(\hat{\delta})=\overline{0}$, then
the triple $(X,\tau;Y)$ does not have the $n$-split BUP and it does not have
the $n$-non-split BUP.

  2. (2)

If $X_{\tau}$ is a closed non-orientable surface,
$\hat{\theta_{\tau}}(\hat{\delta})=\overline{1}$ and $n$ is even, then the
triple $(X,\tau;Y)$ does not have the $n$-non-split BUP.

Lastly, using the same idea of the proof of Proposition 10, we may conclude
that if a triple $(X,\tau;Y)$ has the $1$-BUP, then the same triple has the
$n$-split BUP for all positive integer $n$. Follows directly from this fact
and [20, Theorem 12 and Proposition 32] the following result. Recall that the
homomorphism $\theta_{\tau}$ was defined after the sequence (1.1).

######  Proposition 19.

Let $X$ and $Y$ be closed surfaces different of $\mathbb{S}^{2}$, where $X$ is
eqquiped with a free involution $\tau$ and $Y$ is orientable. In relation to
the presentation of $\pi_{1}(X_{\tau})$ given in Section 5, items (II) and
(III), suppose that one of the following conditions is true:

  1. (1)

$X_{\tau}$ is the Klein bottle and $\theta_{\tau}(u)=\overline{1}$.

  2. (2)

$X_{\tau}$ is the closed non-orientable surface of genus $3$ and
$\theta_{\tau}(c)=\theta_{\tau}(a_{i})=\overline{1}$ for some $i\in\\{1,2\\}$.

Then the triple $(X,\tau;Y)$ has the $n$-split BUP for all $n\geq 1$.

We hope to extend Propositions 18 and 19 and obtain a full classification of
the triples $(X,\tau;Y)$ in relation to the $n$-BUP, and we intend to present
these results soon. In order to accomplish this, we are using the techniques
developed in this article, in particular the intermediary braid groups
$B_{n,n}^{2}(Y)$ and Theorem 3.

##  7\. Acknowledgements

The work on this paper was developed during the Postdoctoral Internship of the
first author at IME-USP from March 2020 to August 2021 that was supported by
the Capes/INCTMat project nº 88887.136371/2017-00-465591/2014-0. The authors
would like to thanks the professors Daciberg Lima Gonçalves and Daniel
Vendrúscolo for their great contributions.

## References

  * [1]
  * [2] E. Artin, Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamb. 4 (1926), 47–72.
  * [3] E. Artin, Theory of braids, Ann. Math. 48 (1947), 101–126.
  * [4] T. Asoh, Classification of Free Involutions on Surfaces, Hiroshima Math J. 6 (1976), 171–181.
  * [5] J.S. Birman, Braids, links and mapping class groups, Ann.Math. Stud. 82, Princeton University Press, 1974.
  * [6] K. Borsuk, Drei Sätze über die $n$-dimensionale Euklidische Sphäre, Fund. Math. 20 (1933), 177–190.
  * [7] R.F. Brown, D.L. Goncalves, On the topology of $n$-valued maps, Adv. Fixed Point Theory, 8 (2018), 205–220.
  * [8] W.L. Chow, On the algebraical braid group, Ann. Math. 49 (1948), 654–658.
  * [9] P.E. Desideri, P.L. Q. Pergher, D. Vendrúscolo, Some generalizations of the Borsuk-Ulam Theorem. Publicationes Mathematicae (Debrecen), 78, (2011), 583–593.
  * [10] E. Fadell, S.Y. Husseini, Geometry and Topology of Configuration Spaces, Springer Monographs in Mathematics, (Springer-Verlag, Berlin 2001).
  * [11] E. Fadell, L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118.
  * [12] R.H. Fox, L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126.
  * [13] F.A. Garside, The braid group and other groups, Quart. J. Math. Oxford 20 (1969), 235–254.
  * [14] R. Gillette and J. Van Buskirk, The word problem and consequences for the braid groups and mapping class groups of the $2$-sphere, Trans. Amer. Math. Soc. 131 (1968), 277–296.
  * [15] C.H. Goldberg, An exact sequence of braid groups, Math. Scand. 33 (1973), 69–82.
  * [16] D.L. Gonçalves, The Borsuk-Ulam theorem for surfaces, Quaest. Math. 29 (2006), 117–123.
  * [17] D.L. Gonçalves, J. Guaschi, The roots of the full twist for surface braid groups, Math. Proc. Cambridge Phil. Soc. 137 (2004), 307–320.
  * [18] D. L. Gonçalves and J. Guaschi, The braid groups of the projective plane, Algebraic & Geometric Topology 4 (2004), 757–780.
  * [19] D.L. Gonçalves, J. Guaschi, The braid group $B_{n,m}(\mathbb{S}^{2})$ and the generalised Fadell-Neuwirth short exact sequence, J. Knot Theory Ramifications 14 (2005), 375–403.
  * [20] D.L. Gonçalves, J. Guaschi, The Borsuk-Ulam theorem for maps into a surface, Top. Appl. 157 (2010), 1742–1759.
  * [21] D.L. Gonçalves, J. Guaschi, Fixed points of $n$-valued maps, the fixed point property and the case of surfaces – a braid approach. Indagationes Mathematicae 29 (2018), 91–124.
  * [22] D.L. Gonçalves, J. Guaschi, V.C. Laass, The Borsuk-Ulam property for homotopy classes of self-maps of surfaces of euler characteristic zero. J. Fixed Point Theory Appl. 21 (2019) 21–65.
  * [23] F.V. Haeseler, G. Skordev, Borsuk-Ulam theorem, fixed point index and chain approximations for maps with multiplicity, Pacific J. Math. 153 (1992), 369–396.
  * [24] V.L. Hansen, Braids and Coverings: selected topics, London Math. Society Student Text 18, Cambridge University Press, 1989.
  * [25] S.T. Hu, Elements of General Topology, Holden-Day Series in Mathematics (1964), Holden-Day, Inc.
  * [26] M. Izydorek, Remarks on Borsuk-Ulam Theorem for multivalued maps, Bull. Polish Acad. Sci. Math. 35 (1987), no. 7-8, 501–504.
  * [27] D.L. Johnson, Presentation of groups, 2nd edition, London Mathematical Society Student Texts, vol.15, Cambridge University Press, 1997.
  * [28] J. Li, J Wu, Artin braid groups and homotopy groups, Proc. London Math. Soc. (3) 99 (2009) 521–556.
  * [29] J. Matoušek, Using the Borsuk-Ulam Theorem, Universitext, Springer–Verlag (2002).
  * [30] D. de Mattos, T.F.M. Monis, E.L. dos Santos, Relative Borsuk-Ulam Theorems for Spaces with a Free $\mathbb{Z}_{2}$-action, Bulletin Polish Acad. Sci. Math. 61 (2013), 71–77.
  * [31] J.R. Munkres, Topology, Prentice Hall, Second Edition (2000).
  * [32] K. Murasugi, Seifert fibre spaces and braid groups, Proc. London Math. Soc. 44 (1982), 71–84
  * [33] H. Schirmer, Fixed points, antipodal points and coincidence of n-acyclic valued multifunctions, Contemp. Math., 21 (1983), 207–212.
  * [34] H. Schirmer, Fix-finite approximations of n-valued multifunctions, Fund. Math. 121 (1984), 73–80.
  * [35] H. Steinlein, Borsuk’s antipodal theorem and its generalizations and applications: a survey, Topological Methods in Nonlinear Analysis, Sém. Math. Sup. 95, Presses Univ. Montréal, Montreal, QC (1985), 166–235.
  * [36]


11institutetext: 1Department of Computer Science and Software Engineering,
Concordia University, Montréal, QC, Canada
2Center for the Developing Brain, Child Mind Institute, New York, NY, USA
11email<EMAIL_ADDRESS>
# Accurate simulation of operating system updates in neuroimaging using Monte-
Carlo arithmetic

Ali Salari1 Yohan Chatelain1 Gregory Kiar2 Tristan Glatard1

###### Abstract

Operating system (OS) updates introduce numerical perturbations that impact
the reproducibility of computational pipelines. In neuroimaging, this has
important practical implications on the validity of computational results,
particularly when obtained in systems such as high-performance computing
clusters where the experimenter does not control software updates. We present
a framework to reproduce the variability induced by OS updates in controlled
conditions. We hypothesize that OS updates impact computational pipelines
mainly through numerical perturbations originating in mathematical libraries,
which we simulate using Monte-Carlo arithmetic in a framework called “fuzzy
libmath” (FL). We applied this methodology to pre-processing pipelines of the
Human Connectome Project, a flagship open-data project in neuroimaging. We
found that FL-perturbed pipelines accurately reproduce the variability induced
by OS updates and that this similarity is only mildly dependent on simulation
parameters. Importantly, we also found between-subject differences were
preserved in both cases, though the between-run variability was of comparable
magnitude for both FL and OS perturbations. We found the numerical precision
in the HCP pre-processed images to be relatively low, with less than $8$
significant bits among the $24$ available, which motivates further
investigation of the numerical stability of components in the tested pipeline.
Overall, our results establish that FL accurately simulates results
variability due to OS updates, and is a practical framework to quantify
numerical uncertainty in neuroimaging.

###### Keywords:

Computational reproducibility Neuroimaging pipelines Monte-Carlo arithmetic.

##  1 Introduction

Numerical round-off and cancellation errors are ubiquitous in floating-point
computations. In neuroimaging, they contribute to results uncertainty along
with other sources of variability, including population selection, scanning
devices, sequence parameters, acquisition noise, and methodological
flexibility [3, 2]. Numerical errors manifest particularly through variations
in elementary mathematical libraries resulting from operating system (OS)
updates. Indeed, due to implementation differences, mathematical functions
available in different OS versions provide slightly different results. The
impact of such epsilonesque differences on image analysis depends on the
conditioning of the problem and the pipeline’s numerical implementation. In
neuroimaging, established image processing pipelines have been shown to be
substantially impacted: for instance, differences in cortical thicknesses
measured by the same Freesurfer version in different execution platforms were
shown to reach statistical significance in some brain regions [9], and Dice
coefficients as low as 0.6 were observed between FSL or Freesurfer
segmentations obtained in different platforms [8, 18]. Such observations
threaten the validity of neuroimaging results by revealing systematic
instabilities.

Despite its possible implications on results validity, the effect of OS
updates remains seldom studied due to (1) the lack of closed-form expressions
of condition numbers for complex pipelines and non-differentiable non-linear
analyses, (2) the technical challenge associated with experimental studies
involving multiple OS distributions and versions, (3) the uncontrolled nature
of OS updates. As a result, the effect of OS updates on neuroimaging analyses
is generally neglected or handled through the use of software containers
(Docker or Singularity), static executable builds, or similar approaches.
While such techniques improve experiment portability, they only mask numerical
instabilities and do not tackle them. Numerical perturbations are bound to
reappear due to security updates [14], obsoleting software [17], or
parallelization. Therefore, the mechanisms through which numerical
instabilities propagate need to be investigated and eventually addressed.

This paper presents “fuzzy libmath” (FL), a framework to simulate OS updates
in controlled conditions, allowing software developers to evaluate the
robustness of their tools with respect to likely-to-occur numerical
perturbations. As we hypothesize that numerical perturbations resulting from
OS updates primarily come from implementation differences in elementary
mathematical libraries, we leverage Monte-Carlo arithmetic (MCA) [16] to
introduce controlled amounts of noise in these libraries. FL enables MCA in
mathematical functions used by existing pipelines without the need to modify
or recompile them. To demonstrate the approach, we study the effect of common
OS updates on the numerical precision of structural MRI pre-processing
pipelines of the Human Connectome Project [19], a major neuroimaging
initiative.

##  2 Simulating OS updates with Monte-Carlo arithmetic

MCA models floating-point roundoff and cancellations errors through random
perturbations, allowing for the estimation of error distributions from
independent random result samples. MCA simulates computations at a given
_virtual precision_ using the following perturbation:

$inexact(x)=x+2^{e_{x}-t}\xi$ (1)

where $e_{x}$ is the exponent in the floating-point representation of $x$, $t$
is the virtual precision and $\xi$ is a random uniform variable of
$(-\frac{1}{2},\frac{1}{2})$.

MCA allows for three perturbation modes: Random Rounding (RR) introduces the
perturbation in function outputs, simulating roundoff errors; Precision
Bounding (PB) introduces the perturbation in function operands, allowing for
the detection of catastrophic cancellations; and, Full MCA combines RR and PB,
resulting in the following perturbation:

$mca\\_mode(x\circ y)=inexact_{RR}(inexact_{PB}(x)\circ inexact_{PB}(y))$ (2)

To simulate OS updates, we introduce random perturbations in the GNU
mathematical library, the main mathematical library in GNU/Linux systems.
Instrumenting mathematical libraries with MCA raises a number of issues as
many functions assume deterministic arithmetic. For instance, applying random
perturbations around a discontinuity or within piecewise approximations
results in large variations and a total loss of significance that are not
relevant in our context. Therefore, we have applied MCA to proxy mathematical
functions wrapping those in the original library, such that only the outputs
of the original functions were perturbed but not their inputs or the
implementations themselves. This technique allows us to control the magnitude
of the perturbation as perceived by the application.

We instrumented the GNU mathematical library with MCA using Verificarlo [5], a
tool that (1) uses the Clang compiler to generate an LLVM (http://llvm.org)
Intermediate Representation (IR) of the source code, (2) replaces floating-
point operations in the IR by a call to the Verificarlo API, and (3) compiles
the modified IR to an executable using LLVM. The perturbation applied by the
Verificarlo API can be configured at runtime, for instance to change the
virtual precision applied to single- and double-precision floating-point
values.

The resulting MCA-instrumented mathematical library, “fuzzy libmath” (FL), is
loaded in the pipeline using LD_PRELOAD, a Linux mechanism to force-load a
shared library into an executable. As a result, functions defined in fuzzy
libmath transparently overload the original ones without the need to modify or
recompile the analysis pipeline. Fuzzy libmath functions call the original
functions through dlsym, a function that returns the memory address of a
symbol. To trigger MCA instrumentation, a floating-point zero is added to the
output of the original function and the result of this sum is perturbed and
returned.

Finally, we measure results precision as the number of significant bits among
result samples, as defined in [16]:

$s=-\log_{2}\left|\frac{\sigma}{\mu}\right|$ (3)

where $\sigma$ and $\mu$ are the observed cross-sample standard deviation and
average.

##  3 HCP Pipelines & Dataset

We apply the methodology described above to the minimal structural pre-
processing pipeline associated with the Human Connectome Project (HCP) dataset
[7], entitled “PreFreeSurfer”. This pipeline consists of many independent
components, including: spatial distortion correction, brain extraction, cross-
modal registration, and alignment to standard space. Each high-level component
of this pipeline (Fig. 1) consists of several function calls using FSL, the
FMRIB Software Library [12]. The pipeline requires T1w and T2w images for each
subject. A full description of the pipeline is available at [7].

It should be noted that the PreFreeSurfer pipeline uses both single and double
precision functions from the GNU mathematical library. Among the pre-
processing steps in the pipeline, it has been shown that linear and non-linear
registrations implemented in FSL FLIRT [13, 11] and FNIRT [1] are the most
sensitive to numerical instabilities [18].

Figure 1: PreFreeSurfer pipeline steps.

We selected $20$ unprocessed subjects from the HCP data release S500 available
in the ConnectomDB repository. We selected these subjects from different
subject types to cover execution paths sufficiently. For each, the available
data consisted of $1$ or $2$ T1w and T2w images each, with spatial dimensions
of $256\times 320\times 320$ and voxel resolution of $0.7$ mm. Acquisition
protocols and parameters are detailed in [19]. Two distinct experimental
configurations were tested:

Operating Systems (OS):

subjects were processed on three different Linux operating systems inside
Docker images: CentOS7 (glibc v.2.17), CentOS8 (glibc v.2.28), and Ubuntu20
(glibc v.2.31).

Fuzzy libmath (FL):

the dataset was processed on an Ubuntu20 system using fuzzy libmath. The
virtual precision ($t$) for the perturbations was swept from $53$ bits (the
full mantissa for double-precision data) down to $1$ bit by steps of $2$. For
$t>=24$ bits, only double-precision was altered and single-precision was set
to 24 bits, and for $t<24$ bits, both double- and single-precision
simultaneously were changed. Three FL-perturbed samples were generated for
each subject and virtual precision, to match the number of OS samples.

After conducting both experiments, we selected the virtual precision that most
closely simulated the variability observed across OSes via the root-mean-
square error (RMSE) between the number of significant bits per voxel in all
subjects and conditions. This precision is referred to as the global nearest
virtual precision and was used to compare results obtained in both the FL and
OS versions.

##  4 Results

The fuzzy libmath source code, Docker image specifications, and analysis code
to reproduce the results are available at https://github.com/big-data-lab-
team/MCA-libmath-paper. All experiments were conducted on the Béluga HPC
computing cluster made available by Compute Canada through Calcul Québec.
Béluga is a general-purpose cluster with $872$ available nodes. All nodes
contain $2\times$ Intel Gold 6148 Skylake @ 2.4 GHz ($40$ cores/node) CPU, and
node memory can range between $92$ to $752$ GB. The average processing time of
the pipeline without FL instrumentation was $69$ minutes (average of 3
executions). The FL perturbation increased it to $93$ minutes.

We ensured that the pipeline does not use pseudo-random numbers by processing
each subject twice on the same operating system. To validate that FL was
correctly instrumented with Verificarlo, we used Veritracer [4], a tool for
tracing the numerical quality of variables over time. For one subject, the
traces showed that the number of significant bits in the function outputs
varied over time, confirming the instrumentation with MCA. Throughout the
pipeline execution, Veritracer reported approximately $4$ billion calls to FL,
with the following ratio of calls: $47.12\%$ log, $40.96\%$ exp, $6.92\%$
expf, $3.39\%$ logf, $1.55\%$ sincosf, and $0.06\%$ of cumulated calls to
atan2f, pow, sqrt, exp2f, powf, log10f, log10, cos, and asin. We also checked
that long double types were not used.

###  4.1 Fuzzy libmath accurately simulates the effect of OS updates

(a) Distribution of significant bits

(b) Significance map (subject average)

Figure 2: Comparison of OS and FL effects on the precision of PreFreeSurfer
results for n=20 subjects. FL samples were obtained at the global nearest
virtual precision of t=37 bits.

Fuzzy libmath accurately reproduced the effect of OS updates, both globally
(Fig. 2(a)) and locally (Fig. 2(b)). The distributions of significant bits in
the atlas registered T1w images were nearly identical ($p>0.05$, KS test) on
the average and individual subject distributions for $15/20$ subjects, after
correcting for multiple comparisons. Locally, the spatial distribution of
significant digits also appeared to be preserved. Losses in significance were
observed mainly at the brain-skull interface and between brain lobes,
indicating spatial dependency of numerical properties.

The average number of significant bits in either the FL or OS conditions were
$7.76$ out of $24$ available, which corresponds to $2.32$ significant (base
$10$) digits. This relatively low precision motivates future investigations of
the stability of pipeline components, in particular for image registration.

###  4.2 Fuzzy libmath preserves between-subjects image similarity

Numerically-perturbed samples remained primarily clustered by individual
subjects (Fig. 3), indicating that neither FL nor OS perturbations were
impactful enough to blur the differences between subjects. Notably, the
similarity between subjects was also preserved by the numerical perturbation,
leading to the same subject ordering in the dendrograms. However, the average
RMSE within samples of a given subject was approximately $13\times$ lower than
the average RMSE between different subjects. The fact that between-subject
variabilities were nearly on the same order of magnitude as OS and FL
variability demonstrates the potential severity of these instabilities.

Figure 3: RMSE-based hierarchical clustering of OS (left) and FL (right)
samples. Colors identify different subjects, showing that similarities between
subjects are preserved by the numerical perturbations. Horizontal gray lines
represent average RMSEs between (top line) and within (bottom line) subject
clusters.

###  4.3 Results are stable across virtual precision

The FL results presented previously were obtained at the global nearest
virtual precision of t=$37$ bits, determined as the precision which minimized
the RMSE between FL and OS average maps of significant bits. We varied the
virtual precision in steps of $2$ between t=$1$ and t=$53$ bits (Fig. 4). On
average, no noticeable RMSE change was observed between the FL and OS
variability for precisions ranging from t=$21$ to t=$53$ bits, which shows
that FL can robustly approximate OS updates.

Figure 4: Comparison of RMSE values computed between OS and FL results for
different virtual precisions.

The observed plateau suggests the existence of an “intrinsic precision” for
the pipeline, above which no improvement in results precision is expected. For
the tested pipeline, this intrinsic precision was observed at t=$21$ bits,
which indicates that the pipeline could be implemented exclusively with
single-precision floating-point representations ($24$ bits of mantissa)
without loss of results precision. This would substantially decrease the
pipeline memory footprint and computational time, as approximately $88\%$ of
operations used in this pipeline made use of double-precision data. In
addition, the presence of such a plateau suggests that numerical perturbations
introduced by OS updates might be in the range of machine error (t=$53$ bits),
although it is also possible that the extent of the plateau results from the
numerical conditioning of the tested pipeline. It is possible in contrast that
the absence of such a plateau would suggest an unstable pipeline that would
benefit either from correction or larger datatypes. The ability to capture
stability across a range of precisions importantly demonstrates a key
advantage of using FL to simulate OS variability.

The relationship between RMSE of individual subjects was generally consistent
with the average line, with the notable exception of subject $18$. The
observed discrepancies between this subject and potential others might be
leveraged for quality control checks and, as a result, inform tool
development.

The pipeline failed to complete for at least one subject below the virtual
precision of t=$13$ bits, also referred to as the tolerance of the pipeline.
Specifically, $51\%$ of pipeline executions crashed among all subjects for
precisions ranging from $1$–$11$ bits, and there was no relationship between
tolerance-level and precision. The error raised was in the Readout Distortion
Correction portion of the pipeline, and appears to stem from the FSL FAST
tissue segmentation. The specific source of the error within this component is
presently unknown, but is an open question for further exploration.

##  5 Conclusion & Discussion

We demonstrated fuzzy libmath as an accurate method to simulate variability in
neuroimaging results due to OS updates. Alongside this evaluation, fuzzy
libmath can be used by pipeline developers or consumers to evaluate the
numerical uncertainty of tools and results. Such evaluations may also help
decrease pipeline memory usage and computational time through the controlled
use of reduced numerical precision. Fuzzy libmath does not require any
modification of the pipeline as it operates on the level of shared libraries.
The accuracy of the simulations were shown to be robust across a wide range of
virtual precisions, which reinforces the applicability of the method.

The proposed technique is directly applicable to MATLAB code executed with GNU
Octave, to Python programs executed on Linux, and to C programs that depend on
GNU libmath. Numerical noise can be introduced in other libraries, such as
OpenBLAS or NumPy, using our https://github.com/verificarlo/fuzzy environment.

A commonly used approach to address instabilities resulting from OS version
updates in practice is to sweep the issue under the rug of software containers
or static linking. While such solutions are undoubtedly helpful to improve
code portability or strict re-executability, a more honest position is to
consider computational results as realizations of random variables depending
on numerical error. The presented technique enables estimating result
distributions, a first step toward making analyses reproducible across
heterogeneous execution environments. While this work did not investigate the
precise cause of numerical instabilities by tracing the system function calls,
this is a topic for future work.

The tested OS versions span a timeframe of 7 years (2012–2020) and focused on
GNU/Linux, a widely-used platform in neuroimaging [10]. Given that our
experiments focused on numerical perturbations applied to mathematical
functions, which are implemented similarly across OSes, our findings are
likely to generalize to OS/X or MS Windows, although future work would be
needed to confirm that. The tested pipeline is the official solution of the
HCP project to pre-process data, and is considered the state-of-the-art. This
pipeline assembles software components from the FSL toolbox consistent with
common practice in neuroimaging, such as in fMRIPrep [6] or the FSL feat
workflow [12], to which fuzzy libmath can be directly applied. Efforts are on-
going to use fuzzy libmath in fMRIPrep software tests, to guarantee that bug
fixes do not perturb results beyond numerical uncertainty.

The fact that the induced numerical variability preserves image similarity
between subjects is reassuring and, in fact, exciting. OS updates provide a
convenient, practical target to define a virtual precision leading to a
detectable but still reasonable numerical perturbation. However, it is also of
importance that OS- and FL-induced variability were on a similar order of
magnitude as subject-level effects. This suggests that the preservation of
relative between-subject differences may not hold in all pipelines, and such a
comparison could be used to evaluate the robustness of a pipeline to OS
instabilities. The fact that the results observed across OS versions and FL
perturbations arise from equally-valid numerical operations also suggests that
the observed variability may contain meaningful signal. In particular, signal
measured from these perturbations might be leveraged to enhance biomarkers, as
suggested in [15] where augmenting a diffusion MRI dataset with numerically-
perturbed samples was shown to improve age classification.

## References

  * [1] Andersson, J.L., Jenkinson, M., Smith, S., et al.: Non-linear registration, aka Spatial normalisation FMRIB. Tech. Rep. TR07JA2, FMRIB Analysis Group of the University of Oxford (2007)
  * [2] Botvinik-Nezer, R., Holzmeister, F., Camerer, C.F., Dreber, A., Huber, J., Johannesson, M., Kirchler, M., Iwanir, R., Mumford, J.A., Adcock, R.A., et al.: Variability in the analysis of a single neuroimaging dataset by many teams. Nature 582(7810), 84–88 (2020)
  * [3] Bowring, A., Maumet, C., Nichols, T.E.: Exploring the impact of analysis software on task fMRI results. Human Brain Mapping 40, 1–23 (2019)
  * [4] Chatelain, Y., de Oliveira Castro, P., Petit, E., Defour, D., Bieder, J., Torrent, M.: Veritracer: Context-enriched tracer for floating-point arithmetic analysis. In: 2018 IEEE 25th Symposium on Computer Arithmetic (ARITH). pp. 61–68. IEEE (2018)
  * [5] Denis, C., de Oliveira Castro, P., Petit, E.: Verificarlo: Checking Floating Point Accuracy through Monte Carlo Arithmetic. In: 2016 IEEE 23nd Symposium on Computer Arithmetic (ARITH). pp. 55–62 (2016)
  * [6] Esteban, O., Markiewicz, C.J., Blair, R.W., Moodie, C.A., Isik, A.I., Erramuzpe, A., Kent, J.D., Goncalves, M., DuPre, E., Snyder, M., et al.: fMRIPrep: a robust preprocessing pipeline for functional MRI. Nature methods 16(1), 111–116 (2019)
  * [7] Glasser, M.F., Sotiropoulos, S.N., Wilson, J.A., Coalson, T.S., Fischl, B., Andersson, J.L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J.R., et al.: The minimal preprocessing pipelines for the Human Connectome Project. Neuroimage 80, 105–124 (2013)
  * [8] Glatard, T., Lewis, L.B., Ferreira da Silva, R., Adalat, R., Beck, N., Lepage, C., Rioux, P., Rousseau, M.E., Sherif, T., Deelman, E., Khalili-Mahani, N., Evans, A.C.: Reproducibility of neuroimaging analyses across operating systems. Frontiers in Neuroinformatics 9, 12 (2015)
  * [9] Gronenschild, E.H.B.M., Habets, P., Jacobs, H.I.L., Mengelers, R., Rozendaal, N., van Os, J., Marcelis, M.: The effects of FreeSurfer version, workstation type, and Macintosh operating system version on anatomical volume and cortical thickness measurements. PloS one 7(6), e38234 (Jan 2012)
  * [10] Hanke, M., Halchenko, Y.O.: Neuroscience runs on gnu/linux. Frontiers in neuroinformatics 5, 8 (2011)
  * [11] Jenkinson, M., Bannister, P., Brady, M., Smith, S.: Improved optimization for the robust and accurate linear registration and motion correction of brain images. Neuroimage 17(2), 825–841 (2002)
  * [12] Jenkinson, M., Beckmann, C.F., Behrens, T.E., Woolrich, M.W., Smith, S.M.: FSL. Neuroimage 62(2), 782–790 (2012)
  * [13] Jenkinson, M., Smith, S.: A global optimisation method for robust affine registration of brain images. Medical image analysis 5(2), 143–156 (2001)
  * [14] Kaur, B., Dugré, M., Hanna, A., Glatard, T.: An analysis of security vulnerabilities in container images for scientific data analysis. GigaScience 10(6), giab025 (2021)
  * [15] Kiar, G., Chatelain, Y., Salari, A., Evans, A.C., Glatard, T.: Data augmentation through monte carlo arithmetic leads to more generalizable classification in connectomics. bioRxiv (2020)
  * [16] Parker, D.S.: Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic. University of California (Los Angeles). Computer Science Department (1997)
  * [17] Perkel, J.M.: Challenge to scientists: does your ten-year-old code still run? Nature 584(7822), 656–658 (2020)
  * [18] Salari, A., Kiar, G., Lewis, L., Evans, A.C., Glatard, T.: File-based localization of numerical perturbations in data analysis pipelines. GigaScience 9(12) (12 2020)
  * [19] Van Essen, D.C., Smith, S.M., Barch, D.M., Behrens, T.E., Yacoub, E., Ugurbil, K., Consortium, W.M.H., et al.: The wu-minn Human Connectome Project: an overview. Neuroimage 80, 62–79 (2013)