# Holomorphic maps between configuration spaces of Riemann surfaces

Lei Chen and Nick Salter  LC: Department of Mathematics, University of
Maryland, 4176 Campus Drive, College Park, MD 20742<EMAIL_ADDRESS>NS:
Department of Mathematics, University of Notre Dame, Hurley Hall, Notre Dame,
IN 46556<EMAIL_ADDRESS>
(Date: January 19, 2023)

###### Abstract.

We prove a suite of results classifying holomorphic maps between configuration
spaces of Riemann surfaces; we consider both the ordered and unordered setting
as well as the cases of genus zero, one, and at least two. We give a complete
classification of all holomorphic maps
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{m}(\mathbb{C})$
provided that $n\geq 5$ and $m\leq 2n$ extending the Tameness Theorem of Lin,
which is the case $m=n$. We also give a complete classification of holomorphic
maps between ordered configuration spaces of Riemann surfaces of genus at most
one (answering a question of Farb), and show that the higher genus setting is
closely linked to the still-mysterious “effective de Franchis problem”. The
main technical theme of the paper is that holomorphicity allows one to promote
group-theoretic rigidity results to the space level.

##  1\. Introduction

Let $\operatorname{PConf}_{n}(X)$ denote the space of $n$ ordered distinct
points on a manifold $X$, and let $\operatorname{Conf}_{n}(X)$ denote the
corresponding space of unordered tuples. If $X$ has a complex structure, then
$\operatorname{PConf}_{n}(X)$ and $\operatorname{Conf}_{n}(X)$ inherit complex
structures from $X$. In this paper we take $X,Y$ to be Riemann surfaces, and
consider the family of problems of classifying holomorphic maps
$h:\operatorname{(P)Conf}_{n}(X)\to\operatorname{(P)Conf}_{m}(Y)$.

As discussed in more detail below, a complete answer in full generality seems
to be out of reach, involving unresolved questions related to the “effective
de Franchis problem” of enumerating and bounding the set of all holomorphic
maps $X\to Y$ as well as delicate group-theoretic considerations. However, we
are able to address a good portion of the general problem, especially in the
ordered setting.

A general phenomenon recurring throughout our results is that of “twisting”.
Given a holomorphic map
$h:\operatorname{(P)Conf}_{n}(X)\to\operatorname{(P)Conf}_{m}(Y)$, suppose a
holomorphic map $A:\operatorname{(P)Conf}_{n}(X)\to\operatorname{Aut}(Y)$ is
given, where $\operatorname{Aut}(Y)$ is the group of holomorphic automorphisms
of $Y$, in our setting itself a complex manifold. Then the twist $h^{A}$ of
$h$ is defined via the formula

$h^{A}(x_{1},\dots,x_{n})=A(x_{1},\dots,x_{n})(h(x_{1},\dots,x_{n})).$

Note that the affine twist of a constant map need not be constant. In the case
$Y=\mathbb{C}$, the relevant automorphism group is the affine group
$\operatorname{Aff}=\\{az+b\mid a\in\mathbb{C}^{*},b\in\mathbb{C}\\}$, in
which case we call this an affine twist. We also note that
$\operatorname{PConf}_{n}(X)$ admits a family of automorphisms given by
permuting the coordinates; for ease of stating our main results, we will also
consider this a kind of twist.

The following summarizes our main results; see the indicated statements for
precise details.

  * •

3.1: For $m\geq 5$ and $n\leq 2m$, up to affine twisting, every holomorphic
map
$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{m}(\mathbb{C})$
is either constant, the identity, or a “root map” (see Section 3).

  * •

4.3: For $m\geq 2$, up to a slight generalization of affine twisting, every
holomorphic map
$h:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$
is either constant or a forgetful map.

  * •

4.4: For $n\geq 3$, up to twisting, every holomorphic map
$h:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\operatorname{PConf}_{m}(\mathbb{CP}^{1})$
is either constant or a forgetful map.

  * •

5.2: Let $X,Y$ be compact Riemann surfaces of genera $g(X)=g(Y)=1$. Then
$X\cong Y$ and up to twisting, every holomorphic map
$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$ is a forgetful
map.

  * •

5.3: Let $X,Y$ be compact Riemann surfaces with $g(X)\geq 2$ and $g(Y)=1$.
Then up to twisting, every holomorphic map
$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$ is constant.

  * •

6.1: Let $X,Y$ be compact Riemann surfaces each of genus at least two. Then up
to twisting, either $X\cong Y$ and $h$ is a forgetful map, or else $h$ factors
as the composition of a forgetful map $p:\operatorname{PConf}_{n}(X)\to X$ and
a holomorphic map $f:X\to\operatorname{PConf}_{m}(Y)$.

We note that in the case $m=n$, 3.1 was previously established by Lin [15,
Theorem 1.4]. Farb [10, Problem 2.4] has asked for a classification of
holomorphic maps
$\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$,
which is resolved in 4.3.

Observe that in 6.1, a holomorphic map $f:X\to\operatorname{PConf}_{m}(Y)$ is
the same thing as an $m$-tuple $f_{1},\dots,f_{m}$ of holomorphic maps
$f_{i}:X\to Y$ with the properties that the graphs are pairwise disjoint in
$X\times Y$. The study of the set $\operatorname{Hol}(X,Y)$ of nonconstant
holomorphic maps $X\to Y$ falls under the purview of the classical de Franchis
theorem, which states that when $g(Y)\geq 2$, the set is finite. Unlike its
cousin the Hurwitz theorem (the case $X=Y$), there is a very large gap between
known upper bounds on $\left|\operatorname{Hol}(X,Y)\right|$ as a function of
the genera $g(X),g(Y)$, and the lower bounds achieved by examples: by [3],
there is an upper bound that is slightly super-exponential in $g(X)$, whereas,
to the authors’ knowledge, there are no families of examples where the number
of holomorphic maps grows faster than linearly with $g(X)$. However, we show
that the extra constraint of having disjoint graphs is highly restrictive,
which may be of independent interest:

  * •

6.3: Let $X,Y$ be compact Riemann surfaces each of genus at least two, and let
$f:X\to\operatorname{PConf}_{m}(Y)$ be holomorphic. Then $m\leq 4g(X)g(Y)$.

Our final main result is of a slightly different flavor, considering instead
the holomorphic rigidity properties of $\operatorname{Conf}_{n}(\mathbb{C})$
into the moduli space of curves $\mathcal{M}_{g}$. One such map is given by
the hyperelliptic embedding
$H:\operatorname{Conf}_{n}(\mathbb{C})\to\mathcal{M}_{g}$ for
$g=[\frac{n-1}{2}]$:

$H(\\{x_{1},...,x_{n}\\})=\text{the algebraic curve
}\\{y^{2}=(x-x_{1})...(x-x_{n})\\}$

  * •

3.2: For $n\geq 26$ and $g\leq n-2$, if
$h:\operatorname{Conf}_{n}(\mathbb{C})\to{\mathcal{M}}_{g}$ is a non-constant
holomorphic map of orbifolds, then $h$ is the hyperelliptic embedding.

The holomorphic landscape. To put the results of this paper into better
context, we describe here what is known about the totality of holomorphic maps
between configuration spaces. We organize the problem along two axes: first,
whether the configurations are ordered or not, and secondly by the genera of
the Riemann surfaces $X$ and $Y$. An entry is left blank if we neither know of
any interesting examples nor a classification. The “covering construction”
mentioned below is the following: let $p:Y\to X$ be an unbranched covering of
compact Riemann surfaces of degree $d$. Then for all $n\geq 1$, there is a
holomorphic map $P:\operatorname{Conf}_{n}(X)\to\operatorname{Conf}_{dn}(Y)$,
given by $P(\\{x_{1},\dots,x_{n}\\})=p^{-1}(\\{x_{1},\dots,x_{n}\\})$.

$\begin{array}[]{|c|c|c|c|}\hline\cr\operatorname{PConf}&g(Y)=0&1&\geq 2\\\
\hline\cr g(X)=0&\mbox{Fully classified:}&\mbox{All constant:}&\mbox{All
constant:}\\\ &\mbox{Theorems \ref{Ccase},\ref{CPcase}}&\mbox{Proposition
\ref{lowtohigh}}&\mbox{Proposition \ref{lowtohigh}}\\\ \hline\cr
1&&\mbox{Fully classified:}&\mbox{All constant:}\\\ &&\mbox{Theorem
\ref{thmtorus}}&\mbox{Proposition \ref{lowtohigh}}\\\ \hline\cr\geq
2&&\mbox{Classified modulo}&\mbox{Classified modulo}\\\ &&\mbox{case
$m=1$:}&\mbox{understanding }\operatorname{Hol}(X,Y):\\\ &&\mbox{Theorem
\ref{thmtorus2}}&\mbox{Theorem \ref{mainpure}}\\\ \hline\cr\end{array}$
$\begin{array}[]{|c|c|c|c|}\hline\cr\operatorname{Conf}&g(Y)=0&1&\geq 2\\\
\hline\cr g(X)=0&\mbox{Partially classified:}&\mbox{All constant:}&\\\
&\mbox{Theorem \ref{main}}&\mbox{Proposition \ref{clowhigh}}&\\\ \hline\cr
1&&\mbox{Abundant:}&\\\ &&\mbox{covering construction}&\\\ \hline\cr\geq
2&&&\mbox{Abundant:}\\\ &&&\mbox{covering construction}\\\
\hline\cr\end{array}$

We think that filling in the remaining entries of these tables is a worthwhile
goal of future research. In particular, we would like to highlight the
following.

######  Question 1.1.

For $g(X),g(Y)\geq 2$, does every holomorphic map
$h:\operatorname{Conf}_{n}(X)\to\operatorname{Conf}_{m}(Y)$ arise via the
covering construction?

Rigidity: holomorphic vs. continuous. While it is hard to find holomorphic
maps between configuration spaces of Riemann surfaces, there are many homotopy
classes of continuous maps between them. For instance, there are continuous
maps
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{n+k}(\mathbb{C})$
by adding $k$ points “near infinity”, a “doubling map”
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{2n}(\mathbb{C})$
which replaces a configuration of points with two juxtaposed copies, and many
more complicated examples. Even in the case when the induced map
$\pi_{1}(\operatorname{Conf}_{n}(\mathbb{C}))\to\pi_{1}(\operatorname{Conf}_{m}(\mathbb{C}))$
factors through $\mathbb{Z}$, there are many possibilities, encompassing all
of the possible Nielsen-Thurston types of the associated mapping class of the
$m$-punctured plane. However, according to 3.1, none of the above can be
induced by a holomorphic map, so long as we are in the range $m\leq 2n$. Work
of the first author, Kordek, and Margalit [6] (cf. 2.1) explores this
phenomenon and gives a complete classification of homotopy classes of
continuous maps
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{m}(\mathbb{C})$ in
the range $m\leq 2n$; this is the source of the restriction $m\leq 2n$ in 3.1.
This is accomplished by giving a classification of homomorphisms $B_{n}\to
B_{m}$, where $B_{n}=\pi_{1}(\operatorname{Conf}_{n}(\mathbb{C}))$ is the
braid group on $n$ strands. Following [6], there is now a conjectural
classification of all homomorphisms $\rho:B_{n}\to B_{m}$ for $n\geq 5$. To
formulate it, we recall that one can also think of the braid group as
$B_{n}\cong\operatorname{Mod}(D_{n})$, the mapping class group of the
$n$-punctured disk.

######  Conjecture 1.2 (Reducibility conjecture).

For $n\geq 5$ and $m>n$, every homomorphism $\rho:B_{n}\to B_{m}$ either has
cyclic image or else is reducible, i.e. there is a nonempty set of disjoint
essential, non-boundary-parallel curves on the $m$-punctured disk invariant
under $\rho(B_{n})$.

As discussed in [6], the reducibility conjecture in fact implies a stronger
classification of homomorphisms, predicting that all homomorphisms are
recursively assembled from a small list of basic ones using a certain set of
operations. Our methods here can be used to show that 1.2 implies a complete
classification of all holomorphic maps between configuration spaces.

######  Theorem 1.3.

If 1.2 holds, then the statement of 3.1 holds for all $n\geq 5$ with no
restriction on $m$.

Prior Results. Within the world of moduli spaces of complex-analytic objects,
there have been various efforts to understand the relationship between their
behavior in the holomorphic and smooth categories. The results here expand on
the work [15] of Lin mentioned above, who obtained 3.1 in the case $m=n$, and
also investigated the corresponding questions in the setting of configuration
spaces on $\mathbb{CP}^{1}$. Our proof is independent, making use of the
developments of [6] that give a classification of maps between braid groups,
as well as employing some powerful results from Teichmüller theory.

Two other results, while not bearing directly on the results of this paper,
were a source of inspiration and merit mention. Antonakoudis–Aramayona–Souto
[1] give a classification of holomorphic maps
$\mathcal{M}_{g}\to\mathcal{M}_{h}$ for $h\leq 2g-2$, showing that the only
non-constant map is the identity. Letting $\mathcal{A}_{h}$ be the moduli
space of $h$-dimensional, principally-polarized abelian varieties, a recent
result of Farb [9] shows that a nonconstant holomorphic map
$f:\mathcal{M}_{g}\to\mathcal{A}_{h}$ exists for $h\leq g$ iff $h=g$ and $f$
is the map sending a curve to its Jacobian. It is interesting to note that in
both these results, every smooth map is homotopic to a holomorphic map. This
is in contrast to the setting of this paper, where smooth maps between
configuration spaces exist in relative abundance (see above).

Proof Strategy. The main theme running through our arguments is the promotion
of group-theoretic rigidity statements to the level of holomorphic maps. In
[5], the first author has established a general rigidity theorem classifying
homomorphisms $f:PB_{n}(X)\to\Lambda$, where $X$ is a Riemann surface,
$PB_{n}(X):=\pi_{1}(\operatorname{PConf}_{n}(X))$ denotes the space of ordered
configurations of $n$ points, and $\Lambda$ is a torsion-free nonelementary
hyperbolic group. When $f$ is induced from a holomorphic map $F$ between
complex manifolds, principles of complex analysis can be used to show that $F$
must have a specific form. In other settings, we make use of rigidity
phenomena for holomorphic maps of Riemann surfaces into the moduli space of
Riemann surfaces, equivalently the rigidity of holomorphically-varying
families of Riemann surfaces over Riemann surfaces. We will use two such
theorems in our proof of 3.1, both concerning the monodromy of such families,
the homomorphism $\rho:\pi_{1}(B)\to\operatorname{Mod}(S)$ from the
fundamental group of the base to the mapping class group of the fiber.
Specifically, we will use a result of Daskalopoulos-Wentworth [8] showing that
the monodromy of a non-isotrivial family of curves is necessarily “rich” in a
certain technical sense, and a result of Imayoshi–Shiga [13] stating that if
two families have the same monodromy, then they are equal to each other.
Inside $\operatorname{Conf}_{n}(\mathbb{C})$, there are many embedded Riemann
surfaces: given distinct points $Y\in\operatorname{Conf}_{n-1}(\mathbb{C})$,
there is an associated embedding
$i_{Y}:\mathbb{C}-Y\to\operatorname{Conf}_{n}(\mathbb{C})$ of the finite-type
surface $\mathbb{C}-Y$ into $\operatorname{Conf}_{n}(\mathbb{C})$. The
holomorphic map
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{m}(\mathbb{C})$
thus equips each such surface with a family of Riemann surfaces (specifically,
the fiber is $\mathbb{C}$ punctured at $m$ points). The monodromy of such
families factors through the homomorphism $B_{n}\to B_{m}$ induced by the map
on the configuration spaces; we exploit the classification of [6] along with
the criterion of Daskalopoulos-Wentworth to rule out many possibilities, and
the rigidity result of Imayoshi–Shiga gives control over the situations where
a map is possible. The phenomenon of affine twisting appears because the
target space $\operatorname{Conf}_{m}(\mathbb{C})$ is not exactly the moduli
space of $m$ points in $\mathbb{C}$, but becomes a finite cover of this after
erasing the action of the affine group $\operatorname{Aff}$.

Acknowledgements. NS is supported by NSF Award No. DMS-2153879. LC is
supported by NSF Award No. 2005409.

##  2\. background

###  2.1. Classification of homomorphisms between braid groups

The basic strategy of the proof of the main results is to first understand all
possible induced maps on the fundamental groups (i.e. braid groups), and then
to contrast this with known rigidity results about holomorphic maps between
the associated moduli spaces. In this section we carry out the first of these
tasks.

2.1 below is a corollary of the work of Chen–Kordek–Margalit [6]. To state it,
we introduce the following terminology. A transvection of a group homomorphism
$f:G\to H$ is a homomorphism $f^{t}:G\to H$ defined via

$f^{t}(g)=f(g)t^{\ell(g)},$

where $\ell:G\to\mathbb{Z}$ is a homomorphism and $t\in H$ centralizes $f(G)$.
A homomorphism $f:G\to B_{n}$ is said to be reducible if the image $f(G)$
preserves a finite set of isotopy classes of essential non-boundary-parallel
curves on the $n$-punctured disk (viewing $B_{n}$ as the mapping class group
of the punctured disk). Lastly, an element of $B_{n}$ is said to have
prefinite order if its image in $B_{n}/Z_{n}$ has finite order.

######  Theorem 2.1.

For $n\geq 5$ and $m\leq 2n$, let $\rho:B_{n}\to B_{m}$ be a homomorphism.
Then exactly one of the following conditions hold:

  1. (1)

$n=m$ and $\rho$ is the identity map up to transvection,

  2. (2)

$\rho$ is either reducible and nontrivial, or else has infinite cyclic image
generated by a pseudo-Anosov,

  3. (3)

$\rho$ has prefinite cyclic image.

###### Proof.

Note first that the statement is equivalent to the assertion that at least one
of the following conditions hold:

  1. (1’)

$n=m$ and $\rho$ is the identity map up to transvection,

  2. (2’)

$\rho$ is reducible (possibly trivial)

  3. (3’)

$\rho$ has cyclic image.

The latter assertion follows more readily from [6, Theorem 1.1], but we will
see in the proof of 3.1 that the former organization corresponds to the
classification of holomorphic maps.

[6, Theorem 1.1] asserts that for $n\geq 5$, any homomorphism $\rho:B_{n}\to
B_{2n}$ is a transvection of one of five “standard homomorphisms”, and [6,
Corollary 1.2] asserts that every $\rho:B_{n}\to B_{m}$ for $m<2n$ is one of
two of the five types possible when $m=2n$. We first argue that if $\rho$
satisfies at least one of the conditions, then so does any transvection of
$\rho$; then we will see that each standard homomorphism satisfies one of the
conditions.

There is nothing to check if $\rho$ satisfies condition (1’). Suppose now that
$\rho$ is reducible. If $\rho$ is the trivial homomorphism, then any
transvection of $\rho$ has cyclic image (note that if $t$ is pseudo-Anosov,
the transvection will not necessarily be reducible). If $\rho$ is reducible
and nontrivial, then $\rho$ has a nonempty “canonical reduction system”
$\\{C_{i}\\}$ of disjoint essential, non-boundary-parallel curves (see e.g.
[2]). Given a transvection $\rho^{t}$, the element $t$ commutes with every
element of $\rho$, from which it follows that $t$ preserves $\\{C_{i}\\}$ as
well, so that $\rho^{t}$ is likewise reducible. If $\rho$ has cyclic image,
then $\rho$ factors through the abelianization map $\ell:B_{n}\to\mathbb{Z}$,
and hence by construction any transvection of $\rho$ does as well.

We now see that each of the five “standard homomorphisms” described in [6]
satisfies one of the above conditions. Each of these is a routine
verification; for brevity’s sake we will assume familiarity with the five, as
given in [6, page 1]. The trivial homomorphism satisfies conditions (2’) and
(3’), the inclusion homomorphism satisfies (1’) if $m=n$ and (2’) if $m>n$,
and the diagonal, flip-diagonal, and $k$-twist cabling maps are all reducible
(condition (2’)). ∎

In anticipation of 3.2, we next consider the case of homomorphisms
$\rho:B_{n}\to\operatorname{Mod}(S_{g})$ (here $\operatorname{Mod}(S_{g})$
denotes the mapping class group of a closed surface of genus $g$). Let $H$ be
the hyperelliptic embedding defined in the introduction; $H_{*}$ is then the
induced homomorphism on orbifold fundamental groups. We have the following
result of Chen–Mukerjea [7, Theorem 1.1].

######  Theorem 2.2.

For $n\geq 26$ and $g\leq n-2$, let $h:B_{n}\to\operatorname{Mod}(S_{g})$ be a
homomorphism. Then up to transvection, $h$ is either trivial or $H_{*}$.

###  2.2. Some results from complex analysis and Teichmüller theory

Before considering the holomorphic theory of families of Riemann surfaces, we
first mention a useful variant of the removable singularity theorem that we
will employ throughout the paper.

######  Proposition 2.3.

Let $X,Y$ be Riemann surfaces of finite type, each of negative Euler
characteristic, and let $f:X\to Y$ be holomorphic. Then $f$ admits a
holomorphic extension $F:\bar{X}\to\bar{Y}$, where $\bar{X},\bar{Y}$ denote
the compact Riemann surfaces associated to $X,Y$.

###### Proof.

Since each of $X,Y$ have negative Euler characteristic, they are uniformized
by $\mathbb{D}$ and hence admit complete hyperbolic metrics. By the Schwarz-
Pick theorem, $f$ is distance non-increasing in these metrics. Let
$x\in\bar{X}-X$ be given, and let $\gamma\subset X$ be a loop encircling $x$.
The homotopy class of $\gamma$ admits representatives of arbitrarily short
length, and hence the same is true for $f(\gamma)$, showing that $f(\gamma)$
is either null-homotopic or else encircles a puncture in $Y$. Thus $f$ admits
a continuous extension $F:\bar{X}\to\bar{Y}$, and by the usual removable
singularity theorem, it follows that $F$ is holomorphic. ∎

We now turn our attention to families of Riemann surfaces. Let
$\mathcal{M}_{g,n}^{\prime}$ be a finite orbifold cover of
$\mathcal{M}_{g,n}$. Imayoshi–Shiga proved the following [13, Section 3].

######  Theorem 2.4 (Imayoshi–Shiga).

Let $B$ be a Riemann surface of finite type. If
$f,h:B\to\mathcal{M}_{g,n}^{\prime}$ are non-constant holomorphic maps and the
monodromy maps $f_{*}=h_{*}:\pi_{1}(B)\to\operatorname{Mod}(S_{g,n})$
coincide, then $f=h$.

A homomorphism $f:G\to\operatorname{Mod}(S_{g,n})$ is said to be sufficiently
large if the image contains two pseudo-Anosov elements with distinct fixed
point sets in $\text{PMF}(S_{g,n})$. We will not need to know the precise
meaning of these terms, only that a sufficiently large subgroup is not
reducible, i.e. there is no globally-invariant finite set of curves.
Daskalopoulos-Wentworth proved the following [8, Theorem 5.7].

######  Theorem 2.5 (Daskalopoulos-Wentworth).

Let $B$ be a Riemann surface of finite type and $f:B\to\mathcal{M}_{g,n}$ be a
non-constant holomorphic map. Then
$f_{*}:\pi_{1}(B)\to\operatorname{Mod}(S_{g,n})$ is sufficiently large.

###  2.3. Relations between $\operatorname{Conf}_{m}(\mathbb{C})$,
$\operatorname{PConf}_{m}(\mathbb{C})$ and $\mathcal{M}_{0,m,1}$

Let $\mathcal{M}_{0,m,1}$ denote the moduli space of $m+1$ points on
$\mathbb{CP}^{1}$, where one of the points is distinguished. We now discuss
the natural projection map
$\pi_{m}:\operatorname{Conf}_{m}(\mathbb{C})\to\mathcal{M}_{0,m,1}$. Let
$\operatorname{Aff}$ be the affine group of $\mathbb{C}$, which induces an
action on $\operatorname{Conf}_{m}(\mathbb{C})$ and on
$\operatorname{PConf}_{m}(\mathbb{C})$.

######  Lemma 2.6.

The preimage $\pi_{m}^{-1}(\pi_{m}(X))$ of a point
$X\in\operatorname{Conf}_{m}(\mathbb{C})$ is the orbit
$\operatorname{Aff}(X)\subset\operatorname{Conf}_{m}(\mathbb{C})$.

###### Proof.

The point $\pi_{m}(X)$ represents $\mathbb{CP}^{1}$ with distinct marked
points $(x_{0},\\{x_{1},...,x_{m}\\})$, the first of which is distinguished.
We first apply a Mobiüs transformation to send the first point into $\infty$.
Now $\pi_{m}(X)$ is represented as $(\infty,\\{x_{1},...,x_{m}\\})$ where we
have $m$ distinct points $x_{1},...,x_{m}\in\mathbb{C}$. Two points
$(\infty,\\{x_{1},...,x_{m}\\})$ and $(\infty,\\{y_{1},...,y_{m}\\})$
represent the same points in $\mathcal{M}_{0,m,1}$ if they are holomorphically
related, i.e. if and only if there is a biholomorphism
$f:\mathbb{C}\to\mathbb{C}$ taking $\\{x_{1}\dots,x_{m}\\}$ to
$\\{y_{1},\dots,y_{m}\\}$. The group of holomorphic automorphisms of
$\mathbb{C}$ is $\operatorname{Aff}$, and so $(\infty,\\{x_{1},...,x_{m}\\})$
and $(\infty,\\{y_{1},...,y_{m}\\})$ represent the same points in
$\mathcal{M}_{0,m,1}$ if and only if there is $f\in\operatorname{Aff}$ such
that $f(\\{x_{1},...,x_{m}\\})=\\{y_{1},...,y_{m}\\}$. ∎

Denote by $P\mathcal{M}_{0,m+1}$ the moduli space of $m+1$ ordered distinct
points in $\mathbb{CP}^{1}$. Similar to 2.6, we have the following (cf. [12,
p. 247 ff.] for the assertion about orbifold fundamental groups).

######  Lemma 2.7.

There is an isomorphism of complex orbifolds

$\operatorname{PConf}_{m}(\mathbb{C})/\operatorname{Aff}\cong
P\mathcal{M}_{0,m+1}.$

The quotient
$\operatorname{PConf}_{m}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})/\operatorname{Aff}$
induces the natural map $PB_{n}\to PB_{n}/Z_{n}$ on orbifold fundamental
groups.

A rotation group in $\operatorname{Aff}$ centered at $c$ of order $p$ is the
cyclic subgroup generated by the element

$g(x)=e^{2\pi/p}(x-c)+c.$

######  Lemma 2.8.

Any finite subgroup of $\operatorname{Aff}$ is a rotation group centered at
some $c\in\mathbb{C}$.

###### Proof.

Let $G<\operatorname{Aff}$ be a finite subgroup. Any orientation-preserving
finite group action on $\mathbb{R}^{2}$ is cyclic by [19], which implies that
$G$ is generated by a single element $f(x)=ax+b$ of order $p$. This implies
that the linear coefficient $a$ must be a $p^{th}$ root of unity, and then the
center can be determined by the formula $c=b/(1-a)$. ∎

##  3\. Holomorphic maps between configuration spaces on $\mathbb{C}$: 3.1

Our first main result, 3.1, gives a classification of holomorphic maps
$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{m}(\mathbb{C})$
in the range $m\leq 2n$. To state the result, we must first discuss one of the
possible archetypes.

Root maps. There are two maps

$r_{p}:\operatorname{Conf}_{k}(\mathbb{C}^{*})\to\operatorname{Conf}_{kp}(\mathbb{C}^{*})$

and

$r_{p}^{\prime}:\operatorname{Conf}_{k}(\mathbb{C}^{*})\to\operatorname{Conf}_{kp+1}(\mathbb{C}),$

where the first takes $p^{th}$ roots of the $k$ distinct nonzero points, and
the second takes the union of the $p^{th}$ roots of the $k$ distinct nonzero
points and $\\{0\\}$. Such maps are called basic root maps.

A map
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{kp+\epsilon}(\mathbb{C})$
(with $\epsilon\in\\{0,1\\}$) is called a root map if it admits a
factorization

$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{k}(\mathbb{C}^{*})\to\operatorname{Conf}_{kp+\epsilon}(\mathbb{C}),$

where the map
$\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{k}(\mathbb{C}^{*})$
is a twist of a constant map by some holomorphic map
$A:\operatorname{Conf}_{n}(\mathbb{C})\to\mathbb{C}^{*}$ and the latter map is
a basic root map as above. By convention, we consider the zero map
$\operatorname{Conf}_{k}(\mathbb{C})\to\operatorname{Conf}_{1}(\mathbb{C})\cong\mathbb{C}$
to be a root map of the second kind with $p=0$.

The main result of this section is the following rigidity result about
holomorphic maps. We note that in the case $m=n$, 3.1 was established by Lin
[15, Theorem 1.4].

######  Theorem 3.1.

For $n\geq 5$ and $m\leq 2n$, if
$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{m}(\mathbb{C})$
is a non-constant holomorphic map, then $h$ is either an affine twist of the
identity map or a root map.

We will also consider a variant of this result, where the target is the moduli
space ${\mathcal{M}}_{g}$. One such holomorphic map is given by the
hyperelliptic embedding
$H:\operatorname{Conf}_{n}(\mathbb{C})\to\mathcal{M}_{g}$ for
$g=[\frac{n-1}{2}]$:

$H(\\{x_{1},...,x_{n}\\})=\text{the algebraic curve
}\\{y^{2}=(x-x_{1})...(x-x_{n})\\}$

######  Theorem 3.2.

For $n\geq 26$ and $g\leq n-2$, if
$h:\operatorname{Conf}_{n}(\mathbb{C})\to{\mathcal{M}}_{g}$ is a non-constant
holomorphic map, then $h$ is the hyperelliptic embedding.

Note that 3.2 does not mention affine twists, in contrast to 3.1. The reason
is that affine twisting gives equivalent points in the moduli space of the
punctured sphere; the hyperelliptic embedding does not depend on the actual
location of the points but only on the holomorphic structure of the punctured
sphere. Taking the orbifold structure on $\mathcal{M}_{g}$ into account, it is
possible to distinguish between two maps, one being a sort of twist of the
other by the hyperelliptic involution; see Section 3.5 for details.

To prove 3.1, we divide into cases according to 2.1. Let

$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Conf}_{n}(\mathbb{C})$

be a holomorphic map, and let $h_{*}$ be the map induced by $h$ on fundamental
group. 2.1 asserts that there are three possibilities for $h_{*}$; we consider
each in turn.

###  3.1. $h_{*}$ is the identity map up to transvection

###### Proof.

By hypothesis, $h_{*}:B_{n}\to B_{n}$ has the form
$h_{*}(\sigma)=\sigma\zeta^{\ell(\sigma)}$, where $\zeta\in Z_{n}$ and
$\ell:B_{n}\to\mathbb{Z}$ is the signed word-length homomorphism. Note that
$Z_{n}\leq PB_{n}$, and so $h_{*}(PB_{n})\leq PB_{n}$. Thus $h$ admits a lift

$\tilde{h}:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{n}(\mathbb{C}),$

and by hypothesis, $\tilde{h}_{*}:PB_{n}\to PB_{n}$ is also the identity map
up to a transvection by an element in the center of $B_{n}$.

Let
$\pi_{n}^{\prime}:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{n}(\mathbb{C})/\operatorname{Aff}$
be the natural quotient map. By 2.7, the space
$\operatorname{PConf}_{n}(\mathbb{C})/\operatorname{Aff}$ is a finite orbifold
cover of $\mathcal{M}_{0,n+1}$ and
$\pi_{1}^{orb}(\operatorname{PConf}_{n}(\mathbb{C})/\operatorname{Aff})\cong
PB_{n}/Z_{n}$, with $\pi_{n,*}^{\prime}$ the natural quotient $PB_{n}\to
PB_{n}/Z_{n}$.

Let $Y=(x_{1},...,x_{n-1})$ be an $n-1$-tuple of ordered distinct points in
$\mathbb{C}$. There is an embedding

$i_{Y}:\mathbb{C}-Y\to\operatorname{PConf}_{n}(\mathbb{C})$

such that $i_{Y}(x)=(Y,x)$. From the above description of
$\pi_{n,*}^{\prime}$, we see that both $\pi_{n}^{\prime}\circ\tilde{h}\circ
i_{Y}$ and $\pi_{n}^{\prime}\circ i_{Y}$ induce the same map on the orbifold
fundamental group. By 2.4, it follows that

$\pi_{n}^{\prime}\circ\tilde{h}\circ i_{Y}=\pi_{n}^{\prime}\circ i_{Y}.$

Therefore $\tilde{h}(Y,x)=g(Y,x)(Y,x)$ for some function
$g:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{Aff}$.

######  Claim 3.3.

The map $g:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{Aff}$ is
holomorphic and factors through $\operatorname{Conf}_{n}(\mathbb{C})$.

###### Proof.

For $X=(x_{1},\dots,x_{n})\in\operatorname{PConf}_{n}(\mathbb{C})$, we write

$\tilde{h}(X)=(\tilde{h}_{1}(X),\dots,\tilde{h}_{n}(X))$

with

$\tilde{h}_{i}(X)=g(X)(x_{i}).$

Writing $g(X)(z)=az+b$, we can solve for $a,b$ using the fact that for any
pair of distinct indices $i,j$, the map $g(X)$ takes $x_{i}$ to
$\tilde{h}_{i}(X)$ and $x_{j}$ to $\tilde{h}_{j}(X)$. This yields the
expressions

$a=\frac{\tilde{h}_{i}(X)-\tilde{h}_{j}(X)}{x_{i}-x_{j}},\qquad
b=\frac{x_{j}\tilde{h}_{i}(X)-x_{i}\tilde{h}_{j}(X)}{x_{i}-x_{j}}.$

Observe that since $\tilde{h}$ is holomorphic, these expressions vary
holomorphically with $X$, and that they are independent of $i$ and $j$ by the
assumption that $\tilde{h}_{i}(X)=g(X)(x_{i})$ for all $i$. ∎

Thus in this case, we have shown that the original map $h$ is the affine twist
of the identity map by the holomorphic map
$g:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Aff}$.

###  3.2. The image of $h_{*}$ is reducible or else infinite cyclic pseudo-
Anosov

Given a set $Y=\\{x_{1},...,x_{n-1}\\}$ of $n-1$ distinct points in
$\mathbb{C}$, there is an embedding

$i_{Y}:\mathbb{C}-Y\to\operatorname{Conf}_{n}(\mathbb{C})$

such that $i_{Y}(x)=Y\cup\\{x\\}$. Composing $i_{Y}$ with the natural
projection
$\pi_{m}:\operatorname{Conf}_{m}(\mathbb{C})\to\mathcal{M}_{0,m,1}$, we obtain
a holomorphic map $h_{Y}:=\pi_{m}\circ h\circ i_{Y}$ from the finite-type
Riemann surface $\mathbb{C}-Y$ into $\mathcal{M}_{0,m,1}$.

Suppose first that $h_{*}$ has cyclic image generated by a pseudo-Anosov. As
the kernel of
$\pi_{m,*}:\pi_{1}(\operatorname{Conf}_{m}(\mathbb{C}))\to\pi_{1}^{orb}(\mathcal{M}_{0,m,1})$
is contained in the center $Z_{n}\leq
B_{n}=\pi_{1}(\operatorname{Conf}_{m}(\mathbb{C}))$ by [12, p. 247 ff.], it
follows that $h_{Y,*}$ also has cyclic image generated by a pseudo-Anosov
element. By 2.5, this is not possible. Likewise, in the case when $h_{*}$ has
reducible image, the induced map $h_{Y*}$ also has nontrivial reducible image,
which is again prohibited by 2.5.

###  3.3. $h_{*}$ has prefinite cyclic image

We will show that in this case $h$ is a root map up to an affine twist.

######  Claim 3.4.

The map $\pi_{m}\circ h$ is a constant map.

###### Proof.

We continue to consider

$h_{Y}:\mathbb{C}-Y\to\mathcal{M}_{0,m,1}$

as in Section 3.2. If $h_{Y}$ were not constant, then it would determine a
locally nontrivial family, which would then have sufficiently large monodromy
by 2.5, contrary to hypothesis. Therefore $\pi_{m}\circ
h(\\{x_{1},...,x_{n}\\})=h_{\\{x_{1},...,x_{n-1}\\}}(x_{n})$, which does not
depend on $x_{n}$. This implies that it does not depend on any coordinate, by
symmetry. The claim follows. ∎

Denote the image of $\pi_{m}\circ h$ by $X\in\mathcal{M}_{0,m,1}$, and let
$X_{0}=\\{x_{1},...,x_{m}\\}$ be a representative of $X$ in
$\operatorname{Conf}_{m}(\mathbb{C})$. By 2.7, the image $\pi_{m}^{-1}(X)$ is
given as the orbit $\operatorname{Aff}(X_{0})$, and by the orbit-stabilizer
theorem, there is an isomorphism of complex manifolds, defining
$G:=\operatorname{Stab}(X_{0})$,

$\operatorname{Aff}(X_{0})\cong\operatorname{Aff}/G.$

Thus $h$ is given as a holomorphic map
$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Aff}/G$. By 2.8, $G$ is
a rotation group of order $p$ with center $c$ and $X_{0}$ is a $G$-invariant
subset of $\mathbb{C}$. By applying an affine twist, we can assume that the
center point of $G$ is $c=0$, so that $G=\mu_{p}$, the $p^{th}$ roots of
unity, and $X_{0}$ consists of all $p^{th}$ roots of some fixed subset
$Y_{0}=\\{y_{1},...,y_{k}\\}\subset\mathbb{C}^{*}$, possibly along with $0$.

There is an isomorphism of complex manifolds

$\operatorname{Aff}/G\cong\mathbb{C}^{*}\times\mathbb{C}$

which sends the coset of $z\mapsto az+b$ to $(a^{p},b)$. Under this
identification, let

$A:\operatorname{Conf}_{n}(\mathbb{C})\to\mathbb{C}^{*}$

be given as the first coordinate of the map
$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Aff}/G$, and likewise
define $B:\operatorname{Conf}_{n}(\mathbb{C})\to\mathbb{C}$ as the second
coordinate. The map of configuration spaces

$h:\operatorname{Conf}_{n}(\mathbb{C})\to\operatorname{Aff}(X_{0})\subset\operatorname{Conf}_{kp+\epsilon}(\mathbb{C})$

is now seen to be an affine twist of a root map, factoring as shown below:

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∎

###  3.4. Proof of 1.3

1.2 asserts that every homomorphism $\rho:B_{n}\to B_{m}$ with $5\leq n<m$ is
reducible, or else has cyclic image. The arguments of Section 3.2 and Section
3.3 then apply to extend the classification to this setting.

###  3.5. Proof of 3.2

Suppose $h:\operatorname{Conf}_{n}(\mathbb{C})\to\mathcal{M}_{g}$ is
holomorphic, with $n,g$ satisfying the bounds of 2.2. We execute the same
strategy as before, now following the cases delineated by 2.2. If
$h_{*}:B_{n}\to\operatorname{Mod}(S_{g})$ is trivial, then we imitate the
proof of 3.4 to see that $h$ is constant. If the image of $h_{*}$ is cyclic,
it is not sufficiently large, and so cannot arise from a holomorphic map by
2.5.

It remains to consider the case where $h_{*}$ is a transvection of $H_{*}$
(the map induced by the hyperelliptic embedding
$H:\operatorname{Conf}_{n}(\mathbb{C})\to\mathcal{M}_{g}$). Note that the
image of $H_{*}$ is the hyperelliptic mapping class group, which has
centralizer $\mathbb{Z}/2\mathbb{Z}$, generated by the hyperelliptic
involution $\iota$. Thus, there is exactly one nontrivial transvection of
$H_{*}$, by $\iota$. If $h_{*}:B_{n}\to\operatorname{Mod}(S_{g})$ is given by
$H_{*}$ , we follow the argument of Section 3.1 to see that on each
submanifold of the form
$\mathbb{C}-Y\subset\operatorname{Conf}_{n}(\mathbb{C})$, the maps $h$ and $H$
coincide, hence coincide globally. If $h_{*}=H_{*}^{\iota}$, the same argument
shows that there is at most one holomorphic map in the homotopy class of $h$;
it remains to give a construction. As a map of sets, the underlying map
$H^{\iota}$ is the same as $H$, sending the configuration
$\\{x_{1},\dots,x_{n}\\}$ to the hyperelliptic curve
$y^{2}=(x-x_{1})\dots(x-x_{n})$. As a map of complex orbifolds, the two differ
in how the marking is specified on the universal covers: $H^{\iota}$ arises
from $H$ by precomposition with an affine twist by some holomorphic map
$\Delta:\operatorname{Conf}_{n}(\mathbb{C})\to\mathbb{C}^{*}$ that induces the
abelianization $\Delta_{*}:B_{n}\to\mathbb{Z}$.

##  4\. Rigidity of holomorphic maps between
$\operatorname{PConf}_{n}(\mathbb{C})$ and
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})$

In this section, we will give a new proof of [15, Theorem 2.5] (classifying
holomorphic maps
$F:\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$) and use this
to give a complete classification of holomorphic maps between
$\operatorname{PConf}_{n}(\mathbb{C})$.

To state the results, we first define some basic ingredients: the maps
$sr,cr,RQ,$ and $NI$. The maps $sr$ and $cr$ are holomorphic maps
$\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$, both arising
from the cross-ratio. The “simple ratio”
$sr(i,j,k):\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$ is
given by

$sr(i,j,k)(x_{1},...,x_{n})=\frac{x_{k}-x_{i}}{x_{j}-x_{i}}.$

The second map is the cross-ratio
$cr(i,j,k,l):\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$,
given by

$cr(i,j,k,l)(x_{1},...,x_{n})=\frac{x_{l}-x_{i}}{x_{l}-x_{k}}\frac{x_{j}-x_{k}}{x_{j}-x_{i}}.$

Another interpretation of the maps $sr(i,j,k)$ and $cr(i,j,k,l)$ are the
following. For $(x_{1},...,x_{n})\in\operatorname{PConf}_{n}(\mathbb{C})$,
there is a unique element in $A\in\operatorname{Aff}$ such that
$A(x_{i})=0,A(x_{j})=1$. We define the value
$sr(i,j,k)(x_{1},...,x_{n})=A(x_{k})$. Likewise, for
$(x_{1},...x_{n})\in\operatorname{PConf}_{n}(\mathbb{C})$, there is a unique
element in $A\in\operatorname{PSL}(2,\mathbb{C})$ such that

$A(x_{i})=0,A(x_{j})=1,A(x_{k})=\infty.$

We define the value $cr(i,j,k,l)(x_{1},...,x_{n})=A(x_{l})$.

For later use, we record the following properties of $cr$ and $sr$ under
permutation of indices; these can be checked by direct inspection.

######  Lemma 4.1.

There are identities

$sr(j,i,k)=1-sr(i,j,k),\qquad sr(j,k,i)=1/sr(i,j,k),$

and

$cr(i,j,k,l)=cr(k,l,i,j),\qquad cr(j,i,k,l)=1-cr(i,j,k,l),\qquad
cr(k,j,i,l)=1/cr(i,j,k,l).$

We will give a new proof of the following result of Lin [15, Theorem 2.15],
which has the virtue of being somewhat shorter than the original.

######  Theorem 4.2.

For $n\geq 3$, any non-constant holomorphic map
$f:\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$ is given by
either $sr(i,j,k)$ or $cr(i,j,k,l)$.

Using this, we will classify holomorphic maps
$h:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$.
We define two basic ingredients. The first is the map
$RQ_{n}:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{n-1}(\mathbb{C})$,
given by the following. For
$(x_{1},...,x_{n})\in\operatorname{PConf}_{n}(\mathbb{C})$, note that
$cr(x_{1},x_{2},x_{3},z)$, viewed as a Möbius transformation, sends
$x_{1},x_{2},x_{3}$ to $0,1,\infty$, respectively. Then define

$RQ_{n}(x_{1},...,x_{n})=(0,1,cr(x_{1},x_{2},x_{3},x_{4}),...,cr(x_{1},x_{2},x_{3},x_{n})).$

The terminology comes from the fact that the classical “resolving the quartic”
map
$\operatorname{PConf}_{4}(\mathbb{C})\to\operatorname{PConf}_{3}(\mathbb{C})$
is affine equivalent to $RQ_{4}$.

The second basic map we consider is the “normalized inversion”
$NI_{n}:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{n}(\mathbb{C})$,
given by the formula

$NI_{n}(x_{1},\dots,x_{n})=\left(0,\frac{1}{x_{2}-x_{1}},\dots,\frac{1}{x_{n}-x_{1}}\right).$

We can now state the main results of the section.

######  Theorem 4.3.

Let $m\geq 2$. Let
$h:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$
be a holomorphic map. Then up to permutation of coordinates in the source and
target and affine twisting, $h$ is a composition of one or more of the
following:

  * •

$h=c$ a constant map,

  * •

$h=RQ_{n}$,

  * •

$h=NI_{n}$,

  * •

$h=\pi_{n}^{m}$ the forgetful map
$(x_{1},\dots,x_{n})\mapsto(x_{1},\dots,x_{m})$ (including $m=n$, the
identity).

In particular, there is no holomorphic map
$\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$
for $1<n<m$.

######  Theorem 4.4.

Let $m\geq 3$. Let
$h:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\operatorname{PConf}_{m}(\mathbb{CP}^{1})$
be a holomorphic map. Up to a permutation on the domain and twisting by a
holomorphic map
$A:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\operatorname{PSL}(2,\mathbb{C})$,
either $F$ is a constant map or a forgetful map.

###  4.1. Rigidity of
$\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$ and
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\mathbb{C}-\\{0,1\\}$

We begin with the following lemma.

######  Lemma 4.5.

Any holomorphic map $f:\operatorname{Aff}\to\mathbb{C}-\\{0,1\\}$ is constant.
Likewise, any holomorphic map
$f:\operatorname{PSL}(2,\mathbb{C})\to\mathbb{C}-\\{0,1\\}$ is a constant map.

###### Proof.

Given $f:\operatorname{Aff}\to\mathbb{C}-\\{0,1\\}$ holomorphic, there is an
induced holomorphic map $F$ between the universal covers. As a complex
manifold, $\operatorname{Aff}$ is isomorphic to
$\mathbb{C}^{*}\times\mathbb{C}$ and hence its universal cover is
$\mathbb{C}^{2}$, while the universal cover of $\mathbb{C}-\\{0,1\\}$ is
$\mathbb{D}$; by Liouville’s theorem, it follows that $F$, and hence $f$, must
be constant.

Now suppose holomorphic
$f:\operatorname{PSL}(2,\mathbb{C})\to\mathbb{C}-\\{0,1\\}$ is given. The
action by Möbius transformation gives a fiber bundle

$\operatorname{Aff}\to\operatorname{PSL}(2,\mathbb{C})\to\mathbb{CP}^{1}$

whose fibers are holomorphic submanifolds. By the previous paragraph, the
restriction of $f$ to each fiber must be constant, and hence $f$ factors
through the base space $\mathbb{CP}^{1}$; the result now follows via the
maximum principle. ∎

We will also make use of the following result of [5, Theorem 1.4]:

######  Theorem 4.6.

For $n\geq 3$ and $m\geq 2$, any homomorphism

$f:PB_{n}\to F_{m}$

factors through a forgetful map $p_{*}:PB_{n}\to PB_{m}$ with $m\in\\{3,4\\}$;
in the case $m=4$, there is a further factorization through
$RQ_{4,*}:PB_{4}\to PB_{3}$.

Our other main tool will be the following lemma:

######  Lemma 4.7.

For $n\geq 3$, let

$f:\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}$

be holomorphic. If $f_{*}:PB_{n}\to F_{2}$ factors through a forgetful map
$p_{*}:PB_{n}\to PB_{m}$, then $f$ factors through a forgetful map
$p:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$.

###### Proof.

Suppose that $p_{*}$ forgets the $i^{th}$ strand. Fixing a configuration of
$n-1$ points

$X_{i}:=x_{1},\dots,\widehat{x_{i}},\dots,x_{n},$

the restriction of $f$ to the subspace
$\mathbb{C}-X_{i}\subset\operatorname{PConf}_{n}(\mathbb{C})$ (where only the
$i^{th}$ coordinate varies) then lifts to the universal cover $\mathbb{D}$ of
$\mathbb{C}-\\{0,1\\}$. However by the removability singularity theorem, this
extends to give a holomorphic map $F:\mathbb{C}\to\mathbb{D}$, which must be
constant by Liouville’s theorem, as desired. ∎

###### Proof of Theorem 4.2.

We proceed by induction on $n$.

Base case 1: $n=3$. For $\tau\in\mathbb{C}-\\{0,1\\}$ fixed, define
$h_{\tau}:\operatorname{Aff}\to\mathbb{C}-\\{0,1\\}$ by

$h_{\tau}(az+b)=h(b,a+b,a\tau+b).$

By 4.5, each $h_{\tau}$ is constant, with value
$c(\tau)\in\mathbb{C}-\\{0,1\\}$. Since $h$ is holomorphic, so too is the
induced map $c:\mathbb{C}-\\{0,1\\}\to\mathbb{C}-\\{0,1\\}$.

By the Great Picard Theorem, the points $0,1,\infty$ are at worst poles of
$c$. Thus, $c$ extends to a holomorphic map
$\hat{c}:\mathbb{CP}^{1}\to\mathbb{CP}^{1}$, a rational function so that
$\hat{c}^{-1}(\\{0,1,\infty\\})\subset\\{0,1,\infty\\}$. If this containment
is strict then $\hat{c}$ has degree zero and hence $c$ is constant; otherwise
we can assume $\hat{c}$ restricts as the identity on the set
$\\{0,1,\infty\\}$, and that $\hat{c}$ has no other singularities.

Writing $\hat{c}(z)=p(z)/q(z)$, it follows that $p(z)$ is divisible by $z$,
there are no other zeros of $p$, that $q$ has no zeroes, and that
$\hat{c}(1)=1$. Thus $\hat{c}(z)=z=sr(1,2,3)(0,1,z)$, and as was discussed
above, this implies that $h=sr(1,2,3)$ over the entire domain.

Base case 2: $n=4$. By 4.6, $h_{*}:PB_{4}\to F_{2}$ either factors through a
forgetful map $PB_{4}\to PB_{3}$ or else through $(RQ_{4})_{*}$. In the former
case, we apply 4.7 to reduce to the case $n=3$, and so we assume that $h_{*}$
factors through $(RQ_{4})_{*}$. For $\tau\in\mathbb{C}-\\{0,1\\}$ fixed,
define the subset $X_{\tau}\subset\operatorname{PSL}(2,\mathbb{C})$ via

$X_{\tau}:=\\{A\in\operatorname{PSL}(2,\mathbb{C})|A(0)\neq\infty,A(1)\neq\infty,A(\infty)\neq\infty,A(\tau)\neq\infty\\}.$

By construction, $X_{\tau}((0,1,\infty,\tau))=cr(1,2,3,4)^{-1}(\tau)$. Like in
the case $n=3$, we define the holomorphic map
$h_{\tau}:X_{\tau}\to\mathbb{C}-\\{0,1\\}$ by

$h_{\tau}(A)=h(A(0,1,\infty,\tau)).$

We claim that $h_{\tau}$ extends to a holomorphic map
$h_{\tau}:\operatorname{PSL}(2,\mathbb{C})\to\mathbb{C}=\\{0,1\\}$. To see
this, we first claim that $h_{\tau}$ lifts to holomorphic map
$H_{\tau}:X_{\tau}\to\mathbb{D}$. This follows from the assumption that
$h_{*}$ factors through $(RQ_{4})_{*}=(0,1,cr(1,2,3,4))_{*}$ and the
characterization $X_{\tau}((0,1,\infty,\tau))=cr(1,2,3,4)^{-1}(\tau)$, which
shows that $h_{\tau}$ is homotopic to a constant map. Thus $H_{\tau}$ is a
bounded holomorphic map. The space $X_{\tau}$ is the complement of
hypersurfaces in the smooth complex variety
$\operatorname{PSL}(2,\mathbb{C})$. By the higher-dimensional removable
singularity theorem [14, Theorem 4.7.2], it follows that $H_{\tau}$, and hence
$h_{\tau}$, can be extended to $\operatorname{PSL}(2,\mathbb{C})$.

By 4.5, each map
$h_{\tau}:\operatorname{PSL}(2,\mathbb{C})\to\mathbb{C}-\\{0,1\\}$ is a
constant map $h_{\tau}=c(\tau)$. We can now apply the argument of the last two
paragraphs in the case $n=3$ to conclude that
$c:\mathbb{C}-\\{0,1\\}\to\mathbb{C}-\\{0,1\\}$ must be an automorphism, and
hence $h=cr(1,2,3,4)$ up to an automorphism of $\mathbb{C}-\\{0,1\\}$ as
claimed.

Inductive step: $n\geq 5$. We proceed by induction on $n$, taking $n=4$ as the
base case. The inductive step follows by 4.6 and 4.7. ∎

The following lemma gives a useful normalization of a map between
configuration spaces.

######  Lemma 4.8 (Normalization).

Every holomorphic map

$h:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$

is equivalent up to affine twisting to a unique holomorphic map

$h_{s}:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$

such that the first two coordinates of $h_{s}(x_{1},...,x_{n})$ are $0,1$.

###### Proof.

Let
$p_{12}:\operatorname{PConf}_{m}(\mathbb{C})\to\operatorname{PConf}_{2}(\mathbb{C})$
be the projection onto the first two coordinates. Then $p_{12}\circ h$ is a
holomorphic map. Let
$A:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{Aff}$ be characterized
by the condition that $A(x_{1},...,x_{n})(p_{12}(x_{1},...,x_{n}))=(0,1)$. Via
the identification of complex manifolds
$\operatorname{PConf}_{2}(\mathbb{C})\cong\operatorname{Aff}$, it follows that
$A$ is holomorphic. Defining $h_{s}$ as the affine twist $h^{A}$, the claim
follows. ∎

We now discuss the common values of $cr(i,j,k,l)$ and $sr(i,j,k)$.

######  Lemma 4.9.

  1. (1)

The function $sr(1,2,3)-sr(i,j,k)$ has no zero in
$\operatorname{PConf}_{n}(\mathbb{C})$ if and only if $(i,j,k)$ is one of
$(1,2,p),(1,p,3),(p,2,3)$ for $p\geq 4$.

  2. (2)

The function $cr(1,2,3,4)-sr(i,j,k)$ always has a zero in
$\operatorname{PConf}_{n}(\mathbb{C})$.

  3. (3)

The function $cr(1,2,3,4)-cr(i,j,k,l)$ has no zero in
$\operatorname{PConf}_{n}(\mathbb{C})$ if and only if $(i,j,k,l)$ is one of
$(p,2,3,4),(1,p,3,4),(1,2,p,4),(1,2,3,p)$ for $p\neq 1,2,3,4$.

###### Proof.

We will prove (3), the proofs of (1),(2) being similar. The function
$cr(1,2,3,4)-cr(i,j,k,l)$ has a zero if and only if the expression

$\frac{x_{4}-x_{1}}{x_{4}-x_{3}}\frac{x_{2}-x_{3}}{x_{2}-x_{1}}=\frac{x_{l}-x_{i}}{x_{l}-x_{k}}\frac{x_{j}-x_{k}}{x_{j}-x_{i}}.$
(1)

has a solution. If $\\{i,j,k,l\\}=\\{1,2,3,4\\}$, we can see by 4.1 that
$cr(i,j,k,l)$ is the image of $cr(1,2,3,4)$ under some element of the dihedral
group $D_{3}$ acting on $\mathbb{C}-\\{0,1\\}$ via $z\mapsto 1-z,z\mapsto
1/z$, and it is straightforward to verify that no element of this group acts
freely, implying that the equation $cr(1,2,3,4)=cr(i,j,k,l)$ admits a solution
in $\operatorname{PConf}_{n}(\mathbb{C})$ in this case.

We therefore assume that index $i$ is not in $\\{1,2,3,4\\}$; without loss of
generality, set $i=5$. If we view the equation as a function of $x_{5}$
(fixing other points), it will have a unique solution
$z_{5}=z_{5}(x_{1},\dots,\widehat{x_{5}},\dots,x_{n})$ in $\mathbb{CP}^{1}$.
If this is not a valid solution in $\operatorname{PConf}_{n}(\mathbb{C})$,
then $z_{5}$ must either be one of the $x_{q}$ for $q\neq 5$ or else $\infty$.
As $z_{5}$ varies continuously with the parameters $x_{i},i\neq 5$, if (1) has
no solutions, then there must be an identity $z_{5}=x_{q}$ for $q\neq 5$ or
else $z_{5}=\infty$.

In case $z_{5}=\infty$, (1) simplifies to the ostensible identity in
$\mathbb{C}(x_{1},\dots,x_{n})$

$\frac{x_{4}-x_{1}}{x_{4}-x_{3}}\frac{x_{2}-x_{3}}{x_{2}-x_{1}}=\frac{x_{j}-x_{k}}{x_{l}-x_{k}},$

which is readily seen to not hold regardless of the values of $j,k,l$. If
$z_{5}=x_{q}$ for some $q\neq 5$, a similar analysis shows that the only way
(1) can be satisfied is if $(p,j,k,l)=(1,2,3,4)$. ∎

We now discuss similar results for
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})$.

######  Theorem 4.10.

Every non-constant holomorphic map
$f:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\mathbb{C}-\\{0,1\\}$ is of the
form $cr(i,j,k,l)$.

###### Proof.

Let $f:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\mathbb{C}-\\{0,1\\}$ be a
holomorphic map. Let
$E_{n}:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{n}(\mathbb{CP}^{1})$
be the natural embedding, which is a holomorphic map. By 4.2, $f\circ E_{n}$
is a cross ratio map, either some $sr$ or $cr$. Since $E_{n}$ has dense image,
the map $f$ is uniquely determined by $f\circ E_{n}$.

If $f\circ E_{n}=sr(i,j,k)$, then letting $x_{i}$ approach $\infty$, the image
under $f\circ E_{n}$ approaches $1$. Thus $sr(i,j,k)$ cannot extend to
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})$ and so $f\circ E_{n}$ must be a
four-term cross ratio map $cr(i,j,k,l)$, each of which extends to
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})$ via the same formula. ∎

Similarly, we have the following counterpart to 4.9.

######  Theorem 4.11.

The equation $cr(1,2;3,4)=cr(i,j,k,l)$ has no zero in
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})$ if and only if $(i,j,k,l)$ is one
of $(p,2,3,4),(1,p,3,4),(1,2,p,4),(1,2,3,p)$ for $p\neq 1,2,3,4$.

###  4.2. Rigidity of
$\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$

###### Proof of 4.3.

Let

$h:\operatorname{PConf}_{n}(\mathbb{C})\to\operatorname{PConf}_{m}(\mathbb{C})$

be a holomorphic map. We can assume the first two coordinates of
$h(x_{1},...,x_{n})$ are $(0,1)$ by 4.8. The remaining coordinates of
$h(x_{1},...,x_{n})$ are functions

$f_{3},...,f_{m}:\operatorname{PConf}_{n}(\mathbb{C})\to\mathbb{C}-\\{0,1\\}.$

Since the image lies in $\operatorname{PConf}_{m}(\mathbb{C})$, the
expressions $f_{i}-f_{j}=0$ have no solutions in
$\operatorname{PConf}_{n}(\mathbb{C})$.

Case 1: $f_{3}=sr(i,j,k)$. By 4.9, if $f_{3}=sr(i,j,k)$, then all other
$f_{i}$’s are also simple ratios $sr$. Up to a permutation of coordinates on
the domain, we assume that $f_{3}=sr(1,2,3)$. By 4.9, $f_{4}$ is either
$sr(1,2,4)$, $sr(1,4,3)$ or $sr(4,2,3)$. Applying 4.1, by applying an affine
twist and/or $NI_{n}$, we can assume $f_{4}=sr(1,2,4)$. We then claim that
$f_{k}=sr(1,2,k)$ for all $k\geq 3$, up to permutation on the domain. This
follows from 4.9: the only tuple $(i,j,k)$ that differs from both $(1,2,3)$
and $(1,2,4)$ in a single entry is $(1,2,p)$ for some other $p$. Then we can
apply a permutation such that $p=k$. Applying the affine twist
$(x_{2}-x_{1})z+x_{1}$ then shows that $h$ is affine-equivalent to the
forgetful map $\pi_{n}^{m}$.

Case 2: $f_{3}=cr(i,j,k,l)$. By 4.9, if $f_{3}=cr(i,j,k,l)$, then all other
$f_{i}$’s are also four-term cross ratio functions $cr$. Applying a
permutation on the domain, we assume that $f_{3}=cr(1,2,3,4)$. By 4.9, $f_{4}$
is a $cr$ function indexed by a tuple where exactly one entry of $(1,2,3,4)$
has been replaced by $5$. As in Case 1, we apply 4.1 so that possibly after an
affine twist and/or an application of $NI_{n}$, we can assume that
$f_{4}=cr(1,2,3,5)$. Arguing as in Case 1, it follows that
$f_{k}=cr(1,2,3,k+1)$ for $3\leq k\leq m$ up to permutation on the domain, and
visibly then $h=\pi_{n}^{m}\circ RQ_{n}$. ∎

###  4.3. Rigidity of
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\operatorname{PConf}_{m}(\mathbb{CP}^{1})$
for $m\geq 3$

###### Proof.

Let

$h:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\operatorname{PConf}_{m}(\mathbb{CP}^{1})$

be a holomorphic map. Similar to 4.8, we can assume that the first three
coordinates of all images of $h$ are $0,1,\infty$ after applying a twist
$A:\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\operatorname{PSL}(2,\mathbb{C})$.
All the holomorphic maps
$\operatorname{PConf}_{n}(\mathbb{CP}^{1})\to\mathbb{C}-\\{0,1\\}$ are given
by cross ratio functions $cr(i,j,k,l)$. By a similar argument as in 4.2, it
can be shown that $h$ is a forgetful map (in this setting, the maps $RQ,NI$
arise as twists and do not need to be considered separately). ∎

##  5\. Pure configuration spaces in genus one

Here we consider the setting of holomorphic maps between pure configuration
spaces of Riemann surfaces where the target $Y$ has genus one. In outline, the
proof is the same as that of the previous section - we construct a
normalization $f_{s}$ of
$f:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$, relative to
which the component functions are tightly constrained enough to be completely
classifiable. In the previous section this classification of component
functions (4.2) relied on the group-theoretic classification 4.6, which was
proved in [5]. The results of [5] also treat the case where $g(X)\geq 2$ (this
is recalled below in 5.9), but we must establish the case $g(X)=1$ ourselves.
It is interesting to note that whereas all three results have a similar flavor
(asserting that maps from configuration spaces to hyperbolic groups factor
through forgetful maps), the precise nature of these forgetful maps reflect
the different genus regimes (for $g(X)=0$ we forget all but three or four, for
$g(X)=1$ we forget all but two, and for $g(X)\geq 2$ we forget all but one).

######  Theorem 5.1.

Any homomorphism $PB_{n}(T^{2})\to F_{m}$ either factors through
$PB_{2}(T^{2})$ or has cyclic image.

Using this, we will establish the main results of the section:

######  Theorem 5.2.

Let $X,Y$ be compact Riemann surfaces with $g(X)=g(Y)=1$. Then up to
permutation of coordinates and twisting, any holomorphic map
$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$ is induced by an
isomorphism $X\to Y$ and a forgetful map.

######  Theorem 5.3.

Let $X,Y$ be compact Riemann surfaces with $g(X)\geq 2$ and $g(Y)=1$. Then up
to twisting, any holomorphic map
$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$ is constant.

We observe that a twist
$A:\operatorname{PConf}_{n}(X)\to\operatorname{Aut}(Y)$ is essentially the
same thing as the case $m=1$ (see Equation 2 below), and so 5.3 really gives a
reduction to the case $m=1$. It remains to give a classification of
holomorphic $h:\operatorname{PConf}_{n}(X)\to Y$. Certainly one possibility is
to factor through a forgetful map $\operatorname{PConf}_{n}(X)\to X$, but
there are more complicated examples, as well. For instance, if $X$ admits $Y$
as one of its isogeny factors (i.e. the Jacobian $\operatorname{Jac}(X)$
admits a finite cover isomorphic to a product $Y\times A$), then it is
possible to use the Abel-Jacobi map to induce a map
$\operatorname{PConf}_{n}(X)\to Y$ this way. We leave the problem of
classifying $m=1$ for future work.

###  5.1. 5.1: from torus braid groups to free groups

We first discuss some facts about the group $PB_{n}(T^{2})$. Given a disk
$D^{2}$ embedded in $T^{2}$, we obtain an embedding of the pure braid group
$i:PB_{n}<PB_{n}(T^{2})$ as a subgroup.

We now introduce a generating set for $PB_{n}$. Recall that $PB_{n}$ is the
pure mapping class group of the disk with $n$ marked points; i.e.,
$\pi_{0}(\operatorname{Diff}(\mathbb{D}_{n}))$, where
$\operatorname{Diff}(\mathbb{D}_{n})$ is the group of diffeomorphisms of
$\mathbb{D}$ fixing $n$ marked points pointwise. Consider the disk with $n$
marked points $\mathbb{D}_{n}$ in Figure 3.

Figure 1. $\mathbb{D}_{n}$.

Figure 2.

Figure 3. $a_{124}$.

Let $L$ be a line segment below all the marked points $x_{1},...,x_{n}$. Let
$L_{1},...,L_{n}$ be line segments connecting $x_{1},...,x_{n}$ to $L$ as in
Figure 3. Similarly, let $U$ be a line segment above all marked points and let
$U_{1},...,U_{n}$ be line segments connecting $x_{1},...,x_{n}$ to $U$ as
shown in Figure 3.

For $I\subset\\{1,...,n\\}$, let $a_{I}$ (resp. $a_{I}^{\prime}$) be the
boundary curve of the tubular neighborhood of $\bigcup_{i\in I}L_{i}\cup L$
(resp. $\bigcup_{i\in I}U_{i}\cup U$). Let $T_{I}$ (resp. $T_{I}^{\prime}$) be
the Dehn twist about $a_{I}$ (resp. $a_{I}^{\prime}$). Figure 3 gives an
example of a curve representing $a_{124}$. The following proposition about
generating sets of $PB_{n}$ is classical and can be found in [17, Theorem
2.3].

######  Proposition 5.4.

Both $\\{T_{ij}|1\leq i<j\leq n\\}$ and $\\{T_{ij}^{\prime}|1\leq i<j\leq
n\\}$ are generating sets for $PB_{n}$.

An embedding $\mathbb{D}_{n}\hookrightarrow T^{2}$ induces an injection
$PB_{n}\hookrightarrow PB_{n}(T^{2})$. We fix one such embedding, and use this
to identify $T_{I}$ and $T_{I}^{\prime}$ with elements of $PB_{n}(T^{2})$.

######  Lemma 5.5.

For $n\geq 2$, any Dehn twist $T_{c}$ about a simple closed curve $c$
surrounding $p_{i},p_{j}$ is conjugate in $PB_{n}(T^{2})$ to $T_{ij}$.

###### Proof.

The Dehn twists $T_{c}$ and $T_{ij}$ can be viewed as point-pushing maps about
based loops $c^{+}$ and $c_{ij}^{+}$ starting from $p_{i}$ around $p_{j}$
following $c$ and $c_{ij}$ respectively. The point-pushing subgroup based at
$p_{i}$ inside $PB_{n}(T^{2})$ is a free group
$\pi_{1}(T^{2}-\\{x_{1},...,\hat{x_{i}},...,x_{n}\\})$. The loops $c^{+}$ and
$c_{ij}^{+}$ are conjugate in
$\pi_{1}(T^{2}-\\{x_{1},...,\hat{x_{i}},...,x_{n}\\})$, since as unbased loops
they both encircle the puncture $x_{j}$. ∎

We now prove the following statement; in anticipation of later use, we
formulate it for a general compact surface, not just tori (the Dehn twists
$T_{ij}$ retain the meaning given above, as loops of the $i^{th}$ point around
the $j^{th}$ inside some topological embedding $\mathbb{D}_{n}\hookrightarrow
X$).

######  Lemma 5.6.

Let $S$ be a compact surface and let
$e_{n}:\operatorname{PConf}_{n}(S)\to\operatorname{PConf}_{n-1}(S)\times S$ be
the natural embedding. The kernel of the induced map $e_{n*}$ on the
fundamental groups is normally generated by $\\{T_{1n},...,T_{n-1,n}\\}$.

###### Proof.

Endow $S$ with a complete Riemannian metric. Define
$W:=\operatorname{PConf}_{n-1}(S)\times S-\operatorname{PConf}_{n}(S)$, noting
that $W$ is a union $W=\coprod W_{i}$ of $n-1$ disjoint embedded copies of
$\operatorname{PConf}_{n-1}(S)$, according to the unique
$i\in\\{1,\dots,n-1\\}$ such that $x_{n}=x_{i}$. Let $N(W)$ be a tubular
neighborhood; likewise there is a decomposition $N(W)=\coprod N(W)_{i}$. Let
$p:N(W)\to W$ be the projection taking $(x_{1},\dots,x_{n})\in N(W)_{i}$ to
$(x_{1},\dots,x_{n-1},x_{i})$. The map $p$ is a homotopy equivalence because
it extends to a deformation retraction by radially contracting $x_{n}$ in to
the associated $x_{i}$, and the set $N(W)_{i}^{\circ}:=N(W)_{i}-W_{i}$ has the
homotopy type of a $S^{1}$-bundle over $W_{i}$, with bundle map given by $p$.

The use of van Kampen to obtain the result is complicated by the fact that $W$
and $N(W)$ are disconnected. Accordingly, define
$Y_{0}:=\operatorname{PConf}_{n}(S)$ and for $i\geq 1$,

$Y_{i}=Y_{i-1}\bigcup_{N(W)_{i}^{\circ}}N(W)_{i};$

let $q_{i}\in N(W)_{i}^{\circ}$ be a basepoint.

Applying the van Kampen theorem to the above decomposition of $Y_{i}$ yields
the following pushout diagram:

$\textstyle{\pi_{1}(N(W)_{i}^{\circ},q_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{i,*}}$$\scriptstyle{p_{i,*}}$$\textstyle{\pi_{1}(Y_{i-1},q_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{i,*}}$$\textstyle{\pi_{1}(N(W)_{i},q_{i})\cong\pi_{1}(W_{i},q_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(Y_{i},q_{i})}$

Therefore the kernel of $e_{i,*}$ is normally generated by the image of
$\ker(p_{i,*})$ under $f_{i,*}$. Observe that $\ker(p_{i,*})$ is normally
generated by the $S^{1}$-fiber, which corresponds under $f_{i,*}$ to the Dehn
twist $T_{i,n}$. Inductively, we see that the kernel of the inclusion map
$\pi_{1}(\operatorname{PConf}_{n}(S),q_{i})\to\pi_{1}(Y_{i},q_{i})$ is
normally generated by the Dehn twists $T_{j,n}$ for $j\leq i$; as
$Y_{n-1}=\operatorname{PConf}_{n-1}(S)\times S$, the result follows. ∎

Let
$P_{i}:=\pi_{1}(T^{2}-\\{x_{1},...,\hat{x_{i}},...,x_{n}\\})<PB_{n}(T^{2})$ be
the point-pushing subgroup based at the point $x_{i}$.

######  Lemma 5.7.

The group $PB_{n}(T^{2})$ is generated by elements in $P_{i}$ for
$i\in\\{1,...,n\\}$

###### Proof.

The subgroup $P_{n}$ is the kernel of the homomorphism $PB_{n}(T^{2})\to
PB_{n-1}(T^{2})$ induced by forgetting the last point. The lemma is deduced by
induction. ∎

We will make use of the result [5, Theorem 2.5]:

######  Theorem 5.8.

Let $G_{1},...,G_{n}$ be groups and let $\Gamma<G_{1}\times...\times G_{n}$ be
a finite index subgroup. Let $\pi_{i}:\Gamma\to G_{i}$ be the $i^{th}$
projection map and let $\Gamma_{i}$ be the image of $\pi_{i}$. Let $\Lambda$
be a torsion-free, non-elementary hyperbolic group. Then any homomorphism
$\phi:\Gamma\to\Lambda$ either factors through $\pi_{i}$ or its image is a
cyclic group.

We now start the proof of 5.1.

###### Proof of 5.1.

Let $\rho:PB_{n}(T^{2})\to F_{m}$ be a homomorphism.

Case 1: $n=3$. Suppose that one of $\rho(T_{12})$, $\rho(T_{13})$ or
$\rho(T_{23})$ is not trivial (we will consider the situation where all three
vanish below). We assume without loss of generality that $\rho(T_{12})\neq 1$.
Let us consider the centralizer of $T_{12}$. The point push of the third point
gives the embedding $P_{3}:\pi_{1}(T^{2}-\\{x_{1},x_{2}\\})\to PB_{3}(T^{2})$.
Denote the based loop at $x_{3}$ corresponding to $T_{12}$ as $c$. Then
$P_{3}(c)=T_{12}^{-1}T_{123}$. The loop $c=[a,b]$ is a commutator in
$\pi_{1}(T^{2}-\\{x_{1},x_{2}\\})$, where $a,b$ are standard generators for
$\pi_{1}(T^{2},x_{3})$ disjoint from $T_{12}$, and so crucially, both
$P_{3}(a)$ and $P_{3}(b)$ commute with $T_{12}$. As the centralizer of any
nontrivial element of $F_{m}$ is cyclic, it follows that
$\rho([P_{3}(a),P_{3}(b)])=\rho(T_{12}^{-1}T_{123})=1$ and hence
$\rho(T_{123})=\rho(T_{12})$. By the same logic, either $\rho(T_{23})=1$ or
$\rho(T_{23})=\rho(T_{123})$, and likewise either $\rho(T_{13})=1$ or else
$\rho(T_{13})=\rho(T_{123})$.

By the lantern relation, $T_{12}T_{23}T_{13}=T_{123}$. If either
$\rho(T_{23})$ or $\rho(T_{13})$ is trivial, this implies that both are. The
other possibility is that all three of
$\rho(T_{12}),\rho(T_{23}),\rho(T_{13})$ equal $\rho(T_{123})$, which implies
that $\rho(T_{123})^{2}=1$; as $F_{m}$ is torsion-free, this implies
$\rho(T_{123})=1$, contrary to the assumption that $\rho(T_{12})\neq 1$. We
conclude that $\rho(T_{13})=\rho(T_{23})=1$, so by 5.6, it follows that $\rho$
factors through the product $PB_{2}(T^{2})\times T^{2}$. By 5.8, we conclude
that $\rho$ either factors through $PB_{2}(T^{2})$ or else has cyclic image.
Note that the argument of this paragraph covers the case where all of
$\rho(T_{12})$, $\rho(T_{13})$, $\rho(T_{23})$ are trivial.

Case 2: $n=4$. As above, we can assume without loss of generality that
$\rho(T_{12})\neq 1$. By 4.6, the restriction $\rho|_{PB_{4}}$ either factors
through a forgetful map to $PB_{3}$ or $RQ_{4}$. In the first case,
$\rho(T_{14})=\rho(T_{24})=\rho(T_{34})=1$, which implies that $\rho$ factors
through $e_{4*}$ by 5.6. Applying 5.8, either the image is cyclic or else we
have reduced to the case $n=3$.

Suppose then that $\rho$ factors through $RQ_{4}$. On the level of $B_{4}$,
the map $RQ_{4}$ sends the standard generators $\sigma_{1},\sigma_{2}$ to
themselves and $\sigma_{3}$ to $\sigma_{1}$. Thus
$\rho(T_{34})=\rho(\sigma_{3}^{2})=\rho(\sigma_{1}^{2})=\rho(T_{12})$.

Subcase 1: $\rho(T_{23})$ does not commute with $\rho(T_{12})$. We first claim
that under this assumption,

$\rho(T_{123})=\rho(T_{1234})=\rho(T_{234})=1.$

To see this, observe that each such element commutes with both
$\rho(T_{12})=\rho(T_{34})$ and $\rho(T_{23})$ and hence lies in the
intersection of their centralizers, which is trivial.

$\gamma$ at 90 91 $\gamma_{3}$ at 75 90 $\gamma_{4}$ at 60 90 $\gamma_{34}$ at
40 90 $\bar{c}$ at 170 30

Figure 4. The configuration of curves used in Subcase 1.

Referring to the left side of Figure 4, we define the elements
$S,S_{3},S_{4},S_{34}\in\operatorname{Mod}(T^{2},\\{x_{1},\dots,x_{4}\\})$ as
the Dehn twists about the curve $\gamma$ with the corresponding subscript.
Then there is a lantern relation of the form

$T_{34}S_{4}S_{3}=SS_{34}$

We rearrange, expressing each side as an element of $PB_{4}(T^{2})$:

$T_{34}S_{4}S^{-1}=S_{3}^{-1}S_{34}$

Since $S_{4}S^{-1}$ commutes with $T_{12}$ and $T_{23}$, it follows that
$\rho(S_{4}S^{-1})=1$ since $\rho(T_{12})$ and $\rho(T_{23})$ do not commute
by hypothesis. Thus $\rho(S_{3}^{-1}S_{34})=\rho(T_{34})$. There is a second
lantern relation $T^{\prime}_{24}T_{34}T_{23}=T_{234}$, from which we conclude
$\rho(T^{\prime}_{24})=\rho(T_{34}T_{23})^{-1}$.

Let $c$ be the curve given as a regular neighborhood of the arc $\bar{c}$
indicated on the right side of Figure 4. By disjointness, $T_{c}$ commutes
with $T^{\prime}_{24}$ and $S_{3}^{-1}S_{34}$. However
$\rho(S_{3}^{-1}S_{34})=\rho(T_{34})=\rho(T_{12})$ and
$\rho(T^{\prime}_{24})=\rho(T_{34}T_{23})^{-1}=\rho(T_{12}T_{23}^{-1})$ don’t
commute, which implies that $\rho(T_{c})=1$. By 5.5, $\rho(T_{c})$ is
conjugate to $\rho(T_{13})$, implying that $\rho(T_{13})=1$ as well. However,
there is a lantern relation $T_{12}T_{23}T_{13}=T_{123}$, and since
$\rho(T_{123})=1$, this implies that $\rho(T_{13})=\rho(T_{12}T_{23})^{-1}\neq
1$, a contradiction.

Figure 5. Subcase 2: generators for $P_{1}$ and $P_{2}$.

Subcase 2: $\rho(T_{23})$ commutes with $\rho(T_{12})$. In this case,
$\rho(PB_{4})$ is in the centralizer of $\rho(T_{12})$. We now show that
$\rho(PB_{4}(T^{2}))$ is in also in the centralizer of $\rho(T_{12})$, and
hence $\rho$ has cyclic image. By 5.7, it suffices to show that $\rho(P_{i})$
is in the centralizer of $\rho(T_{12})$ for $i=1,\dots,4$. The subgroup
$P_{1}$ is the fundamental group of $T^{2}-\\{x_{2},x_{3},x_{4}\\}$ and is
generated by point-push maps about the paths indicated in Figure 5. Expressing
a point push as the product of Dehn twists about the boundary components of a
regular neighborhood, one observes that each such curve is disjoint from
either $a_{12}$ or $a_{34}$ with the exception of $T_{23}$. By hypothesis,
$\rho(T_{23})$ commutes with $\rho(T_{12})$, and it follows that $\rho(P_{1})$
commutes with $\rho(T_{12})=\rho(T_{34})$ as claimed. A similar analysis of
the generating sets for $P_{2},P_{3},P_{4}$ show that the same result holds,
completing the argument.

Case 3: general $n$. This proceeds as in the first step of the case $n=4$. If
all $\rho(T_{ij})=1$, then the image is abelian and hence cyclic. Otherwise,
$\rho(T_{12})\neq 1$ without loss of generality. By 4.6, the restriction
$\rho|_{PB_{n}}$ factors through a forgetful map, so that (without loss of
generality) $\rho(T_{i,n})=1$ for $1\leq i\leq n-1$. By 5.6, it follows that
$\rho$ factors through $e_{n,*}$, and applying 5.8 implies that either $\rho$
has cyclic image or else we have reduced to the case $m=n-1$. ∎

###  5.2. 5.2: holomorphic maps between configuration spaces on elliptic
curves

To begin, we recall the classification of maps $f:PB_{n}(X)\to F_{m}$
established (in greater generality) in [5, Theorem 1.1].

######  Theorem 5.9.

Let $X$ be a compact Riemann surface with $g(X)\geq 2$, and let $F_{m}$ be a
free group of rank $m\geq 2$. Then every homomorphism $f:PB_{n}(X)\to F_{m}$
either has cyclic image or else factors through a forgetful map
$p_{i}:PB_{n}(X)\to X$.

Using this and 5.1, we establish the following analogue of 4.2.

######  Lemma 5.10.

Let $X,Y$ be compact Riemann surfaces, with $g(Y)=1$. Choose a base point
$O\in Y$, thereby endowing $Y$ with a group structure. Let
$h:\operatorname{PConf}_{n}(X)\to Y-\\{O\\}$ be a holomorphic map. Then either
$h$ is constant or there is an isomorphism $I:X\cong Y$, and
$h=I(x_{i}-x_{j})$ for some $i,j$.

###### Proof.

We proceed by induction, the case $n=1$ being trivial (as every holomorphic
map $h:X\to Y-\\{O\\}$ is constant). Now let $n\geq 2$ be given. By 5.1 and
5.9, either $h_{*}$ has cyclic image or $h_{*}$ factors through a forgetful
map: either $PB_{n}(X)\to PB_{2}(X)$ in the case of $g(X)=1$ or $PB_{n}(X)\to
X$ in the case of $g(X)\geq 2$.

Firstly, suppose that $h_{*}$ has cyclic image. Fix a configuration
$C=(x_{1},...,x_{n-1})\in\operatorname{PConf}_{n-1}(X)$, and for $1\leq k\leq
n$, consider the natural embedding
$i_{C,k}:X-C\subset\operatorname{PConf}_{n}(X)$ where all but the $k^{th}$
coordinate is fixed at $C$; note this is a holomorphic map. By 2.3, $h\circ
i_{C,k}$ extends to a holomorphic map $H:X\to Y$. If $H$ is not constant, then
Hodge theory asserts that $H^{1}(Y;\mathbb{C})$ is spanned by holomorphic
differentials and their conjugates. Since nonzero holomorphic differentials
pull back to nonzero holomorphic differentials along holomorphic maps, it
follows that $H^{*}:H^{1}(Y;\mathbb{C})\to H^{1}(X;\mathbb{C})$ is injective,
and dually $H_{*}:H_{1}(X;\mathbb{C})\to H_{1}(Y;\mathbb{C})$ is surjective.
This contradicts the assumption that $h\circ i_{C,k}$ has cyclic image.

Next, suppose that that $g(X)\geq 2$ and $h_{*}$ factors through a forgetful
map $p:PB_{n}(X)\to X$; without loss of generality, assume that $p$ is the
projection onto the $n^{th}$ factor. As in the previous paragraph, we fix a
configuration $C=(x_{1},...,x_{n-1})\in\operatorname{PConf}_{n-1}(X)$ and
consider $h\circ i_{C,k}:X-C\to Y-\\{O\\}$. By hypothesis, for $1\leq k\leq
n-1$, the induced map $(h\circ i_{C,k})_{*}$ on fundamental group is trivial
and so there is a lift $H:X-C\to\mathbb{D}$. By the removable singularity
theorem, this extends to a holomorphic map $H:X\to\mathbb{D}$, but this must
be constant by the maximum principle. We conclude that
$h:\operatorname{PConf}_{n}(X)\to Y-\\{O\\}$ factors through
$p_{n}:\operatorname{PConf}_{n}(X)\to X$. The induced map $h:X\to Y$ then has
degree zero (since it misses $O\in Y$ by hypothesis) and hence is constant, as
claimed.

Finally, suppose that $g(X)=1$ and $h_{*}$ factors through a forgetful map
$PB_{n}(X)\to PB_{2}(X)$; for notational simplicity, we may therefore assume
that $n=2$. Let $x_{1}\in X$ and
$i_{x_{1}}:X-\\{x_{1}\\}\to\operatorname{PConf}_{2}(X)$ be the natural
embedding. Then the induced map $h\circ i_{x_{1}}$ is also holomorphic. By the
removable singularity theorem (2.3), $h\circ i_{x_{1}}$ can be uniquely
extended to a map $\overline{h\circ i_{x_{1}}}:X\to Y$. Every holomorphic map
between elliptic curves is either constant or else a covering map. Since $O$
has at most a single preimage, $\overline{h\circ i_{x_{1}}}$ is either an
isomorphism or a constant map.

If $\overline{h\circ i_{x_{1}}}$ is a constant map, then this holds for all
$x_{1}\in X$. Thus $h$ factors through the forgetful map to the first
coordinate $X$, reducing further to the previous case $n=1$, from which we
conclude that $h$ is constant.

If $\overline{h\circ i_{x_{1}}}$ is an isomorphism, then we obtain a family of
isomorphisms $\overline{h\circ i_{x_{1}}}:X\to Y$ such that $\overline{h\circ
i_{x_{1}}}(x_{1})=O$. Thus we obtain a global holomorphic map $H:X\times X\to
Y$ such that $H(x_{1},x_{1})=O$. Choose a basepoint $O_{X}\in X$, and let
$I=\overline{h\circ i_{O_{X}}}$. Thus $H(O_{X},x_{2})=I(x_{2})$. A holomorphic
map $X\to Y$ is uniquely determined by the map on the fundamental group and
the image of a single point. Thus $H(x_{1},x_{2})=I(x_{2}-x_{1})$ for any
other $x_{1}$ since this map satisfies that $H(x_{1},x_{1})=O$ and is
compatible on fundamental groups. ∎

###### Proof of 5.2.

Let $h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$ be a
holomorphic map, and fix a basepoint $O\in Y$. Let $y_{1}\in Y$ denote the
first coordinate of $h(x_{1},\dots,x_{n})$; note that $y_{1}$ is a holomorphic
function of $x_{1},\dots,x_{n}$. Exploiting the group structure, define the
normalization

$h_{s}(x_{1},...,x_{n})=h(x_{1},...,x_{n})-(y_{1},...,y_{1}).$

Thus,
$h_{s}(x_{1},...,x_{n})=(O,f_{2}(x_{1},...,x_{n}),...,f_{m}(x_{1},...,x_{n}))$
where $f_{i}:\operatorname{PConf}_{n}(X)\to Y-\\{O\\}$ are holomorphic maps.
By 5.10, it follows that $f_{p}(x_{1},...,x_{n})=I_{p}(x_{i}-x_{j})$ for some
isomorphism $I_{p}:X\to Y$ and $1\leq i<j\leq n$.

To proceed, we consider the set of all isomorphisms $I:X\to Y$. This set is a
torsor for $\operatorname{Aut}(Y)$, the group of automorphisms of $Y$ as a
Riemann surface. There is a semi-direct product structure

$\operatorname{Aut}(Y)\cong(Y,O)\rtimes\operatorname{Aut}(Y,O),$ (2)

where $(Y,O)$ indicates the elliptic curve structure on $Y$ with $O$ as the
origin, and $\operatorname{Aut}(Y,O)$ indicates the subgroup of
$\operatorname{Aut}(Y)$ fixing $O$, equivalently the subgroup of group
automorphisms. The group $\operatorname{Aut}(Y,O)$ is always finite, and
always contains the negation map $-id:x\mapsto-x$.

Fix the identification $I_{2}:X\to Y$ associated with the first nontrivial
coordinate $f_{2}$ once and for all; accordingly, we suppress $I_{2}$ from the
notation. For $p\geq 3$, the torsor structure then gives expressions
$I_{p}=(\epsilon_{p},\alpha_{p})\circ I_{2}$, where $\epsilon_{p}\in Y$ and
$\alpha_{p}\in\operatorname{Aut}(Y,O)$. Succinctly,
$I_{p}(x)=\alpha_{p}(x)+\epsilon_{p}$, so that we can write (after a
permutation of coordinates in the domain)

$f_{s}(x_{1},\dots,x_{n})=(O,x_{2}-x_{1},\alpha_{3}(x_{i_{3}})-\alpha_{3}(x_{j_{3}})+\epsilon_{3},\dots,\alpha_{m}(x_{i_{m}})-\alpha_{m}(x_{j_{m}})+\epsilon_{m}).$

By hypothesis, the difference
$\alpha_{p}(x_{i_{p}})-\alpha_{p}(x_{j_{p}})+\epsilon_{p}-(x_{2}-x_{1})$ does
not have $O$ in its image, so that by 5.10, there is an identity

$\alpha_{p}(x_{i_{p}})-\alpha_{p}(x_{j_{p}})+\epsilon_{p}-(x_{2}-x_{1})=\beta(x_{k})-\beta(x_{l})+\delta,$

for distinct indices $k,l$ and $(\beta,\delta)\in\operatorname{Aut}(Y)$, or
equivalently

$x_{1}+\alpha_{p}(x_{i_{p}})+\beta(x_{l})-(x_{2}+\alpha_{p}(x_{j_{p}})+\beta(x_{k}))=\delta-\epsilon_{p}.$

The right-hand side above is constant, and so the left-hand side must exhibit
cancellation. If either $\beta$ or $\alpha_{p}$ is not $\pm id$, then this is
not possible (e.g. lifting to the universal cover $\mathbb{C}$ of $Y$, the
derivative of the left-hand side would necessarily be everywhere nontrivial).
Up to an exchange of indices, we can assume that $\alpha_{p}=\beta=id$, so
that

$x_{1}+x_{i_{p}}+x_{l}-(x_{2}+x_{j_{p}}+x_{k})=\delta-\epsilon_{p}.$

It follows that exactly one of the conditions $i_{p}=2$ or $j_{p}=1$ holds
(they cannot both hold simultaneously since then $k=l$). By a permutation of
coordinates, we can assume $j_{3}=1$ and $i_{3}=3$. At this point, our
expression for $f_{s}$ has become

$f_{s}(x_{1},\dots,x_{n})=(O,x_{2}-x_{1},x_{3}-x_{1}+\epsilon_{3},\dots,x_{i_{m}}-x_{j_{m}}+\epsilon_{m}),$

subject to the condition that exactly one of the identities $i_{p}=2$ or
$j_{p}=1$ holds for all $p$. By comparing the second and $p^{th}$ entries, we
conclude that $\epsilon_{p}=O$ for all $p\geq 3$.

We claim that necessarily $j_{p}=1$ must hold for all $p\geq 4$. Suppose to
the contrary; then (without loss of generality)

$f_{s}(x_{1},\dots,x_{n})=(O,x_{2}-x_{1},x_{3}-x_{1},x_{2}-x_{j_{4}},\dots).$

By the previous analysis, comparing the third and fourth component forces
$j_{4}=1$, but then the fourth component equals the second, a contradiction.
Finally, we can re-normalize $f_{s}$ by translation by $x_{1}$, showing that
$f$ is given by a forgetful map as claimed. ∎

###  5.3. 5.3: from higher genus to genus one

###### Proof of 5.3.

Write $h=(f_{1},\dots,f_{m})$, with each $f_{k}:\operatorname{PConf}_{n}(X)\to
Y$ holomorphic. Choose a basepoint $O\in Y$, and, as in 5.2, normalize
$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$ by the twist

$A(x_{1},\dots,x_{n})(y)=y-f_{1}(x_{1},\dots,x_{n})$

so that $h=(O,f_{2},\dots,f_{m})$ with $f_{k}:\operatorname{PConf}_{n}(X)\to
Y-O$ holomorphic. By 5.10, each $f_{k}$ must be constant. ∎

##  6\. Pure configuration spaces in higher genus

In this section we consider the problem of classifying holomorphic maps
$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$, where $X,Y$ are
Riemann surfaces of higher genus. We find that the situation is quite rigid -
6.1 shows that either $X=Y$ and the map is forgetful, or else $h$ is induced
from a family of maps $f_{i}:X\to Y$ which pairwise have disjoint graphs. In
6.3, we consider the question of how many such $f_{i}$ can exist, which is
essentially a variant of the de Franchis theorem from classical algebraic
geometry.

###  6.1. Classification of holomorphic maps

######  Theorem 6.1.

Let $X$ and $Y$ be compact Riemann surfaces, with $g(X),g(Y)\geq 2$. Suppose

$h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$

is a nonconstant holomorphic map. Then up to permutation of coordinates and
the actions of $\operatorname{Aut}(X)$ and $\operatorname{Aut}(Y)$, either
$X=Y,n\geq m$, and $h$ is a forgetful map
$\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(X)$, or else $h$
factors as the composition of a forgetful map $\operatorname{PConf}_{n}(X)\to
X$ and a holomorphic map $X\to\operatorname{PConf}_{m}(Y)$.

###### Proof.

Consider the composition $p_{i}\circ h:\operatorname{PConf}_{n}(X)\to Y$,
where $p_{i}:\operatorname{PConf}_{m}(Y)\to Y$ is the projection onto the
$i^{th}$ factor.

######  Claim 6.2.

Each $p_{i}\circ h$ is nonconstant and induces a surjection
$H_{1}(\operatorname{PConf}_{n}(X);\mathbb{Q})\to H_{1}(Y;\mathbb{Q})$.

###### Proof.

We first show that each $p_{i}\circ h$ is nonconstant. If $m=1$ and $p_{1}$ is
constant then $h$ is constant, contrary to hypothesis. If $m>1$, then
supposing that any $p_{i}\circ h$ is constant (say with value $y_{0}\in Y$)
let $j\neq i$ be some other index, and consider $p_{j}\circ h$. If this is
nonconstant, then let $C\subset X$ be a configuration of $n-1$ distinct
points, and consider the inclusion
$i_{C,k}:X-C\subset\operatorname{PConf}_{n}(X)$ as before. Here $k$ is chosen
so as to make the composition $p_{j}\circ h\circ i_{C,k}$ nonconstant. By the
removable singularity theorem (2.3), this extends to give a nonconstant
holomorphic map $f_{C}:X\to Y$ and by the de Franchis theorem, the space of
such maps is discrete, and hence $f:=f_{C}$ is independent of $C$. Since
$f:X\to Y$ is nonconstant, it has positive degree and hence is surjective.
Thus, there is some $(x_{1},\dots,x_{n})\in\operatorname{PConf}_{n}(X)$ such
that $p_{j}h(x_{1},\dots,x_{n})=y_{0}=p_{i}h(x_{1},\dots,x_{n})$, i.e. the map
fails to have codomain $\operatorname{PConf}_{n}(Y)$, contrary to assumption.

To see that $(p_{i}\circ h)_{*}$ is surjective, it suffices to see that
$f_{*}:H_{1}(X;\mathbb{Q})\to H_{1}(Y;\mathbb{Q})$ is surjective, where $f$ is
as in the above paragraph, but this is a general property of nonconstant
holomorphic maps between compact Riemann surfaces, following from the
existence of a transfer homomorphism $f^{!}:H_{1}(Y;\mathbb{Q})\to
H_{1}(X;\mathbb{Q})$ with the property that
$f^{!}f_{*}=\deg(f)I_{H_{1}(X;\mathbb{Q})}$ (see, e.g. [18, Definition 2.2]
for a dual cohomological formulation). ∎

We consider the induced map $(p_{i}\circ
h)_{*}:\operatorname{PB}_{n}(X)\to\pi_{1}(Y)$. By [4, Lemma 2.5], either
$(p_{i}\circ h)_{*}$ factors as $(p_{i}\circ h)_{*}=f_{*}\circ p_{j,*}$ for
some $f_{*}:\pi_{1}(X)\to\pi_{1}(Y)$, where
$p_{j,*}:\operatorname{PB}_{n}(X)\to\pi_{1}(X)$ is induced by projecting onto
the $j^{th}$ factor, or else $(p_{i}\circ h)_{*}$ has cyclic image, possibly
trivial. The latter case cannot happen, since such maps would not induce a
surjection on $H_{1}$, contrary to 6.2.

Defining $[k]:=\\{1,\dots,k\\}$, we next claim that there is a function
$j:[m]\to[n]$ and nonconstant holomorphic maps $\alpha_{i}:X\to Y$ for which
the diagram below commutes for all $i$:

$\textstyle{\operatorname{PConf}_{n}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{p_{j(i)}}$$\textstyle{\operatorname{PConf}_{m}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{i}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\textstyle{Y}$
(3)

The function $j$ can be defined as follows: by the above, $(p_{i}\circ
h)_{*}:\operatorname{PB}_{n}(X)\to\pi_{1}(Y)$ factors through some projection
$p_{j,*}:\operatorname{PB}_{m}(X)\to\pi_{1}(X)$; let $j(i)$ be this $j$. We
observe that $h$ is determined by its values on the collection of submanifolds
$(X-C)_{k}$. For $k\neq j(i)$, the restriction of the holomorphic map
$p_{i}\circ h$ to $(X-C)_{k}$ is nullhomotopic by the preceding paragraph, and
hence constant. Thus $p_{i}\circ h$ factors through $p_{j(i)}$ as required.

By construction, if $j$ is constant, then $h$ factors through some projection
$p_{j}:\operatorname{PConf}_{n}(X)\to X$. It remains to show that if $j$ is
nonconstant, then $Y=X,n\geq m$, and $h$ is a forgetful map up to permutation
of coordinates and the application of some $\alpha\in\operatorname{Aut}(X)$ to
each component of $\operatorname{PConf}_{n}(X)$.

We claim that if $j$ is nonconstant, then $\alpha_{i_{1}}=\alpha_{i_{2}}$ for
all pairs of indices. Supposing to the contrary, without loss of generality,
we may take $i=1,j=2$, and $j(1)=1,j(2)=2$. Let $x_{1}\in X$ be given such
that $\alpha_{1}(x_{1})\neq\alpha_{2}(x_{1})$, and let $x_{2}\in X$ satisfy
$\alpha_{2}(x_{2})=\alpha_{1}(x_{1})$. Completing $x_{1},x_{2}$ to a point
$(x_{1},x_{2},\dots,x_{n})\in\operatorname{PConf}_{n}(X)$, we see that
$h(x_{1},\dots,x_{n})=(\alpha_{1}(x_{1}),\alpha_{2}(x_{2}),\dots,\alpha_{n}(x_{j(n)}))$
has a repeated entry $\alpha_{1}(x_{1})=\alpha_{2}(x_{2})$, a contradiction.

To see that $X=Y$ in this case, we show that the fixed map $\alpha:X\to Y$ has
degree $1$. If not, let $x_{1}\neq x_{2}\in X$ satisfy
$\alpha(x_{1})=\alpha(x_{2})=y$; then $h(x_{1},x_{2},\dots)=(y,y,\dots)$ has a
repeated entry. Thus $X=Y$, and by adjusting by $\alpha^{-1}$ if necessary, we
may assume that $\alpha=id$. Now it is clear that $n\geq m$, otherwise
$h(x_{1},\dots,x_{n})=(x_{j(1)},\dots,x_{j(n)})$ would have a repeated entry.
∎

###  6.2. Bounds

To complement 6.1, we consider the problem of determining the maximal $m$ for
which there is a nonconstant holomorphic map
$h:X\to\operatorname{PConf}_{m}(Y)$. This is closely related to the effective
de Franchis problem, which asks for bounds on the number of distinct
holomorphic maps $f_{i}:X\to Y$ (known to be finite for $g(X),g(Y)\geq 2$ by
the classical de Franchis theorem); here we add the condition that the images
of $f_{i}$ be pairwise-disjoint (pairwise have no coincidences, in the
terminology of [18]). The general effective de Franchis problem is far from
conclusively resolved - Chamizo [3] obtains an upper bound that is slightly
larger than exponential in $g(X)$, while the largest known examples are linear
in the genus (arising when $X$ and/or $Y$ has a large automorphism group which
can be used to enlarge the number of morphism by pre/post composition).

Here, we find that the condition that the morphisms pairwise have no
coincidences imposes a strong constraint, greatly reducing the upper bound,
although in practice there is still a gap between the upper bound of 6.3 and
the largest known examples (arising when $Y$ is equipped with a group of free
automorphisms).

######  Theorem 6.3.

Let $X,Y$ be compact Riemann surfaces each of genus at least $2$, and let
$h:X\to\operatorname{PConf}_{m}(Y)$ be a nonconstant holomorphic map. Then
$m\leq 4g(X)g(Y)$.

###### Proof.

Each holomorphic map $f:X\to Y$ induces
$f_{*}\in\operatorname{Hom}(J(X),J(Y))$, the induced map on Jacobians; Martens
observes [16] that distinct morphisms $f,g$ induce distinct maps $f_{*},g_{*}$
so long as $g(Y)\geq 2$. Tanabe [18, Definition 2.9], following ideas of
Fuertes – González-Diez [11], Martens [16] and ultimately Weil [20],
introduces a certain positive-definite inner product
$\langle\cdot,\cdot\rangle$ on $\operatorname{Hom}(J(X),J(Y))$. According to
[18, Theorem 4.1], if $f,g:X\to Y$ are holomorphic and have no coincidences
(i.e. $f(x)\neq g(x)$ for all $x\in X$), then $\deg(f)=\deg(g)$ and
$\cos(f_{*},g_{*})=g(Y)^{-1}$, where $\cos(v,w):=\langle v,w\rangle/\lVert
v\rVert\lVert w\rVert$ is defined as in any inner product space. If
$h:X\to\operatorname{PConf}_{m}(Y)$ is given, the component functions
$h_{1},\dots,h_{m}$ pairwise have no coincidences, and thus determine a
configuration of vectors
$h_{1,*},\dots,h_{m,*}\in\operatorname{Hom}(J(X),J(Y))$ where the angles
$\cos(h_{i,*},h_{j,*})=g(Y)^{-1}$ are pairwise fixed and equal.

As $\operatorname{Hom}(J(X),J(Y))$ is a subgroup of
$\operatorname{Hom}(H_{1}(X;\mathbb{Z}),H_{1}(Y,\mathbb{Z}))\cong\mathbb{Z}^{4g(X)g(Y)}$,
to prove the claim, it suffices to show that such a configuration of vectors
must be linearly independent. It suffices to consider the associated unit
vectors $v_{1},\dots,v_{m}$. Let $A$ be the matrix with $A_{ij}=\langle
v_{i},v_{j}\rangle$; linear-independence of $\\{v_{1},\dots,v_{m}\\}$ is
equivalent to the nonsingularity of $A$. By hypothesis,

$A=(1-g(Y)^{-1})I+C,$

where $C$ is the matrix where every entry is given by $g(Y)^{-1}$. The
eigenvalues of $C$ are $0$ and $mg(Y)^{-1}$, and hence the eigenvalues of $A$
are $1-g(Y)^{-1}$ and $1+(m-1)g(Y)^{-1}$. As $g(Y)\geq 2$, both of these are
nonzero, which proves the claim. ∎

##  7\. Distinct genus regime

In this section, we examine what happens when $g(X)$ and $g(Y)$ belong to
distinct genus regimes (i.e. $g=0,1,$ or $g\geq 2$) - this is the setting in
which there are very few holomorphic maps. The results largely follow from
basic principles, but we include the proofs for the sake of completeness.

######  Proposition 7.1.

Let $X,Y$ be Riemann surfaces of finite type with $g(Y)>g(X)$. Then every
holomorphic map $h:\operatorname{PConf}_{n}(X)\to\operatorname{PConf}_{m}(Y)$
is constant.

###### Proof.

Fix a configuration of distinct points $C=\\{x_{1},\dots,x_{n-1}\\}\subset X$,
and for $1\leq k\leq n$, consider the inclusions
$(X-C)_{k}\hookrightarrow\operatorname{PConf}_{n}(X)$ as in the proof of 6.1.
Composing with the projection $p_{i}:\operatorname{PConf}_{m}(Y)\to Y$ onto
the $i^{th}$ factor, we obtain a holomorphic map $f_{ik}:X-C\to Y$. As
$g(Y)>g(X)$, any such map must be constant; varying $i,k$, and $C$ then shows
that $h$ itself is constant. ∎

######  Proposition 7.2.

Let $Y$ be a compact Riemann surface of genus $g(Y)=1$. Then every holomorphic
map $h:\operatorname{Conf}_{n}(\mathbb{CP}^{1})\to\operatorname{Conf}_{m}(Y)$
is constant.

###### Proof.

We consider the induced map on fundamental group

$h_{*}:B_{n}(\mathbb{CP}^{1})\to\pi_{1}(Y);$

note that as the target is abelian, this factors through
$H_{1}(B_{n}(\mathbb{CP}^{1});\mathbb{Z})$. It is well-known that
$B_{n}(\mathbb{CP}^{1})=B_{n}(S^{2})$ is a quotient of $B_{n}$ by the word
$\sigma_{1}\sigma_{2}\dots\sigma_{n-1}\sigma_{n-1}\dots\sigma_{1}$. Thus,
$H_{1}(B_{n}(\mathbb{CP}^{1});\mathbb{Z})\cong\mathbb{Z}/(2n-2)\mathbb{Z}$. As
$\pi_{1}(Y)\cong\mathbb{Z}^{2}$ is torsion-free, it follows that $h_{*}$ is
the trivial map. Applying 5.6, we conclude that $h$ extends to a holomorphic
map

$H:(\mathbb{CP}^{1})^{n}\to Y.$

Every such map is constant (e.g. $H$ lifts to the universal cover $\mathbb{C}$
of $Y$ and is therefore constant via the maximum principle). ∎

We remark that in the setting of $g(Y)\geq 2$, the same argument (replacing
$Y$ with its Jacobian) shows that any holomorphic map
$h:\operatorname{Conf}_{n}(\mathbb{CP}^{1})\to\operatorname{Conf}_{m}(Y)$ has
image contained in a fixed linear system on $Y$, but this by itself is not
enough to conclude that $h$ itself is constant.

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