We answer the question of when a new point can be added in a continuous way to configurations of n distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of n points if and only if n ≠ 1. On the other hand, when the points are not ordered and the dimension of the ball is at least 2, a point can be added continuously if and only if n = 2. These results generalize the Brouwer fixed-point theorem, which gives the negative answer when n = 1. We also show that when n = 2, there is a unique solution to both the ordered and unordered versions of the problem up to homotopy

Chen, Lei

Gadish, Nir

Lanier, Justin

English

arXiv

Caltech Authors - Main

ar
X
iv
:1
80
9.
06
94
6v
2 
 [m
ath
.G
T]
  7
 M
ay
 20
19
ADDING A POINT TO CONFIGURATIONS IN CLOSED BALLS
LEI CHEN, NIR GADISH, AND JUSTIN LANIER
Abstract. We answer the question of when a new point can be added in a continuous way to configurations
of n distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered
configuration of n points if and only if n 6= 1. On the other hand, when the points are not ordered and the
dimension of the ball is at least 2, a point can be added continuously if and only if n = 2. These results
generalize the Brouwer fixed-point theorem, which gives the negative answer when n = 1. We also show
that when n = 2, there is a unique solution to both the ordered and unordered versions of the problem up
to homotopy.
1. Introduction
Let Bm be the closed ball of dimension m, with m ≥ 1. This paper answers the following basic question:
Is there a continuous rule that adds a new distinct point to every configuration of n distinct
points in Bm?
The challenge here is that the new point must be distinct from all of the existing ones, and it is not clear
whether such a choice can be made continuously.
Example 1 (Case n = 1: Brouwer fixed-point theorem). Given one point in Bm, a continuous choice
of a second distinct point can be thought of as a continuous function from the closed ball to itself with no
fixed points. By the Brouwer fixed-point theorem [Bro11] no such continuous function exists, and therefore
introducing a second distinct point continuously is impossible.
When extending the Brouwer fixed-point theorem to n > 1, the question splits into two versions: either
the n points are given with an ordering (p1, . . . , pn), or the points instead form an unordered set. The first
author addressed both versions of this question with respect to point configurations lying in the infinite
plane, in the 2-sphere, and in closed surfaces Sg with g ≥ 2 in [Che17] and in her joint work with Salter
[CS17].
It is perhaps surprising that having more points to avoid does not necessarily make the task of adding a
point harder.
Example 2 (Case n = 2: midpoint). Given two distinct points in Bm, their midpoint provides a continuous
way to introduce a third point, distinct from the existing two. In fact, in §4 we show that this midpoint
construction is the unique way to add a point when n = 2 up to homotopy.
Let us state more formally the problem of continuously adding a point to a configuration of n points
in Bm, which from here on will be abbreviated to B. Let PConfn(B) denote the pure configuration space
of n distinct ordered points in B, topologized as a subspace of (B)n. A continuous map PConfn(B) → B
is said to ‘add a point’ if the image of every configuration (p1, . . . , pn) is a point p0 distinct from all pi
with i ≥ 1. Equivalently, consider the map gm,n : PConfn+1(B) → PConfn(B) that forgets the 0th point
(p0, p1, . . . , pn) 7→ (p1, . . . , pn). The problem of continuously introducing a new point is precisely the question
of finding a continuous section for gm,n.
1
Next, the symmetric group Σn acts on ordered configurations by permuting the indices, and the quotient
Confn(B) := PConfn(B)/Σn is the configuration space of n distinct unordered points. The map gm,n
forgetting the point p0 is Σn-equivariant and descends to a map g¯m,n : PConfn+1(B)/Σn → Confn(B). With
this, our problem of adding a point distinct from a given unordered set of n points is the problem of finding
a section for g¯m,n.
The easy positive result for n = 2 in Example 2 suggests that the problem for larger n might also have
a solution. However, this is where the two versions of the problem part ways: the ordered version extends
the n = 2 case by admitting similar solutions, while the unordered version reverts back to the behavior at
n = 1.
Theorem A (Ordered). In dimension m ≥ 1 and with n ≥ 1 points, the forgetful map gm,n has a section
if and only if n 6= 1. That is, one can continuously add a new point to an ordered configuration exactly when
it consists of 2 or more ordered points.
Theorem B (Unordered). In dimension m ≥ 2 and with n ≥ 1 points, the forgetful map g¯m,n has a
section if and only if n = 2. That is, one cannot continuously add a new point to an unordered configuration
unless it consists of precisely 2 unordered points.
As for the exceptional case of n = 2,
Theorem C (Uniqueness for n = 2). In every dimension m ≥ 1, the midpoint construction of Example 2
homotopically unique. That is, every section of gm,2 or g¯m,2 is homotopic to the midpoint construction via
a homotopy through sections.
Note that the case of points on a line segment (m = 1) is excluded from Theorem B. In this case, the
unordered version coincides with the ordered one, as the points in an unordered configuration are nevertheless
forced into a linear order. We may therefore add a point continuously to a configuration of n points in B1
as long as n 6= 1.
Recasting Theorem B as a direct generalization of Brouwer’s fixed-point theorem one can say that, except
for when n = 2, every continuous map f : Confn(B)→ B has a ‘fixed point’. This is meant in the sense that
there must exist some configuration S = {p1, . . . , pn} whose image under f lies inside S.
We remark that the negative results in Theorem B contrasts with the analogous problem of introducing
a new point to an unordered configuration of points in Rm (or equivalently, on the open ball). This latter
problem always has a solution: add a point ‘at infinity’, i.e., place it very far away from all the others
[Che17]. Such a construction is of course not possible on the closed ball. However, since the configuration
spaces of the open and closed balls are homotopy equivalent (see §3), there is no purely homotopy theoretic
obstruction to finding a section in the case of a closed ball. This means that a different approach is needed.
Even more, many standard tools of algebraic topology fail in our context since the forgetful map gm,n is not
a fibration: it has fibers of distinct homotopy types, depending on how many points lie on the boundary of
the ball.
Outline. We treat the easy ordered version of the problem in §2 by briefly proving Theorem A. The
main argument of our proof of Theorem B proceeds by contradiction. Any section of g¯m,n induces an Σn-
equivariant section of gm,n : PConfn+1(B) → PConfn(B). Theorem B is then proved by pulling back a
cohomology class in two different ways and arriving at a contradiction. We conclude by proving Theorem C
in §4.
2
Acknowledgements. We thank Benson Farb and Dan Margalit for providing useful comments and for their
continued encouragement and support. We also thank the anonymous referee for a number of suggestions
that improved the paper. JL is supported by the NSF grant DGE-1650044.
2. Case of ordered configurations
In this section we prove Theorem A by generalizing the midpoint construction of Example 2.
Proof of Theorem A. The case n = 1 is covered by Example 1. Otherwise, fix n ≥ 2. For any pair (i, j)
with 1 ≤ i, j ≤ n and i 6= j, the line segment [pi, pj] is contained within B. Let vij := pj − pi be the vector
pointing from pi to pj. Further, let di := min {‖pk − pi‖ : k 6= i} be the minimal distance between pi and
any other point in the configuration. One can then introduce a new point “close” to pi, lying at a distance
di/2 from pi along the interval [pi, pj ]: explicitly, p0 := pi +
di
2‖vij‖
vij . The added point p0 is clearly distinct
from all other points, and it depends on the configuration continuously since di and vij do so. 
We remark that the first author gave a prior classification of sections between configuration spaces (see
the discussion preceding [Che17, Theorem 1.1]). Under this classification, the above construction of adding
a point is of the type ‘add close to pi’. Let us also call attention to the fact that the above construction
relies on singling out two of the points, which cannot be done continuously in the unordered version of the
problem when n > 2.
3. Case of unordered configurations
In this section we prove Theorem B, which says that in dimension m ≥ 2 there is no section of g¯m,n
except when n = 2. We begin with a few preliminary observations.
First, generators and relations for the integral cohomology ring of PConfn(R
m) were produced by Arnol’d
in dimension m = 2 and by Cohen for general m [Arn69, Coh88]. This cohomology ring is generated by
classes
Gab ∈ H
m−1(PConfn(R
m);Z)
for distinct 1 ≤ a < b ≤ n, where the generator Gab measures the winding of point a around point b. The
induced Σn-action is given by σ
∗(Gab) = Gσ(a)σ(b), where Gba = (−1)
mGab. This description applies equally
well to the closed ball as the following argument shows. Since Rm is homeomorphic to the open unit ball U,
their configuration spaces are also equivariantly homeomorphic. Then scaling by 0 < t < 1 gives a sequence
of inclusions
tU ⊂ tB ⊂ U ⊂ B
with compositions isotopic to the identity. It follows that U and B have equivariantly homotopy equivalent
configuration spaces, and in particular they have the same cohomology.
Our second observation is that any section s¯ of g¯m,n for the unordered configuration spaces lifts to a
Σn-equivariant section s of gm,n for the ordered spaces. This follows from the lifting criterion for connected
coverings.
Third, we leverage the fact that an equivariant section s of gm,n induces a solution to a related section
problem where the configurations have the added restriction that the point p1 is constrained to the boundary
sphere. This solution gives us a map, whose pullback on cohomology we compute in two ways, leading to a
contradiction. More precisely, let U ⊂ B denote the interior of the closed ball and consider the subspace of
PConfn(B) in which only the 1st point lies on the boundary sphere S
m−1:
B1n := {(p1, . . . , pn) ∈ PConfn(B) | p1 ∈ ∂ B, (p2, . . . , pn) ∈ PConfn(U)}
∼= Sm−1 × PConfn−1(U)
3
Define B1n+1 ⊆ PConfn+1(B) similarly. Lastly, let E
1
n+1 denote the preimage g
−1
m,n(B
1
n) and consider the
inclusion B1n+1 →֒ E
1
n+1. The difference between these two spaces is that in the larger space the additional
point p0 may lie on the boundary ∂ B, while in the smaller space this is not allowed. Despite this apparent
difference, we have the following.
Lemma 3.1. The inclusion B1n+1 →֒ E
1
n+1 is a homotopy equivalence.
Proof. Choosing an inward-pointing vector field that vanishes on the point p1 ∈ ∂ B, one can push any other
point into the interior of B. Explicitly, the vector field −(p− p1) on B gives rise to a smooth vector field on
PConfn+1(B). Its flow produces an isotopy Φ
1
t such that for all t > 0,
Φ1t (B
i
n+1) ⊂ Φ
1
t (E
i
n+1) ⊂ B
1
n+1 ⊂ E
1
n+1
thus establishing the homotopy equivalence. 
Consider the compatible projections onto the 1st coordinate
B1n+1
##●
●
●
●
●
●
●
●
gm,n

B1n // S
m−1.
Pulling back an orientation class [Sm−1] ∈ Hm−1(Sm−1), we get a class 0 6= X1 ∈ H
m−1(B1n) whose pullback
will also be abusively denoted by X1 ∈ H
m−1(B1n+1). These classes measure how many times the point p1
wraps around the boundary sphere.
Lemma 3.2. Under the inclusion ι : B1n+1 →֒ PConfn+1(B) we have
ι∗(G1a) = X1 for all 1 < a
and the class Gab for 1 < a, b pulls back to Gab ∈ H
m−1(PConfn(U)). The same is true for B
1
n ⊆ PConfn(B).
Via the homotopy equivalence B1n+1 ≃ E
1
n+1, we consider Lemma 3.2 as a statement about E
1
n+1 as well.
In particular we shall keep the notation X1 for the corresponding class in H
m−1(E1n+1).
Proof of Lemma 3.2. These facts are geometrically obvious: the class Gab is pulled back from S
m−1 under
the ‘Gauss map’, sending a configuration to the direction vector from pa to pb. When 1 < a, b, this Gauss
map factors through the projection B1n+1 → PConfn(U), as claimed.
Otherwise, if 1 < a then since p1 lies on the boundary and pa is internal, the Gauss map is homotopic
to a map in which pa is fixed at the origin. But when pa = 0 the Gauss map coincides with the projection
which records only p1. 
With these facts in hand, we can now complete our proof.
Proof of Theorem B. Let m ≥ 2. If n = 1, we have no section by Example 1. If n = 2, we have a section
by Example 2. Otherwise, let n ≥ 3. Assume that s¯ is a section of g¯m,n and let s be its Σn-equivariant
lift to a section of gm,n. The assumption that s is a section forces s(B
1
n) ⊆ E
1
n+1, thus it restricts to a
section s′ : B1n → E
1
n+1 of gm,n. From this one observes that (s
′)∗X1 = X1. Let us show that this leads to a
contradiction.
Since the classes Gab span H
m−1(PConfn(B);Z), there is an expansion
s∗(G01) =
∑
1<a≤n
λaG1a +
∑
1<a<b≤n
δabGab
4
for some integer coefficients. Equivariance implies that a permutation σ ∈ Σn fixing 1 will preserve this
expansion, and therefore λa = λb for all 1 < a < b ≤ n. Denote this constant value by λ. Similarly, it also
follows that all δab must be equal, say to the constant δ. We thus get
s∗(G01) = λ
∑
1<a≤n
G1a + δ
∑
1<a<b≤n
Gab. (1)
Next, we have a commutative diagram
E1n+1
ι // PConfn+1(B)
B1n
s′
OO
ι // PConfn(B)
s
OO
through which the pullback of the class G01 ∈ H
m−1(PConfn+1(B)) along the two different paths must agree.
Pulling it back through the top-left corner,
G01
ι∗
7−→ X1
(s′)∗
7−→ X1.
But pulling back through the bottom right corner, one first applies s∗ for which we have the expansion (1).
Since the restriction of Gab with 1 < a, b to B
1
n gives the class Gab again and each G1a restricts to X1, we
obtain the following:
ι∗s∗(G01) = λ(n− 1)X1 + δ
∑
1<a<b≤n
Gab.
The above equation has to be equal to the pulling back from the top-left corner, which is X1. But since the
Ku¨nneth formula for the product B1n
∼= Sm−1 × PConfn−1(U) implies that Xi is linearly independent from
the other classes Gab, such an equality is possible only if δ = 0 and λ(n − 1) = 1. But λ is an integer and
n > 2 which gives a contradiction. 
4. Uniqueness of the midpoint construction for n = 2
We now show that the midpoint section from Example 2 is unique up to a homotopy through sections.
Proof of Theorem C. Let us denote the midpoint section by M : PConf2(B)→ PConf3(B) and suppose that
s is another section of gm,2, possibly Σ2-equivariant. We construct a homotopy between s and M through
sections, such that if s was equivariant then so will be the homotopy.
If for every configuration s(p1, p2) = (p0, p1, p2), the added point p0 lies either between p1 and p2 or off
of the line connecting them, then the straight-line homotopy Ht := (1 − t)s + tM demonstrates the claim.
With this, the uniqueness problem is reduced to finding a homotopy to a section possessing this property.
The idea goes as follows. Given a configuration (p1, p2) we let the points repel each other, moving
outwards along the line connecting them until they hit the boundary. By applying the section s to this
motion, one gets a path of configurations of 3 points, where at the end of the process we have p1 and p2 on
the boundary and p0 somewhere in B. Thus if p0 is on the line containing p1 and p2, it must lie between
them. On such a configuration one can perform the straight-line homotopy to the midpoint. However, to
remain within sections of gm,2, we must apply the aforementioned process while globally scaling B down so
that overall the points p1 and p2 do not move.
Now more explicitly, for any configuration c = (p1, p2), let L
c be the straight line passing through it.
Then Lc intersects ∂ B at two distinct points q1 and q2, labeled so that qi is closer to pi. Let x
c be the
5
unique point on the line Lc from which the ratios of ‖qi− xc‖ to ‖pi− xc‖ are equal for i = 1, 2, and denote
this common ratio by rc. Then the isotopy
hct : v 7→ ((1 − t) + r
ct)(v − xc) + xc
is a scaling of Rm out from xc at a linear rate, and hc0 = Id while h
c
1 : pi 7→ qi for i = 1, 2. Note that since
hct is scaling out from a point inside B, the image of B contains B at every time t.
Now, since hct and (h
c
t)
−1 are injective, they induce Σn-equivariant isotopies of the configuration spaces
(Hct )
±1 : PConfn(R
m)→ PConfn(R
m) by applying them to a configuration diagonally. Lastly, it is clear that
Hct (p1, p2) is contained in B at all times 0 ≤ t ≤ 1, and that everything in sight depends on c continuously
(even algebraically).
A homotopy through sections is then given by
st(c) := (H
c
t )
−1 ◦ s ◦Hct (c)
This map acts by expanding the ball, using s to add a point at every time t and then contracting the ball
back – thus producing a path of configurations in which (p1, p2) never moves. The section s1 has the property
that allows us to connect it to M by a straight-line homotopy.
Lastly, if s was Σ2-equivariant, then st is a composition of equivariant maps and is thus itself equivariant.

References
[Arn69] V. I. Arnol’d. The cohomology ring of the group of dyed braids. Mat. Zametki, 5:227–231, 1969.
[Bro11] L. E. J. Brouwer. U¨ber Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71(1):97–115,
Mar 1911.
[Che17] L. Chen. Section problems for configuration spaces of surfaces. Pre-print,
https://arxiv.org/abs/1708.07921, 2017.
[Coh88] Frederick R Cohen. Artin’s braid groups, classical homotopy theory, and sundry other curiosities.
Braids (Santa Cruz, CA, 1986), 78:167–206, 1988.
[CS17] L. Chen and N. Salter. Section problems for configurations of points on the Riemann sphere.
Pre-print, https://arxiv.org/abs/1807.10171, 2017.
6


Adding a point to configurations in closed balls

ar
X
iv
:1
80
9.
06
94
6v
2 
 [
m
at
h.
G
T
] 
 7
 M
ay
 2
01
9
ADDING A POINT TO CONFIGURATIONS IN CLOSED BALLS
LEI CHEN, NIR GADISH, AND JUSTIN LANIER
Abstract. We answer the question of when a new point can be added in a continuous way to configurations
of n distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered
configuration of n points if and only if n 6= 1. On the other hand, when the points are not ordered and the
dimension of the ball is at least 2, a point can be added continuously if and only if n = 2. These results
generalize the Brouwer fixed-point theorem, which gives the negative answer when n = 1. We also show
that when n = 2, there is a unique solution to both the ordered and unordered versions of the problem up
to homotopy.
1. Introduction
Let Bm be the closed ball of dimension m, with m ≥ 1. This paper answers the following basic question:
Is there a continuous rule that adds a new distinct point to every configuration of n distinct
points in Bm?
The challenge here is that the new point must be distinct from all of the existing ones, and it is not clear
whether such a choice can be made continuously.
Example 1 (Case n = 1: Brouwer fixed-point theorem). Given one point in Bm, a continuous choice
of a second distinct point can be thought of as a continuous function from the closed ball to itself with no
fixed points. By the Brouwer fixed-point theorem [Bro11] no such continuous function exists, and therefore
introducing a second distinct point continuously is impossible.
When extending the Brouwer fixed-point theorem to n > 1, the question splits into two versions: either
the n points are given with an ordering (p1, . . . , pn), or the points instead form an unordered set. The first
author addressed both versions of this question with respect to point configurations lying in the infinite
plane, in the 2-sphere, and in closed surfaces Sg with g ≥ 2 in [Che17] and in her joint work with Salter
[CS17].
It is perhaps surprising that having more points to avoid does not necessarily make the task of adding a
point harder.
Example 2 (Case n = 2: midpoint). Given two distinct points in Bm, their midpoint provides a continuous
way to introduce a third point, distinct from the existing two. In fact, in §4 we show that this midpoint
construction is the unique way to add a point when n = 2 up to homotopy.
Let us state more formally the problem of continuously adding a point to a configuration of n points
in Bm, which from here on will be abbreviated to B. Let PConfn(B) denote the pure configuration space
of n distinct ordered points in B, topologized as a subspace of (B)n. A continuous map PConfn(B) → B
is said to ‘add a point’ if the image of every configuration (p1, . . . , pn) is a point p0 distinct from all pi
with i ≥ 1. Equivalently, consider the map gm,n : PConfn+1(B) → PConfn(B) that forgets the 0th point
(p0, p1, . . . , pn) 7→ (p1, . . . , pn). The problem of continuously introducing a new point is precisely the question
of finding a continuous section for gm,n.
1
Next, the symmetric group Σn acts on ordered configurations by permuting the indices, and the quotient
Confn(B) := PConfn(B)/Σn is the configuration space of n distinct unordered points. The map gm,n
forgetting the point p0 is Σn-equivariant and descends to a map ḡm,n : PConfn+1(B)/Σn → Confn(B). With
this, our problem of adding a point distinct from a given unordered set of n points is the problem of finding
a section for ḡm,n.
The easy positive result for n = 2 in Example 2 suggests that the problem for larger n might also have
a solution. However, this is where the two versions of the problem part ways: the ordered version extends
the n = 2 case by admitting similar solutions, while the unordered version reverts back to the behavior at
n = 1.
Theorem A (Ordered). In dimension m ≥ 1 and with n ≥ 1 points, the forgetful map gm,n has a section
if and only if n 6= 1. That is, one can continuously add a new point to an ordered configuration exactly when
it consists of 2 or more ordered points.
Theorem B (Unordered). In dimension m ≥ 2 and with n ≥ 1 points, the forgetful map ḡm,n has a
section if and only if n = 2. That is, one cannot continuously add a new point to an unordered configuration
unless it consists of precisely 2 unordered points.
As for the exceptional case of n = 2,
Theorem C (Uniqueness for n = 2). In every dimension m ≥ 1, the midpoint construction of Example 2
homotopically unique. That is, every section of gm,2 or ḡm,2 is homotopic to the midpoint construction via
a homotopy through sections.
Note that the case of points on a line segment (m = 1) is excluded from Theorem B. In this case, the
unordered version coincides with the ordered one, as the points in an unordered configuration are nevertheless
forced into a linear order. We may therefore add a point continuously to a configuration of n points in B1
as long as n 6= 1.
Recasting Theorem B as a direct generalization of Brouwer’s fixed-point theorem one can say that, except
for when n = 2, every continuous map f : Confn(B) → B has a ‘fixed point’. This is meant in the sense that
there must exist some configuration S = {p1, . . . , pn} whose image under f lies inside S.
We remark that the negative results in Theorem B contrasts with the analogous problem of introducing
a new point to an unordered configuration of points in Rm (or equivalently, on the open ball). This latter
problem always has a solution: add a point ‘at infinity’, i.e., place it very far away from all the others
[Che17]. Such a construction is of course not possible on the closed ball. However, since the configuration
spaces of the open and closed balls are homotopy equivalent (see §3), there is no purely homotopy theoretic
obstruction to finding a section in the case of a closed ball. This means that a different approach is needed.
Even more, many standard tools of algebraic topology fail in our context since the forgetful map gm,n is not
a fibration: it has fibers of distinct homotopy types, depending on how many points lie on the boundary of
the ball.
Outline. We treat the easy ordered version of the problem in §2 by briefly proving Theorem A. The
main argument of our proof of Theorem B proceeds by contradiction. Any section of ḡm,n induces an Σn-
equivariant section of gm,n : PConfn+1(B) → PConfn(B). Theorem B is then proved by pulling back a
cohomology class in two different ways and arriving at a contradiction. We conclude by proving Theorem C
in §4.
2
Acknowledgements. We thank Benson Farb and Dan Margalit for providing useful comments and for their
continued encouragement and support. We also thank the anonymous referee for a number of suggestions
that improved the paper. JL is supported by the NSF grant DGE-1650044.
2. Case of ordered configurations
In this section we prove Theorem A by generalizing the midpoint construction of Example 2.
Proof of Theorem A. The case n = 1 is covered by Example 1. Otherwise, fix n ≥ 2. For any pair (i, j)
with 1 ≤ i, j ≤ n and i 6= j, the line segment [pi, pj] is contained within B. Let vij := pj − pi be the vector
pointing from pi to pj. Further, let di := min {‖pk − pi‖ : k 6= i} be the minimal distance between pi and
any other point in the configuration. One can then introduce a new point “close” to pi, lying at a distance
di/2 from pi along the interval [pi, pj ]: explicitly, p0 := pi +
di
2‖vij‖
vij . The added point p0 is clearly distinct
from all other points, and it depends on the configuration continuously since di and vij do so. 
We remark that the first author gave a prior classification of sections between configuration spaces (see
the discussion preceding [Che17, Theorem 1.1]). Under this classification, the above construction of adding
a point is of the type ‘add close to pi’. Let us also call attention to the fact that the above construction
relies on singling out two of the points, which cannot be done continuously in the unordered version of the
problem when n > 2.
3. Case of unordered configurations
In this section we prove Theorem B, which says that in dimension m ≥ 2 there is no section of ḡm,n
except when n = 2. We begin with a few preliminary observations.
First, generators and relations for the integral cohomology ring of PConfn(R
m) were produced by Arnol’d
in dimension m = 2 and by Cohen for general m [Arn69, Coh88]. This cohomology ring is generated by
classes
Gab ∈ H
m−1(PConfn(R
m);Z)
for distinct 1 ≤ a < b ≤ n, where the generator Gab measures the winding of point a around point b. The
induced Σn-action is given by σ
∗(Gab) = Gσ(a)σ(b), where Gba = (−1)
mGab. This description applies equally
well to the closed ball as the following argument shows. Since Rm is homeomorphic to the open unit ball U,
their configuration spaces are also equivariantly homeomorphic. Then scaling by 0 < t < 1 gives a sequence
of inclusions
tU ⊂ tB ⊂ U ⊂ B
with compositions isotopic to the identity. It follows that U and B have equivariantly homotopy equivalent
configuration spaces, and in particular they have the same cohomology.
Our second observation is that any section s̄ of ḡm,n for the unordered configuration spaces lifts to a
Σn-equivariant section s of gm,n for the ordered spaces. This follows from the lifting criterion for connected
coverings.
Third, we leverage the fact that an equivariant section s of gm,n induces a solution to a related section
problem where the configurations have the added restriction that the point p1 is constrained to the boundary
sphere. This solution gives us a map, whose pullback on cohomology we compute in two ways, leading to a
contradiction. More precisely, let U ⊂ B denote the interior of the closed ball and consider the subspace of
PConfn(B) in which only the 1st point lies on the boundary sphere S
m−1:
B1n := {(p1, . . . , pn) ∈ PConfn(B) | p1 ∈ ∂ B, (p2, . . . , pn) ∈ PConfn(U)}
∼= Sm−1 × PConfn−1(U)
3
Define B1n+1 ⊆ PConfn+1(B) similarly. Lastly, let E
1
n+1 denote the preimage g
−1
m,n(B
1
n) and consider the
inclusion B1n+1 →֒ E
1
n+1. The difference between these two spaces is that in the larger space the additional
point p0 may lie on the boundary ∂ B, while in the smaller space this is not allowed. Despite this apparent
difference, we have the following.
Lemma 3.1. The inclusion B1n+1 →֒ E
1
n+1 is a homotopy equivalence.
Proof. Choosing an inward-pointing vector field that vanishes on the point p1 ∈ ∂ B, one can push any other
point into the interior of B. Explicitly, the vector field −(p− p1) on B gives rise to a smooth vector field on
PConfn+1(B). Its flow produces an isotopy Φ
1
t such that for all t > 0,
Φ1t (B
i
n+1) ⊂ Φ
1
t (E
i
n+1) ⊂ B
1
n+1 ⊂ E
1
n+1
thus establishing the homotopy equivalence. 
Consider the compatible projections onto the 1st coordinate
B1n+1
##●
●
●
●
●
●
●
●
gm,n

B1n // S
m−1.
Pulling back an orientation class [Sm−1] ∈ Hm−1(Sm−1), we get a class 0 6= X1 ∈ H
m−1(B1n) whose pullback
will also be abusively denoted by X1 ∈ H
m−1(B1n+1). These classes measure how many times the point p1
wraps around the boundary sphere.
Lemma 3.2. Under the inclusion ι : B1n+1 →֒ PConfn+1(B) we have
ι∗(G1a) = X1 for all 1 < a
and the class Gab for 1 < a, b pulls back to Gab ∈ H
m−1(PConfn(U)). The same is true for B
1
n ⊆ PConfn(B).
Via the homotopy equivalence B1n+1 ≃ E
1
n+1, we consider Lemma 3.2 as a statement about E
1
n+1 as well.
In particular we shall keep the notation X1 for the corresponding class in H
m−1(E1n+1).
Proof of Lemma 3.2. These facts are geometrically obvious: the class Gab is pulled back from S
m−1 under
the ‘Gauss map’, sending a configuration to the direction vector from pa to pb. When 1 < a, b, this Gauss
map factors through the projection B1n+1 → PConfn(U), as claimed.
Otherwise, if 1 < a then since p1 lies on the boundary and pa is internal, the Gauss map is homotopic
to a map in which pa is fixed at the origin. But when pa = 0 the Gauss map coincides with the projection
which records only p1. 
With these facts in hand, we can now complete our proof.
Proof of Theorem B. Let m ≥ 2. If n = 1, we have no section by Example 1. If n = 2, we have a section
by Example 2. Otherwise, let n ≥ 3. Assume that s̄ is a section of ḡm,n and let s be its Σn-equivariant
lift to a section of gm,n. The assumption that s is a section forces s(B
1
n) ⊆ E
1
n+1, thus it restricts to a
section s′ : B1n → E
1
n+1 of gm,n. From this one observes that (s
′)∗X1 = X1. Let us show that this leads to a
contradiction.
Since the classes Gab span H
m−1(PConfn(B);Z), there is an expansion
s∗(G01) =
∑
1<a≤n
λaG1a +
∑
1<a<b≤n
δabGab
4
for some integer coefficients. Equivariance implies that a permutation σ ∈ Σn fixing 1 will preserve this
expansion, and therefore λa = λb for all 1 < a < b ≤ n. Denote this constant value by λ. Similarly, it also
follows that all δab must be equal, say to the constant δ. We thus get
s∗(G01) = λ
∑
1<a≤n
G1a + δ
∑
1<a<b≤n
Gab. (1)
Next, we have a commutative diagram
E1n+1
ι // PConfn+1(B)
B1n
s′
OO
ι // PConfn(B)
s
OO
through which the pullback of the class G01 ∈ H
m−1(PConfn+1(B)) along the two different paths must agree.
Pulling it back through the top-left corner,
G01
ι∗
7−→ X1
(s′)∗
7−→ X1.
But pulling back through the bottom right corner, one first applies s∗ for which we have the expansion (1).
Since the restriction of Gab with 1 < a, b to B
1
n gives the class Gab again and each G1a restricts to X1, we
obtain the following:
ι∗s∗(G01) = λ(n− 1)X1 + δ
∑
1<a<b≤n
Gab.
The above equation has to be equal to the pulling back from the top-left corner, which is X1. But since the
Künneth formula for the product B1n
∼= Sm−1 × PConfn−1(U) implies that Xi is linearly independent from
the other classes Gab, such an equality is possible only if δ = 0 and λ(n − 1) = 1. But λ is an integer and
n > 2 which gives a contradiction. 
4. Uniqueness of the midpoint construction for n = 2
We now show that the midpoint section from Example 2 is unique up to a homotopy through sections.
Proof of Theorem C. Let us denote the midpoint section by M : PConf2(B) → PConf3(B) and suppose that
s is another section of gm,2, possibly Σ2-equivariant. We construct a homotopy between s and M through
sections, such that if s was equivariant then so will be the homotopy.
If for every configuration s(p1, p2) = (p0, p1, p2), the added point p0 lies either between p1 and p2 or off
of the line connecting them, then the straight-line homotopy Ht := (1 − t)s + tM demonstrates the claim.
With this, the uniqueness problem is reduced to finding a homotopy to a section possessing this property.
The idea goes as follows. Given a configuration (p1, p2) we let the points repel each other, moving
outwards along the line connecting them until they hit the boundary. By applying the section s to this
motion, one gets a path of configurations of 3 points, where at the end of the process we have p1 and p2 on
the boundary and p0 somewhere in B. Thus if p0 is on the line containing p1 and p2, it must lie between
them. On such a configuration one can perform the straight-line homotopy to the midpoint. However, to
remain within sections of gm,2, we must apply the aforementioned process while globally scaling B down so
that overall the points p1 and p2 do not move.
Now more explicitly, for any configuration c = (p1, p2), let L
c be the straight line passing through it.
Then Lc intersects ∂ B at two distinct points q1 and q2, labeled so that qi is closer to pi. Let x
c be the
5
unique point on the line Lc from which the ratios of ‖qi − xc‖ to ‖pi − xc‖ are equal for i = 1, 2, and denote
this common ratio by rc. Then the isotopy
hct : v 7→ ((1 − t) + r
ct)(v − xc) + xc
is a scaling of Rm out from xc at a linear rate, and hc0 = Id while h
c
1 : pi 7→ qi for i = 1, 2. Note that since
hct is scaling out from a point inside B, the image of B contains B at every time t.
Now, since hct and (h
c
t)
−1 are injective, they induce Σn-equivariant isotopies of the configuration spaces
(Hct )
±1 : PConfn(R
m) → PConfn(R
m) by applying them to a configuration diagonally. Lastly, it is clear that
Hct (p1, p2) is contained in B at all times 0 ≤ t ≤ 1, and that everything in sight depends on c continuously
(even algebraically).
A homotopy through sections is then given by
st(c) := (H
c
t )
−1 ◦ s ◦Hct (c)
This map acts by expanding the ball, using s to add a point at every time t and then contracting the ball
back – thus producing a path of configurations in which (p1, p2) never moves. The section s1 has the property
that allows us to connect it to M by a straight-line homotopy.
Lastly, if s was Σ2-equivariant, then st is a composition of equivariant maps and is thus itself equivariant.

References
[Arn69] V. I. Arnol’d. The cohomology ring of the group of dyed braids. Mat. Zametki, 5:227–231, 1969.
[Bro11] L. E. J. Brouwer. Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71(1):97–115,
Mar 1911.
[Che17] L. Chen. Section problems for configuration spaces of surfaces. Pre-print,
https://arxiv.org/abs/1708.07921, 2017.
[Coh88] Frederick R Cohen. Artin’s braid groups, classical homotopy theory, and sundry other curiosities.
Braids (Santa Cruz, CA, 1986), 78:167–206, 1988.
[CS17] L. Chen and N. Salter. Section problems for configurations of points on the Riemann sphere.
Pre-print, https://arxiv.org/abs/1807.10171, 2017.
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https://authors.library.caltech.edu/101990/2/1809.06946.pdf

Adding a point to configurations in closed balls

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