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# Symmetric products of Galois-Maximal varieties

Javier Orts<EMAIL_ADDRESS>Departamento de Matemática
Instituto Superior Técnico
Av. Rovisco Pais
1049-001 Lisbon
Portugal

(Date: March 2024)

###### Abstract.

The main result of this paper is the proof that all the symmetric products of
a finite Galois-Maximal space are Galois-Maximal spaces. This applies to the
special case of real algebraic varieties, solving the problem first stated by
Biswas and D’Mello in [6] about symmetric products of Maximal curves. We also
give characterisations of these spaces and state a new definition that
generalises to a larger class of spaces. Then, we extend the characterisation
in terms of the Borel cohomology given in [10] to the new family. Finally, we
introduce the notion of cohomological stability and cohomological splitting,
provide a systematic treatment and relate them with the properties of being a
Maximal or Galois-Maximal space. These cohomological properties play an
important role in the proof of our main theorem.

###### Key words and phrases:

Real variety, Maximal variety, Galois-Maximal variety, Borel cohomology,
Symmetric products

###### 2020 Mathematics Subject Classification:

14F45, 14P25, 55N91

###### Contents

  1. 1 Introduction
    1. 1.1 Notation and conventions
  2. 2 Maximal spaces
  3. 3 Characterisation of maximal spaces
  4. 4 Generalisation of maximal spaces
  5. 5 Cohomological properties
  6. 6 Properties of the symmetric products
  7. 7 Main result

##  1\. Introduction

If $G$ is a $p$-group acting on a well-behaved space $X$, then Smith theory
(_cf._ [8, Ch. III]) provides a relation between the cohomology of $X$ and of
$X^{G}$ when considering coefficients in the field of $p$ elements
$\mathbf{F}_{p}$. In particular, for $G=C_{2}$, one has:

(S–T)
$\sum_{q}\dim_{\mathbf{F}_{2}}H^{q}(X^{C_{2}};\mathbf{F}_{2})\leqq\sum_{q}\dim_{\mathbf{F}_{2}}H^{q}(X;\mathbf{F}_{2}).$

This inequality is known as the Smith-Thom inequality and defines _Maximal
spaces_ : spaces for which the equality is attained.

If $\sigma\in C_{2}$ denotes the non-trivial element, then its pullback
$\sigma^{*}$ defines an action of $C_{2}$ on $H^{*}(X;\mathbf{F}_{2})$. This
allows a refinement of the Smith–Thom inequality, sometimes referred to as the
Harnack–Krasnov inequality:

(H–K)
$\sum_{q}\dim_{\mathbf{F}_{2}}H^{q}(X^{C_{2}};\mathbf{F}_{2})\leqq\sum_{q}\dim_{\mathbf{F}_{2}}H^{1}(C_{2},H^{q}(X;\mathbf{F}_{2})).$

Spaces for which the equality in this last relation is attained are referred
to as _Galois-Maximal spaces_.

Maximal and Galois-Maximal spaces are more common in real algebraic geometry,
where the $C_{2}$-space is the complex locus of a real algebraic variety, with
the action defined by complex-conjugation.

A natural question to ask is what operations preserve these properties. For
example, it has been known for a long time that the Cartesian product of
Maximal (resp. Galois-Maximal) spaces is again a Maximal (resp. Galois-
Maximal) space. Biswas and D’Mello considered in [6] the symmetric products of
Maximal curves. Later on, Franz [11] addressed the general case of Maximal
spaces, while Baird [3] considered symmetric product of Galois-Maximal curves.
In all cases symmetric products preserved these properties.

In the work of this paper we deal with the general case of Galois-Maximal
spaces, showing the following:

######  Theorem.

Let $X$ be a Galois-Maximal space. Then, $SP_{n}X$ is a GM-space for every
$n\in\mathbf{N}$.

In fact, we establish a new definition for Maximal and Galois-Maximal spaces
that also applies to spaces for which the sums of (S–T) and (H–K) are not
defined. We then extend the characterisation of these spaces in terms of Borel
cohomology provided in [10].

Finally, we formulate a rigorous definition of two _cohomological properties_
based on some properties of the symmetric products. Suppose given a sequence

$X_{0}\overset{f_{0}}{\longrightarrow}X_{1}\overset{f_{1}}{\longrightarrow}X_{2}\longrightarrow\cdots\longrightarrow
X_{n}\overset{f_{n}}{\longrightarrow}X_{n+1}{\longrightarrow}\cdots$

of spaces and maps in some category $\mathscr{T}$. By a _cohomological
property_ we mean a relation that (some of) the induced maps
$f_{n}^{*}:H^{*}(X_{n+1})\to H^{*}(X_{n})$ on cohomology posses with respect
to some coefficient.The inspiration for this idea comes from the results of
Nakaoka [28] and Steenrod [30] about the cohomology of symmetric products. We
make the passage to $G$-spaces, focusing on the group $C_{2}$ and relating
these properties with those of being a Maximal or Galois–Maximal space. That
will help us prove the previous theorem.

###  1.1. Notation and conventions

We use the notation $\bigcup_{n}X_{n}$ for the colimit of a tower of spaces.
More precisely, given a sequence

$X_{0}\overset{f_{0}}{\longrightarrow}X_{1}\overset{f_{1}}{\longrightarrow}X_{2}\longrightarrow\cdots\longrightarrow
X_{n}\overset{f_{n}}{\longrightarrow}X_{n+1}{\longrightarrow}\cdots$

where each map $f_{n}$ is a closed embedding, then

$\bigcup_{n}X_{n}\overset{def}{=}\varinjlim_{n}X_{n},$

where the topology is the final topology with respect to the natural
inclusions $j_{n}:X_{n}\to\operatorname{colim}_{n}X$: A set $U$ is open in the
colimit if and only if each set $j_{n}^{-1}(U)$ is open in $X_{n}$. When each
space $X_{n}$ is a $G$-space, the union or colomit $X$ inherits a natural
$G$-space structure, to which the inclusions $j_{n}:X_{n}\hookrightarrow X$
are equivariant.

We write $X_{+}$ to denote the pointed space obtained by adding an isolated
point $+$ to the space $X$, $X\smash{Y}$ to denote the smash product of the
pointed spaces $(X,x_{0})$ and $(Y,y_{0})$,

$X\wedge Y=X\times Y\bigg{/}X\vee Y,$

and, in the case of $G$-spaces, $X\simeq_{G}Y$ to indicate that there exists a
homotopy equivalence between $X$ and $Y$ which is also equivariant.

Finally, we denote the Borel cohomology of the $G$-space $X$ by
$H_{G}^{*}(X)$. As it is a $\mathbf{Z}$-graded theory, we can consider shifted
modules, that we call _suspensions_ , and that we denote by $\Sigma^{q}M$. The
reason for this is the suspension axiom of reduced Borel cohomology:

$\tilde{H}_{G}(S^{p}\wedge X)\cong\Sigma^{p}\tilde{H}_{G}^{*}(X).$

In particular, given an $\mathbf{F}_{2}$-vector space $V$, we denoted by
$\Sigma^{q}V$ the $\mathbf{Z}$-graded $\mathbf{F}_{2}[z]$-module which is $V$
concentrated in degree $q$.

##  2\. Maximal spaces

We have mentioned in the introduction that Smith theory requires well-behaved
spaces. One such example is a finite _$G$ -CW complex_. $G$-CW complexes are
the equivalent objects to CW-complexes in equivariant topology. For the
purpose of this work, it is enough to address only the case in which the
equivariance group $G$ is finite. We refer to [24] and [22] for the general
definition and their properties.

######  Definition 2.1.

A _$G$ -CW complex_ is a CW-complex $X$ on which $G$ acts in such a way that
the following hold:

  1. (1)

The action of $G$ is cellular.

  2. (2)

Invariant cells are point-wise fixed.

A $G$-CW complex $X$ whose underline CW-structure has finitely many cells will
be called a _finite $G$-CW complex_. The dimension of one of its top cells
will be refer to as the dimension of $X$ and denoted $\dim X$.

######  Definition 2.2.

Let $X$ be a $C_{2}$-space with the equivariant homotopy type of a finite
$C_{2}$-CW complex. We say that:

  1. (1)

$X$ is a _Maximal space_ or an _M-space_ if

(M) $\sum_{q=0}^{\dim
X^{C_{2}}}\dim_{\mathbf{F}_{2}}H^{q}(X^{C_{2}};\mathbf{F}_{2})=\sum_{q=0}^{\dim
X}\dim_{\mathbf{F}_{2}}H^{q}(X;\mathbf{F}_{2}).$

  2. (2)

$X$ is a _Galois-Maximal space_ or a _GM-space_ if

(GM) $\sum_{q=0}^{\dim
X^{C_{2}}}\dim_{\mathbf{F}_{2}}H^{q}(X^{C_{2}};\mathbf{F}_{2})=\sum_{q=0}^{\dim
X}\dim_{\mathbf{F}_{2}}H^{1}(C_{2},H^{q}(X;\mathbf{F}_{2})).$

A space satisfying either of these conditions will be generically called
_maximal_ (without capital letters).

######  Remark 2.3.

Let $j$ denote the inclusion of the fibre $X\hookrightarrow X_{C_{2}}$. It is
a fact (_cf._ [34, III(1.18)]), that being an $M$-space is equivalent to the
forgetful map $j^{*}:H_{C_{2}}^{*}(X;\mathbf{F}_{2})\to
H^{*}(X;\mathbf{F}_{2})$ being surjective. Such spaces are sometimes referred
to as _totally nonhomologous to zero spaces_ [7] or _equivariantly formal
spaces_ [11].

The term $H^{1}(C_{2},A)$ in (GM) refers to the group cohomology of $C_{2}$
with coefficients in $A$. We are interested in the case when $A$ is an
$\mathbf{F}_{2}$-vector space equipped with an involution $\sigma$, for which

$H^{p}(C_{2},A)=A^{C_{2}}\Big{/}\left\\{a+\sigma a\mid a\in A\right\\}$

(_cf._ [9, pp. 58–59]). Hence
$\dim_{\mathbf{F}_{2}}H^{1}(C_{2},A)\leqq\dim_{\mathbf{F}_{2}}A$. In
particular

$\sum_{q=0}^{n}\dim_{\mathbf{F}_{2}}H^{1}(C_{2},H^{q}(X;\mathbf{F}_{2}))\leqq\sum_{q=0}^{n}\dim_{\mathbf{F}_{2}}H^{q}(X;\mathbf{F}_{2}),$

which means that every Maximal space is Galois-maximal. The necessary and
sufficient conditions under which the converse holds are given in the
following statement.

######  Proposition 2.4 (Krasnov, [19]).

A GM-space is an M-space if and only if the induced action of $C_{2}$ on the
cohomology $H^{*}(X;\mathbf{F}_{2})$ is trivial.

Maximal spaces are more common in real algebraic geometry, where one speaks
about _maximal varieties_. If $V$ is a real variety, _i.e._ , an algebraic
variety over the field of real numbers, then its set of complex points
$V(\mathbf{C})$ endowed with the analytic topology and with the $C_{2}$-action
induced from complex conjugation is a finite $C_{2}$-CW complex (_cf._ [21, p.
131]). For non-singular, complete varieties the fixed points of this action
are exactly the real points of the variety, _i.e._ ,
$V(\mathbf{C})^{C_{2}}\equiv V(\mathbf{R})$.

######  Definition 2.5.

Let $V$ be a non-singular complete real variety. We call $V$ a _Maximal
variety_ or _M-variety_ (resp.  _Galois-Maximal variety_ or _GM-variety_) if
the $C_{2}$-space $V(\mathbf{C})$ with the analytic topology as explained
before is an M-space (resp. GM-space). Maximal varieties of dimension one are
called maximal curves, maximal varieties of dimension two are called maximal
surfaces, etc.

######  Example 2.6 (Maximal spaces).

  1. (1)

Let $\mathbf{R}^{p,q}$ denote the real $p$-dimensional representation of
$C_{2}$ such that

$\sigma\cdot(x_{1},\dots,x_{p})=(x_{1},\dots,x_{p-q},-x_{p-q+1},\dots,-x_{p}),$

and let $S^{p,q}=\mathbf{R}^{p,q}\cup\\{\infty\\}$ be its one-point
compactification. We called spheres of this form _representation spheres_. All
representation spheres are Maximal.

  2. (2)

If $X$ is a topological space, then its _multiplicative induction_ , denoted
by $N^{C_{2}}X$, is the $C_{2}$-space $X\times X$ with the action of the non-
trivial element given by transposition:

$\sigma(x,x^{\prime})=(x^{\prime},x).$

Whenever $X$ has the homotopy type of a finite CW-complex, $N^{C_{2}}X$ is a
GM-space.

  3. (3)

Similarly, there exists an _additive induction_ ,
$M^{C_{2}}X\overset{def}{=}X\wedge(C_{2})_{+}$, where $C_{2}$ is the free
orbit and $\wedge$ is the smash product of Subsection 1.1.

  4. (4)

The Cartesian product and the wedge sum of two maximal spaces, equipped with
the diagonal action, are also maximal.

  5. (5)

Any Riemann surface with at least one real point is a GM-curve (_cf._ Krasnov
[19]). Also, all Abelian varieties with real points are GM-varieties (_cf._
_loc. cit._).

  6. (6)

A smooth real threefold $Y\subset\mathbf{P}_{\mathbf{R}}^{4}$ containing a
real line $L$, such as a real cubic (_cf._ [10]).

  7. (7)

The blow-up of a smooth M-variety along a smooth real M-subvariety is an
M-variety (_cf._ Fu [12]).

  8. (8)

If $V$ is an M-variety (resp. GM-variety) and $E\to V$ is a Real vector
bundle, then the projective bundle $\mathbf{P}(E)\to V$ is an M-variety (resp.
GM-variety) (_cf._ Fu [12] for the Maximal case).

  9. (9)

A curve is Maximal if and only if its Picard variety is Maximal (_cf._ Biswas
and D’Mello [5]).

  10. (10)

Non-singular complete toric varieties are Maximal (_cf._ Bihan _et al._ [4]).

The main problem that motivated this paper is the following result of Biswas
and D’Mello proven in [6].

######  Theorem 2.7 (Symmetric products M-curves).

Let $X_{g}$ be an M-curve of genus $g$. Then $SP_{n}X_{g}$ is an M-variety for
$n=2,3$ and $n\geqq 2g-1$.

The authors could not provide an answer for the cases $4<n<2g-1$, and so the
problem remained open until Franz proved a much more general result.

######  Theorem 2.8 (Franz, [11]).

Let $X$ be a space and let $\Gamma\subset\mathfrak{S}_{n}$ be a subgroup. The
$\Gamma$-product of $X$ is the quotient $X^{n}/\Gamma$. For each $n$ and each
$\Gamma\subset\mathfrak{S}_{n}$, the $\Gamma$-product of an $M$-space is
another M-space. In particular, setting $\Gamma=\mathfrak{S}_{n}$, we see that
all the symmetric products of a Maximal space are Maximal too.

Finally, Baird showed in [3] that a similar theorem for the case of symmetric
products of GM-curves holds

######  Theorem 2.9 (Symmetric products GM-curves).

All the symmetric products of a GM-curve are GM-varieties.

##  3\. Characterisation of maximal spaces

In this section we provide two characterisation theorems for maximal spaces.
We start with one based on _zero-cycles_ : Given a based space $(X,x_{0})$ and
an Abelian group $\Gamma$, a _zero-cycle in $X$ with coefficients in $\Gamma$_
is an element of topological Abelian group with coefficients in $\Gamma$ that
$X$ generates. This spaces is defined by McCord in [26] and denoted by
$B(\Gamma,X)$. Other notations such as $\Gamma\tilde{\otimes}X$ or
$\tilde{\Gamma}[X]$ are also common. We shall write $\Gamma[X]$ to denote the
space of zero-cycles in $X$ with coefficients in $\Gamma$.

In the situation where the space $X$ is a $G$-space, this action is carried
out on to $\Gamma[X]$ by linearity, turning it into another $G$-space. We will
be interested in the case $\Gamma=\mathbf{F}_{2}$ and $G=C_{2}$. The following
characterisation of maximal spaces in terms of zero-cycles will be used in the
proof of the main theorem Theorem 7.1.

######  Theorem 3.1.

Let $X$ be a finite $C_{2}$-CW complex. Then:

  1. (1)

It is an M-space if and only if

$\mathbf{F}_{2}[X]\simeq_{C_{2}}\bigoplus_{i\in
I}\mathbf{F}_{2}[S^{p_{i},q_{i}}],$

where $I$ is a finite set and $p_{i}\geqq q_{i}\geqq 0$ for all $i\in I$.

  2. (2)

It is a GM-space if and only if

$\mathbf{F}_{2}[X]\simeq_{C_{2}}\left(\bigoplus_{i\in
I}\mathbf{F}_{2}[S^{p_{i},q_{i}}]\right)\oplus\left(\bigoplus_{j\in
J}\mathbf{F}_{2}[S^{r_{j}}\wedge(C_{2})_{+}]\right),$

where $I$ and $J$ are finite sets and $p_{i}\geqq q_{i}\geqq 0$ for all $i\in
I$ and $r_{j}\geqq 0$ for all $j\in J$.

In both cases the equivariant homotopy is also a group homomorphism.

  * Proof.—

The results of Theorem 6.13 and Remark 6.14 in [23] show that every
$C_{2}$-module over the equivariant Eilenberg–Mac Lane spectrum
$H\underline{\mathbf{F}}_{2}$ must decompose as wedges of copies of
$H\underline{\mathbf{F}}_{2}$ and of antipodal spheres $S_{a}^{n}$. Then, the
proof of this result can be adapted to the case of zero-cycles, as these
spaces represent the spectrum. Finally, the (M) and (GM) properties impose
further restrictions on the splitting of $\mathbf{F}_{2}[X]$. We suggest that
the reader also consult [10]. $\blacksquare$

Another characterisation of maximal spaces in terms of _Borel cohomology_ was
given in [10]. If $G$ is a topological group and $X$ is a $G$-space, then the
homotopy quotient or _Borel construction_ of $X$ is the space $X_{G}=(X\times
EG)/G$, where $EG\to BG$ is a representative of the universal bundle of $G$.
This quotient sits in a fibration

(3.1) $X\lhook\joinrel\longrightarrow X_{G}\relbar\joinrel\twoheadrightarrow
BG,$

known as the _Borel fibration_. The Borel cohomology of $X$ with coefficients
in a ring $R$ is the singular cohomology of the total space $X_{G}$:

$H_{G}^{*}(X;R)\overset{def}{=}H^{*}(X_{G};R).$

In the case of $G=C_{2}$, we can identify $H_{C_{2}}^{*}(pt;\mathbf{F}_{2})$
with $\mathbf{F}_{2}[z]$, where $z\in H_{C_{2}}^{1}(pt;\mathbf{F}_{2})$ is the
generator. Then, the cohomology of every $C_{2}$-space becomes a module over
$\mathbf{F}_{2}[z]$. We identify the quotient
$H_{C_{2}}^{*}(pt;\mathbf{F}_{2})/(z)$ with the field $\mathbf{F}_{2}$,
regarded as an $\mathbf{F}_{2}[z]$-module.

######  Theorem 3.2.

Let $X$ be a finite $C_{2}$-CW complex. Then:

  1. (1)

It is an M-space if and only if $H_{C_{2}}^{*}(X;\mathbf{F}_{2})$ is free and
finitely generated over $H_{C_{2}}^{*}(pt;\mathbf{F}_{2})$.

  2. (2)

It is a GM-space if and only if its Borel cohomology is a finite direct sum of
suspensions only of the following two type of modules: the free module
$H_{C_{2}}^{*}(pt;\mathbf{F}_{2})$ and the torsion module
$H_{C_{2}}^{*}(pt;\mathbf{F}_{2})/(z)$, $z\in
H_{C_{2}}^{1}(pt;\mathbf{F}_{2})$.

The result for Maximal spaces was already known (_cf._ [34, III(4.16)]). In
fact, as Maximal spaces can be characterised using the forgetful map $j^{*}$
(see Remark 2.3), the Leray–Hirsch theorem (_cf._ [16, Theorem 4D.1] or [34,
III(1.14)]) ensures that the Borel cohomology of Maximal spaces is always
free, even when the underlying $C_{2}$-CW complexes are infinite (but locally
finite, see the next section).

##  4\. Generalisation of maximal spaces

Maximal spaces, as defined until now, must have finite cohomology, so that (M)
and (GM) are defined. This can be surpassed by using a characterisation of
maximal spaces in terms of a spectral sequence, first due to Krasnov [19].

We will consider the group $G=C_{2}$ and define maximal spaces using the
Leray–Serre spectral sequence of the corresponding Borel fibration (3.1) for a
space $X$. It thus follows that the only technical requirement on the spaces
that one needs to impose is that their homotopy quotient be a Serre fibration
over $BC_{2}$. Nevertheless, for computational purposes, we shall restrict
ourselves to spaces with the equivariant homotopy type of a _locally finite_
$G$-CW complex. This is a $G$-CW-complex $X$ such that the $n$-skeleton
$X^{(n)}$ of the underlying CW-complex is finite for every $n\geqq 0$. From
now on, and until the end of this section, all our spaces will be assumed to
have the equivariant homotopy type of a locally finite $G$-CW complex. Notice
that a locally finite $G$-CW-complex is countable, and hence the following
important result due to Milnor [27].

######  Property 4.1.

If $X$ and $Y$ are two locally finite $G$-CW-complexes, then the product
topology and the weak topology on $X\times Y$ agree, and therefore their
product has a natural $G$-CW-complex structure.

######  Remark 4.2.

This definition of locally finite CW-complexes differs from the one provided
in other texts. For example, in [13] and [33], the definition of a locally
finite CW-complex is equivalent to the property that the closed covering
$\\{\bar{e}_{\alpha}\\}_{\alpha}$ consisting of the _closure_ of the all the
cells that define the CW-complex is locally finite. (Our convention is that of
Hatcher [16]: a cell is the homeomorphic image of the _interior_ of the disc
under the attaching map; this differs from [13].) An example of a CW-complex
which is locally finite in this sense but not in ours is the real line
$\mathbf{R}$. Conversely, the infinite wedge $W=\bigvee_{n\geqq 1}S^{n}$
satisfies our criteria of local finiteness but the base-point $\ast\in
W^{(0)}$ meets all the closed cells $\bar{e}_{n}=S^{n}$.

We set the following notation:

(L–S) For a $C_{2}$-space $X$, let $\\{E_{*}(X),d_{*}\\}$ denote the
Leray–Serre spectral of the Borel fibration $X_{C_{2}}\to BC_{2}$ with
coefficients in $\mathbf{F}_{2}$.

######  Definition 4.3.

A _GM-space_ is a $C_{2}$-space with the equivariant homotopy type of a
locally finite $C_{2}$-CW complex satisfying the following two conditions:

  1. (1)

There is at least one fixed point.

  2. (2)

The Leray-Serre spectral sequence $\\{E_{*}(X),d_{*}\\}$ degenerates at the
second page.

An _M-space_ is a GM-space in this sense such that $C_{2}$ acts trivially on
$H^{*}(X;\mathbf{F}_{2})$. Finally, maximal spaces whose underlying
topological space have the homotopy type of a finite $C_{2}$-CW complex will
be referred to as _finite maximal spaces_.

As mentioned at the beginning of the section, the use of a similar111As the
formalism of [19] is in terms of algebraic varieties, the equivariant
cohomological theory considered there is with coefficients in a $G$-sheaf.
Therefore, the appropriate spectral sequence is the second Grothendieck
spectral sequence (_cf._ [15, (5.2.5)]). Nevertheless, for a finite group $G$,
that is equivalent to the Leray–Serre spectral sequence that arises from the
Borel construction (_cf._ [31]). spectral sequence to characterise Galois-
Maximal varieties had already been considered by Krasnov in [19]. We have only
used this idea to provide a more general definition. Of course, whenever (M)
and (GM) are defined, both definitions are equivalent (_cf._ [34, III(1.18)],
[19, Proposition 2.3]) and Proposition 2.4). Nevertheless, we warn the reader
that some of the constructions mentioned in Example 2.6 may not preserve the
property of being a maximal space when the underlying $C_{2}$-spaces do not
have the homotopy type of a finite $C_{2}$-CW. In particular, it remains
unknown whether the product of two GM-spaces in this sense is a GM-space. This
is because the Leray–Serre spectral sequence $E_{*}(X\times Y)$ for the
product is an Eilenberg–Moore spectral sequence (_cf._ [25, Chapters 7 and 8])
over a non-simply connected space, and no222Several attempts to generalise the
conditions under which the Eilenberg–More spectral sequence converges have
been made. We highlight Larry Smith [29], Hodgkin [17] and Greenless [14]. In
fact, with the work of Larry Smith, and using the generalisation of Theorem
3.2, it is possible to prove that the Cartesian product of an M-spaces and a
GM-space is another GM-space, and that the product of two M-spaces is an
M-space.  results for the convergence in this case are known, as far as I
know.

To conclude this section, we would like to extend the characterisation of
Theorem 3.2 so that it includes infinite maximal spaces as in Definition 4.3.
We start by noting that the Leray–Serre spectral sequence associated to the
Borel fibration (3.1) computes the Borel cohomology of the space. In the case
of maximal spaces, this spectral sequence degenerates at the second page and,
as we are considering $\mathbf{F}_{2}$ coefficients, there is no extension
problem. Thus

(4.1)
$H_{C_{2}}^{n}(X;\mathbf{F}_{2})\cong\bigoplus_{p+q=n}E_{2}^{p,q}=\bigoplus_{p+q=n}H^{p}(C_{2},H^{q}(X;\mathbf{F}_{2}))$

as Abelian groups. This is in fact an isomorphism of
$\mathbf{F}_{2}[z]$-modules, as the Leray–Serre spectral sequence has a module
structure over $H_{C_{2}}^{*}(pt;\mathbf{F}_{2})=\mathbf{F}_{2}[z]$ inherited
by the $E_{2}$-term. Actually, all the $E_{r}$-terms contain a copy of
$\mathbf{F}_{2}[z]$ that comes from the fixed points, that appears first in
$E_{2}^{*,0}$ and that survive through all the stages (infinite cocycles).
Moreover, the differentials are module homomorphisms. These facts, all
together, lead to the following property.

######  Property 4.4.

Let $a\in E_{r}^{p,q}$ be non-trivial. Then, there exists $a_{0}\in
E_{r}^{0,q}$ such that $a=z^{p}a_{0}$, and it follows that $d_{r}a=0$ if and
only if $d_{r}a_{0}=0$.

When proving the more general version of Theorem 3.2, the crucial observation
is that the page at which the spectral sequence degenerates determines the
types of modules that can appear in the decomposition (4.1). They can be
worked out by looking at the _torsion order_ of the Borel cohomology. By this,
we mean the following: Let $M$ be a graded module over
$H_{C_{2}}^{*}(pt;\mathbf{F}_{2})$ and let $m\in M$ be a homogeneous torsion
element (non-trivial); the _torsion order_ of $m$ is the integer $r$ such that
$z^{r}m=0$ but $z^{r-1}m\neq 0$. We will refer to $m$ as a torsion element of
_order $r$_. The module $\mathbf{F}_{2}[z]/(z^{r})$ contains elements of
torsion orders $1,2,\dots,r$.

######  Proposition 4.5.

Let $X$ be a locally finite $C_{2}$-CW complex. If the Leray–Serre spectral
sequence for the Borel fibration of $X$ does not degenerate at the second
page, then the Borel cohomology of $X$ must have torsion elements of order
strictly bigger than one.

  * Proof.—

Let $r_{0}\geqq 2$ be the least integer such that $d_{r_{0}}\not\equiv 0$. As
Property 4.4 shows, it is enough to consider the restriction of $d_{r_{0}}$ to
the zeroth column $E_{r_{0}}^{0,*}$. Let

$q_{0}=\min\left\\{q\mid d_{r_{0}}:E_{r_{0}}^{0,q}\to
E_{r_{0}}^{r_{0},q-r_{0}+1}\mbox{ is non-trivial}\right\\}.$

Due to dimensional reasons, $q_{0}\geqq r_{0}-1$ and, in fact,
$q_{0}>r_{0}-1$, for $q_{0}=r_{0}-1$ contradicts the fact that
$E_{2}^{*,0}=E_{\infty}^{*,0}$. Regardless of its value, there will be an
element $a_{1}\in E_{r_{0}}^{0,q_{0}-r+1}$ such that
$z^{r_{0}}a_{1}=d_{r_{0}}a_{0}\equiv 0$ in $E_{r_{0}+1}$. If $a_{1}$ is an
infinite cocycle, then we are done. Otherwise there exists $r_{1}>r_{0}$ such
that $d_{r_{1}}a_{1}\neq 0$. Let $a_{2}\in E_{r_{1}}^{0,q_{0}-r_{0}+1}$ be
such that $d_{r_{1}}a_{1}=z^{r_{1}}a_{2}$. Once again, either it is an
infinite cocycle and the proof is completed, or it is not and there exists
$r_{2}>r_{1}$ such that $d_{r_{2}}a_{2}\neq 0$.

As $q_{0}$ is finite, this must stop eventually: after $n$ steps we will get
an element

$a_{n}\in E_{r_{n}}^{0,q_{0}+n-r_{0}-\dots-r_{n-1}}$

with $d_{r_{n}}a_{n}=0$ because of dimensional reasons. That element will have
torsion order $r_{n}>1$.

The proof becomes clearer with the figure below. $\blacksquare$

$\dots$$\vdots$${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}p}$${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}q}$$E_{*}^{p,q}$$a_{0}$$a_{1}$$d_{r_{0}}a_{0}=z^{r_{0}}a_{1}$$r_{0}$$a_{2}$$d_{r_{1}}a_{1}=z^{r_{1}}a_{2}$$r_{1}$$a_{n-2}$$r_{n-2}$$a_{n-1}$$d_{r_{n-2}}a_{n-2}=z^{r_{n-2}}a_{n-1}$$r_{n-1}$$a_{n}$$d_{r_{n-1}}a_{n-1}=z^{r_{n-1}}a_{n}$$d_{r_{n}}a_{n}$
Figure 1. Illustration of the torsion phenomenon due to the existence of
$r\geqq 2$ such that $d_{r}$ is non-trivial.

##  5\. Cohomological properties

In this section we give a systematic study of the two cohomological properties
satisfied by the symmetric products and which play an important role in the
proof of Theorem 7.1. One of these properties is known as _cohomological
stability_ ; it is well known in group theory and some results about it for
topological spaces can be found in the paper of Jiménez and Wilson [18]. The
results of Steenrod [30] about the cohomology of the symmetric products
constitute the archetypal example. The other cohomological feature, first
suggested by the work of Nakaoka in [28] also for the cohomology symmetric
products, has been tentatively called _cohomological splitting_.

We need some preliminary results on the inverse limits of groups. The reader
is refer to Section 3.5 in Weibel [35], Chapter 7 in Switzer [32] and
additional topic 3F in Hatcher [16] for further details.

A _tower of groups_ [35, p. 80] is a family of groups and homomorphisms of the
form

$\cdots\longrightarrow A_{n+1}\longrightarrow
A_{n}\longrightarrow\cdots\longrightarrow A_{2}\longrightarrow
A_{1}\longrightarrow A_{0}.$

We will denote it by $\\{A_{n}\\}$.

######  Definition 5.1.

A tower $\\{A_{n}\\}$ of Abelian groups satisfies the _Mittag-Leffler
condition_ (_cf._ [35, Definition 3.5.6]) if for each $m$, there exists some
$n_{0}>m$ such that the image of $A_{n}\to A_{m}$ equals the image of
$A_{n_{0}}\to A_{m}$ for all $n>n_{0}$. (The images of the different $A_{n}$
in $A_{m}$ satisfy the descending chain condition.)

The Mittag-Leffler condition is satisfied if, for example, all the maps
$A_{n+1}\to A_{n}$ in the tower $\\{A_{n}\\}$ are onto.

######  Proposition 5.2 (See [35, Proposition 3.5.7]).

If a tower $\\{A_{n}\\}$ of Abelian groups satisfies the Mittag-Leffler
condition, then

$\varprojlim^{1}A_{n}=0.$

######  Definition 5.3.

Let

(5.1)
$X_{0}\overset{f_{0}}{\longrightarrow}X_{1}\overset{f_{1}}{\longrightarrow}X_{2}\longrightarrow\cdots\longrightarrow
X_{n}\overset{f_{n}}{\longrightarrow}X_{n+1}{\longrightarrow}\cdots$

be a sequence of spaces and maps in some appropriate category $\mathscr{T}$.
We say that the sequence (5.1) exhibits:

  1. (1)

_Cohomological stability_ with respect to the ring $R$ if, for each $q>0$,
there exists $n_{0}\in\mathbf{N}$ (depending on $q$) such that the maps

$f_{n}^{*}:H^{q}(X_{n+1};R)\to H^{q}(X_{n};R)$

are all isomorphisms for $n>n_{0}$.

  2. (2)

The _cohomological splitting property_ with respect to the ring $R$ if each
morphism

$f_{n}^{*}:H^{*}(X_{n+1};R)\to H^{*}(X_{n};R)$

admits a section.

In the case when $\mathscr{T}$ is a $G$-category, the sections of the
cohomological splitting property must be equivariant with respect to the
induced action of $G$ on cohomology.

######  Property 5.4.

Products, wedge sums and additive and multiplicative inductions of sequences
(performed term-wise) preserve cohomological stability and the cohomological
splitting property (some of these results might require to consider
coefficients in a field).

From now on we shall restrict ourselves to sequences whose spaces are finite
pointed $C_{2}$-CW complexes and whose morphisms are equivariant pointed
cellular maps, although some of the results might hold in a more general
setting. We further assume that each map $f_{n}:X_{n}\to X_{n+1}$ is a closed
embedding and that the quotient

$X_{n}/X_{n-1}=\bigvee_{i}S^{\alpha_{i}}$

satisfies $\alpha_{i}>\dim X_{n-1}$ for all $i$ (this is to ensure that the
union is a locally finite $C_{2}$-CW complex). Finally, we will consider
cohomological properties with respect to the field $\mathbf{F}_{2}$ and write
$H^{*}(X)$ instead of $H^{*}(X;\mathbf{F}_{2})$.

######  Lemma 5.5.

Let

$X_{0}\overset{f_{0}}{\longrightarrow}X_{1}\overset{f_{1}}{\longrightarrow}X_{2}\longrightarrow\cdots\longrightarrow
X_{n}\overset{f_{n}}{\longrightarrow}X_{n+1}{\longrightarrow}\cdots$

be a sequence as above and let

$X=\bigcup_{n}X_{n}.$

Denote by $j_{n}:X_{n}\to X$ the natural inclusion.

  1. (1)

If the sequence exhibits cohomological stability, then:

    1. (a)

For each $q>0$, there exists $n_{0}$ such that $j_{n}^{*}:H^{q}(X)\to
H^{q}(X_{n})$ is an isomorphism for all $n>n_{0}$. In particular the
inclusions $j_{n}$ induced an isomorphism

$H^{*}(X)\cong\varprojlim_{n}H^{*}(X_{n}).$

    2. (b)

For each $q>0$, there exists a space $X_{n}$ such that $j_{n}^{*}:H^{r}(X)\to
H^{r}(X_{n})$ is an isomorphism for all $r\leqq q$.

  2. (2)

If, instead, the sequence possesses the cohomological splitting property,
then:

    1. (a)

There is an isomorphism

$H^{*}(X)\cong\varprojlim_{n}H^{*}(X_{n}).$

induced by the inclusions $j_{n}:X_{n}\to X$.

    2. (b)

For each $n$, the morphism $j_{n}^{*}:H^{*}(X)\to H^{*}(X_{n})$ admits a
section, which is equivariant.

  * Proof.—

We first explain how the isomorphism
$H^{*}(X)\cong\varprojlim_{n}H^{*}(X_{n})$ is obtained. The Milnor exact
sequence (_cf._ [16, Theorem 3F.8] or [32, 7.66]) yields

(5.2) $0\longrightarrow\varprojlim_{n}^{1}H^{q-1}(X_{n})\longrightarrow
H^{q}(X)\longrightarrow\varprojlim_{n}H^{q}(X_{n})\longrightarrow 0.$

Either of the cohomological properties implies that the Mittag-Leffler
condition is satisfied and therefore

$\varprojlim_{n}^{1}H^{q-1}(X_{n})\equiv 0,$

as desired. It is a fact (_cf._ [32, 7.66 Proposition]) that this isomorphism
is induced by the inclusions $j_{n}:X_{n}\to X$. As these maps is equivariant,
the above is an isomorphism of $C_{2}$-groups.

Furthermore, in the case of cohomological stability, the inverse limit
stabilises for each $q$, meaning that

$\varprojlim_{n}H^{q}(X_{n})\cong H^{q}(X_{n_{0}})$

for some $n_{0}\gg 0$. Now statement (1b) follows from the first by
considering the space $X_{n}$, where

$n=\max_{0\leqq r\leqq q}\left\\{m\mid
j_{n}^{*}:H^{r}(X)\overset{\cong}{\to}H^{r}(X_{m})\right\\}.$

Finally, the section $s_{n}:H^{*}(X_{n})\to H^{*}(X)$ in the case of the
cohomological splitting property is defined as follows: For each $n\geqq 0$,
let $t_{n}$ denote the section of $f_{n}^{*}$; then the image of an element
$a\in H^{*}(X_{n})$ under $s_{n}$ is the element

$(a_{m})_{m}=\begin{cases}f_{n,m}^{*}(a),&\mbox{if }m\leqq n,\\\ a,&\mbox{if
}m=n,\\\ t_{n,m}(a),&\mbox{if }m>n.\end{cases}$

Here $f_{n,m}^{*}$ denotes the composite morphism $f_{n}^{*}f_{n-1}^{*}\cdots
f_{m}^{*}$. Similarly, $t_{n,m}=t_{m}t_{m+1}\cdots t_{n}$. It is a
straightforward exercise to verify that the image of each $s_{n}$ is indeed
contained in the inverse limit of the $H^{*}(X_{n})$. $\blacksquare$

The main result of this section is the following.

######  Theorem 5.6.

Let

$X_{0}\overset{f_{0}}{\longrightarrow}X_{1}\overset{f_{1}}{\longrightarrow}X_{2}\longrightarrow\cdots\longrightarrow
X_{n}\overset{f_{n}}{\longrightarrow}X_{n+1}{\longrightarrow}\cdots$

be as above and let

$X=\bigcup_{n}X_{n}.$

  1. (1)

If the sequence has cohomological stability and each $X_{n}$ is maximal, then
$X$ is also a maximal.

  2. (2)

Conversely, assume that $X_{n}^{C_{2}}$ is non-empty for every $n$. Then, if
the sequence possesses the cohomological splitting property and $X$ is a
maximal, every $X_{n}$ is also a maximal.

  * Proof.—

We will consider first the (GM) property and show that the corresponding
Leray–Serray spectral sequences degenerate charactersiign this property
degenerate at the second page. Recall that for a $C_{2}$-space $X$, we denote
this spectral sequence by $E_{*}(X)$ (see notation (L–S)). For the results
about Maximal spaces we will make use of Lemma 5.5 accordingly.

We start with (1). Given $p$ and $q$, we will show that
$E_{2}^{p,q}=E_{\infty}^{p,q}(X)$, for which we will prove that all the
relevant differentials $d_{r}^{p,q}:E_{r}^{p,q}(X)\to E_{r}^{p+r,q-r+1}$ and
$d_{r}^{p-r,q+r-1}:E_{r}^{p-r,q+r-1}(X)\to E_{r}^{p,q}$ vanish for $r\geqq 2$.
In order to do so, we will use the morphism of spectral sequences induced by
the natural inclusion $j_{n}:X_{n}\hookrightarrow X$, for some appropriate
$n$. We denote this morphism $\varphi_{n}:E_{*}(X)\to E_{*}(X_{n})$. At the
second page, $\varphi_{n}$ is given by

$\varphi_{n}=H^{p}(C_{2},-)j_{n}^{*}:E_{2}^{p,q}(X)=H^{p}(C_{2},H^{q}(X;\mathbf{F}_{2}))\longrightarrow
E_{2}^{p,q}(X_{n})=H^{p}(C_{2},H^{q}(X;\mathbf{F}_{2})),$

and due to cohomological stability we can choose $n$ such that
$j_{n}^{*}:H^{q}(X;\mathbf{F}_{2})\to H^{q}(X_{n};\mathbf{F}_{2})$ becomes an
isomorphism (see the first part of Lemma 5.5). This provides linear maps
$\psi_{n}^{p,q}:E_{2}^{p,q}(X)\to E_{2}^{p,q}(X_{n})$ which are inverses for
the $\varphi_{n}^{p,q}$. It is extremely important to notice that these
inverses occur at the level of vector spaces; the map $\\{\psi_{n}^{p,q}\\}$
might not commute with the differentials, and therefore does not define a
morphism of spectral sequences. In particular, this does not make
$\varphi_{n}$ an isomorphism of spectral sequences.

Our choice for $n$ is such that $j_{n}^{*}:H^{r}(X)\to H^{r}(X_{n})$ is an
isomorphisms for all $r\leqq p^{p}q^{q}$. Let $d_{2}\equiv d_{2}^{p,q}$ and
let $d_{2}^{n}$ denote the differential $E_{2}^{p,q}(X_{n})\to
E_{2}^{p+2,q-1}(X_{n})$. Given $a\in E_{2}^{p,q}(X)$ we have:

$d_{2}a=\psi^{p+2,q-1}(d_{2}^{n}\varphi_{n}^{p,q}(a))=0,$

as the space $X_{n}$ is maximal.

Not only that, but due to the choice of $n$, the same is true for all $s>0$
and for $t\leqq p^{p}q^{q}$. In particular,
$d_{2}^{p-2,q+1}:E_{2}^{p-2,q+1}(X)\to E_{2}^{p,q}(X)$ also vanishes, so that
$E_{3}^{p,q}(X)\equiv E_{2}^{p,q}(X)$. Therefore, $\psi_{n}$ is still defined
and so each $\varphi_{n}^{p,q}$ is a linear isomorphism. Hence, we can repeat
the same strategy: For $a\in E_{3}^{p,q}(X)$, we have

$d_{3}a=\psi^{p+3,q-2}(d_{3}^{n}\varphi_{n}^{p,q}(a))=0,$

and the choice of $n$ also ensures that $d_{3}^{p-3,q+2}:E_{3}^{p-3,q+2}(X)\to
E_{r}^{p,q}(X)$ is trivial. Actually, our choice guarantees that those
differentials $d_{r}^{s,t}$ which can hit $E_{r}^{p,q}$ will be trivial as
long as $s\geqq 0$, and for those values of $t$ over which $n$ has no
influence will have $s<0$, so that $E_{r}^{p,q}(X)$ will never be hit by any
non-zero differential. Similarly, all the differentials $d_{r}^{p,q}$ must
also be trivial, proving that $E_{2}^{p,q}(X)=E_{\infty}^{p,q}(X)$.

Finally, in the case all the spaces were Maximal spaces, So would be $X$.
Indeed, Lemma 5.5(1b) implies that, for each $q$, we can choose a space
$X_{n}$ such that $H^{q}(X)\cong H^{q}(X_{n})$ as $C_{2}$ groups; as $X_{n}$
is maximal the action of $C_{2}$ on cohomology must be trivial, and varying
$q$ we have proven that $C_{2}$ acts trivially $H^{*}(X)$, _i.e._ , $X$ is a
Maximal space.

The proof for (2) follows a similar strategy. The cohomological splitting
property guarantees the existence of a section $s_{n}$ to each
$j_{n}^{*}:H^{*}(X)\to H^{*}(X_{n})$ (second part of Lemma 5.5). In
particular, each morphism $\varphi_{n}$ is onto at the $E_{2}$-term. By this,
we mean that each linear map $\varphi_{n}^{p,q}:E_{2}^{p,q}(X)\to
E_{2}^{p,q}(X_{n})$ is surjective, with a section given by
$\sigma_{n}^{p,q}=H^{p}(C_{2},-)s_{n}$.

This situation is enough to prove that any differential $d_{2}^{n}$ of
$E_{2}(X_{n})$ vanishes. Indeed, given $a_{n}\in E_{2}^{p,q}(X_{n})$, let
$a\in E_{2}^{p,q}(X)$ such that $a_{n}=\varphi_{n}(a)$. Then:

(5.3) $d_{2}^{n}a_{n}=d_{2}^{n}(\varphi_{n}a)=\varphi_{n}(d_{n}a)=0,$

as $X$ is maximal by hypothesis.

This proves that $E_{3}(X_{n})\equiv E_{2}(X_{n})$, and we can start the same
process again: $\sigma_{n}^{p,q}$ is still well-defined, which makes each
$\varphi_{n}^{p,q}:E_{3}^{p,q}(X)\to E_{r}^{p+3,q-2}(X)$ onto; then we can
proceed as in (5.3) to show that all the differentials
$d_{3}^{p,q}:E_{3}^{p,q}(X_{n})\to E_{3}^{p+3,q-2}(X_{n})$ are trivial. The
result is that $d_{r}^{p,q}\equiv 0$ for all $p$ and $q$ and for all $r\geqq
2$.

Finally, if $X$ were a Maximal space, then the action of $C_{2}$ on the
cohomology of each $X_{n}$ would also be trivial, for given $a_{n}\in
H^{*}(X_{n})$, it is enough to consider $a\in H^{*}(X)$ with
$a_{n}=j_{n}^{*}a$, and

$\sigma^{*}a_{n}=\sigma^{*}(j_{n}^{*}a)=j_{n}^{*}(\sigma^{*}a)=j_{n}^{*}a=a.$

$\blacksquare$

##  6\. Properties of the symmetric products

In this section we review some of the most important properties that the
symmetric products posses and that will be necessary for the proof of the main
theorem. Recall that the $n$-fold symmetric product of a space $X$, $n\geqq
2$, is the orbit space $X^{n}/\mathfrak{S}_{n}$, where $\mathfrak{S}_{n}$
denotes the symmetric group in $n$-letters and acts on $X^{n}$ in the obvious
way:

$\pi\cdot(x_{1},\dots,x_{n})=(x_{\pi(1)},\dots,x_{\pi(n)}),\qquad\pi\in\mathfrak{S}_{n}.$

We denote the equivalence class of the point $(x_{1},\dots,x_{n})$ as
$\\{x_{1},\dots,x_{n}\\}$. The convention is that $SP_{0}X$ is a singleton and
$SP_{1}X=X$.

We endow each $SP_{n}X$ with the quotient topology with respect to the
canonical map $X^{n}\to SP_{n}X$. When $X$ is a CW-complex with countably many
cells, there exists a CW-complex structure on $SP_{n}X$ (_cf._ [2, Note
5.2.2]). Moreover, in the case of based CW-complexes, each $n$-fold symmetric
product becomes a pointed space too, and then $SP_{n}X$ lies in $SP_{n+1}X$
canonically. This inclusion, that we denote by $i_{n,n+1}$, is a closed
embedding, so that $SP_{n}X$ can be regarded as subcomplex of $SP_{n+1}X$.
Finally, in this case, we define the infinite symmetric product $SP_{\infty}X$
as the union or colimit of all the finite symmetric products:

$SP_{\infty}X=\bigcup_{n}SP_{n}X$

with the union topology, as explained in Subsection 1.1. There are inclusions
$i_{n}:SP_{n}X\hookrightarrow SP_{\infty}X$, for every $n$.

The above transcripts, mutatis mutandis, to the case where $X$ is a countable
based $G$-CW complex (with $G$ a finite group). In this case, the inclusions
$i_{n,n+1}:SP_{n}X\hookrightarrow SP_{n+1}X$ and $i_{n}:SP_{n}X\hookrightarrow
SP_{n+1}X$ become equivariant. The next results summarises the cohomological
properties of the symmetric products of a finite, based $C_{2}$-CW complex.

######  Lemma 6.1.

Let $(X,x_{0})$ be a finite pointed $C_{2}$-CW complex. Then:

  1. (1)

The inclusion $i_{n,n+1}$ is given by the assignment

$\\{x_{1},\dots,x_{n}\\}\longmapsto\\{x_{0},x_{1},\dots,x_{n}\\}.$

  2. (2)

The sequence

$SP_{0}X\overset{i_{0,1}}{\longrightarrow}SP_{1}X\overset{i_{1,2}}{\longrightarrow}SP_{2}X\longrightarrow\cdots\longrightarrow
SP_{n}X\overset{i_{n,n+1}}{\longrightarrow}SP_{n+1}X\longrightarrow\cdots$

is a sequence in the category of finite based $C_{2}$-CW complexes and
equivariant base-preserving maps and satisfies all the conditions imposed in
Section 5. It also exhibits _both_ cohomological stability _and_ cohomological
splitting.

  * Proof.—

The cohomological splitting property was shown in Nakaoka [28] for simplicias
complexes, and a similar reasoning works for $G$-CW complexes ($G$ any finite
group).

Regarding cohomological stability, it is a classical result of Steenrod [30]
that

$H^{q}(SP_{n}X;\mathbf{F}_{2})\cong H^{q}(SP_{\infty}X;\mathbf{F}_{2})$

(among other coeffcients) when $q<n$. The results of Nakaoka imply that this
isomorphism is induced by the inclusion $i_{n}:SP_{n}X\rightarrow
SP_{\infty}X$, and therefore it also holds in the case of $G$-spaces.
$\blacksquare$

Another important result about symmetric products is that they preserve
homology, meaning that if $H_{q}(X)\cong H_{q}(Y)$, then

$H_{q}(SP_{n}X)\cong H_{q}(SP_{n}Y)$

(_cf._ [2, Corollary 5.2.10]). This is often abbreviated by saying that the
symmetric products are homological functors. A similar result holds for zero-
cycles.

######  Lemma 6.2.

The functors $SP_{n}$, $n\in\mathbf{N}$, are functors of zero-cycles. More
precisely: If $X$ and $Y$ are $C_{2}$-spaces and
$f:\mathbf{F}_{2}[X]\to\mathbf{F}_{2}[Y]$ is an equivariant homotopy
equivalence which is also a group homomorphism, then for each
$n\in\mathbf{N}$, there exists an equivariant homotopy equivalence
$\tilde{f}_{n}:\mathbf{F}_{2}[SP_{n}X]\to\mathbf{F}_{2}[SP_{n}Y]$ which is
also a group homomorphism.

  * Proof.—

The map $\tilde{f}_{n}:\mathbf{F}_{2}[SP_{n}X]\to\mathbf{F}_{2}[SP_{n}Y]$ is
defined as the linear extension of the long composite
$SP_{n}X\to{\mathbf{F}}_{2}[SP_{n}Y]$ in the diagram

${SP_{n}{\mathbf{F}}_{2}[Y]}$${{\mathbf{F}}_{2}[SP_{n}Y]}$${SP_{n}X}$${SP_{n}{\mathbf{F}}_{2}[X]}$${{\mathbf{F}}_{2}[SP_{n}X],}$$\scriptstyle{\epsilon_{Y}}$$\scriptstyle{SP_{n}f}$$\scriptstyle{\epsilon_{X}}$

where, for a space $Z$,
$\epsilon_{Z}:SP_{n}{\mathbf{F}}_{2}[Z]\to{\mathbf{F}}_{2}[SP_{n}Z]$ is a
continuous map defined by the following two properties:

    1. (1)

For $z_{1},\dots,z_{n}\in Z$,

$\epsilon_{Z}\\{z_{1},\dots,z_{n}\\}=\\{z_{1},\dots,z_{n}\\}.$

    2. (2)

It is multi-linear, _i.e._

$\epsilon_{Z}\\{z_{1},\dots,z_{i}+z_{i}^{\prime},\dots,z_{n}\\}=\epsilon_{Z}\\{z_{1},\dots,z_{i},\dots,z_{n}\\}+\epsilon_{Z}\\{z_{1},\dots,z_{i}^{\prime},\dots,z_{n}\\}.$

We suggest that the reader check that this «tilde» ($\sim$) construction
defines indeed a continuous map and that it is functorial, so that if
$g:\mathbf{F}_{2}[Y]\to\mathbf{F}_{2}[X]$ is a homotopy inverse for $f$ and
$h:\mathbf{F}_{2}[X]\times I\to\mathbf{F}_{2}[Y]$ is the corresponding
homotopy, then $\tilde{g}_{n}$ is a homotopy inverse for $\tilde{f}_{n}$ and
$\tilde{h}$ is a valid homotopy given by

$\tilde{h}_{t}=\mathclap{\phantom{h_{t}}\mathclap{\widetilde{\mathclap{\phantom{\mathbf{W}}\mathclap{h_{t}\,}}\phantom{\mathbf{W}}}}}\phantom{h_{t}}.$

$\blacksquare$

######  Property 6.3 (Aguilar _et al._ , [2]).

Let $(X,x_{0})$ be a pointed space. Then

$SP_{\infty}M^{C_{2}}X\cong N^{C_{2}}SP_{\infty}X,$

as $C_{2}$-spaces, where $M^{C_{2}}$ and $N^{C_{2}}$ denote the additive and
multiplicative inductions introduced in Example 2.6.

######  Remark 6.4.

The previous property makes an identification between the limit of a product
and a product of the limit. This is possible due to the assumption of locally
finiteness for the complexes, so that the product is automatically a
$C_{2}$-CW complex (_cf._ Property 4.1), and the following categorical fact:

> In the category of topological spaces, finite limits commute with filtered
> colimits. In particular, if $\\{(X_{n},f_{n})\\}_{n}$ and
> $\\{(Y_{n},g_{n})\\}_{n}$ are two family of topological spaces and maps such
> that all the maps $f_{n}:X_{n}\to X_{n+1}$ and $g_{n}:Y_{n}\to Y_{n+1}$ are
> closed embeddings, then
>
>
> $\left(\bigcup_{n}X_{n}\right)\times\left(\bigcup_{n}Y_{n}\right)\cong\bigcup_{n}(X_{n}\times
> Y_{n}).$

Indeed, it is well known (_cf._ [20, IX.2]) that finite limits commute with
filtered colimits when the category is cocomplete. And the category of
topological spaces and continuous maps is an example of a cocomplete category
(_cf._ [1, 12.6 Examples]). We will use this result again in the proof of
Theorem 7.1.

##  7\. Main result

######  Theorem 7.1.

Let $X$ be a finite GM-space. Then, $SP_{n}X$ is a GM-space for every
$n\leqq\infty$.

  * Proof.—

We start by proving the result in the particular case when $X$ is of the form

$\left(\bigvee_{i\in I}S^{p_{i},q_{i}}\right)\vee\left(\bigvee_{j\in
J}[S^{r_{j}}\wedge(C_{2})_{+}]\right),$

with $I$ and $J$ finite sets. Once this is solved, we will explain how to
address the general case.

To avoid confusion, let us use another letter:

(7.1) $W=\left(\bigvee_{i\in I}S^{p_{i},q_{i}}\right)\vee\left(\bigvee_{j\in
J}[S^{r_{j}}\wedge(C_{2})_{+}]\right).$

This is a GM-space, as shown in Example 2.6(4). The aim is to show that
$SP_{\infty}W$ is a GM-space too and then use the results about cohomological
stability and cohomological splitting of Theorem 5.6 for the particular case
of symmetric products, as in Lemma 6.1, to obtain the result for each finite
symmetric power $SP_{n}W$.

First, notice that

$SP_{\infty}W\cong\left(\prod_{i\in
I}SP_{\infty}S^{p_{i},q_{i}}\right)\times\left(\prod_{j\in
J}N_{C_{2}}SP_{\infty}S^{r_{j}}\right)\cong\bigcup_{n}\left\\{\left(\prod_{i\in
I}SP_{n}S^{p_{i},q_{i}}\right)\times\left(\prod_{j\in
J}N^{C_{2}}SP_{n}S^{r_{j}}\right)\right\\}$

due to Property 6.3 (we are using the commutativity properties of products and
colimits again as explained in Remark 6.4). Each $SP_{n}S^{p_{i},q_{i}}$ is an
M-space by the results of Franz given in Theorem 2.8, and so is their product;
similarly, $N^{C_{2}}SP_{n}S^{r_{j}}$ is a GM-space for all $n$ and $j$, as
mentioned in Example 2.6(2). Hence, each product

$Z_{n}=\left(\prod_{i\in I}SP_{n}S^{p_{i},q_{i}}\right)\times\left(\prod_{j\in
J}N^{C_{2}}SP_{n}S^{r_{j}}\right)$

is a finite GM-space. Moreover, the sequence

$Z_{0}\longrightarrow Z_{1}\longrightarrow
Z_{2}\longrightarrow\cdots\longrightarrow Z_{n}\longrightarrow
Z_{n+1}\longrightarrow\cdots$

with the obvious maps

$\displaystyle\varphi_{n}=\left(\prod_{i\in
I}\iota^{i}_{n,n+1}\right)\times\left(\prod_{j\in
J}N^{C_{2}}\lambda^{j}_{n,n+1}\right),\qquad$
$\displaystyle\iota^{i}_{n,n+1}:SP_{n}S^{p_{i},q_{i}}\hookrightarrow
SP_{n}S^{p_{i},q_{i}},$
$\displaystyle\lambda_{n,n+1}^{j}:SP_{n}S^{r_{j}}\hookrightarrow
SP_{n+1}S^{r_{j}},$

exhibits cohomological stability, as follows from Property 5.4. Thus,

$Z=\bigcup_{n}Z_{n}\cong SP_{\infty}W$

is a GM-space, by virtue of Theorem 5.6(1) (notice that the isomorphism is
only possible after swapping products and colimits).

Now that $SP_{\infty}W$ is known to be a GM-space, we can use Theorem 5.6(2)
to retrieve the result for each $SP_{n}W$. We only need to make sure that the
hypotheses of this statement are verified, but this is exactly Lemma 6.1.

Finally, we address the general case. If $X$ is any finite GM-space, then
Theorem 3.1 applies and

$\mathbf{F}_{2}[X]\simeq_{C_{2}}\left(\bigoplus_{i\in
I}\mathbf{F}_{2}[S^{p_{i},q_{i}}]\right)\oplus\left(\bigoplus_{j\in
J}\mathbf{F}_{2}[S^{r_{j}}\wedge(C_{2})_{+}]\right)\equiv\mathbf{F}_{2}[W],$

where $W$ is of the form (7.1). We also apply the same characterisation
theorem to each $SP_{n}W$:

$\mathbf{F}_{2}[SP_{n}W]\simeq_{C_{2}}\left(\bigoplus_{k\in
K_{n}}\mathbf{F}_{2}[S^{p_{k},q_{k}}]\right)\oplus\left(\bigoplus_{l\in
L_{n}}\mathbf{F}_{2}[S^{r_{l}}\wedge(C_{2})_{+}]\right)$

for some finite sets $K_{n}$ and $L_{n}$. As
$\mathbf{F}_{2}[X]\simeq_{C_{2}}\mathbf{F}_{2}[W]$, Lemma 6.2 implies that
$\mathbf{F}_{2}[SP_{n}X]\simeq_{C_{2}}\mathbf{F}_{2}[SP_{n}W]$, _i.e._ ,

$\mathbf{F}_{2}[SP_{n}X]\simeq_{C_{2}}\left(\bigoplus_{k\in
K_{n}}\mathbf{F}_{2}[S^{p_{k},q_{k}}]\right)\oplus\left(\bigoplus_{l\in
L_{n}}\mathbf{F}_{2}[S^{r_{l}}\wedge(C_{2})_{+}]\right).$

Hence, by Theorem 3.1 again, we conclude that $SP_{n}X$ is also a GM-space.
Now, the case $n=\infty$ follows from cohomological stability once again.
$\blacksquare$

## Acknowledgments

The results of the paper are part of the author’s thesis. During his PhD, the
author was supported by a scholarship granted by the Portuguese institution
Fundação para a Ciência e a Tecnologia (FCT), to which the author expresses
his gratitude. The author would also like to thank his advisors Pedro F. dos
Santos and Carlos Florentino for all their guidance and support, and Erwan
Brugallé, Florent Schaffhauser and Clover for the useful conversations and
ideas that they shared.

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# Analyzing Hate Speech Data along Racial, Gender and Intersectional Axes

Antonis Maronikolakis  Equal contribution. Philip Baader∗ Hinrich Schütze
Center for Information and Language Processing (CIS), LMU Munich, Germany
<EMAIL_ADDRESS>
###### Abstract

Warning: This work contains strong and offensive language, sometimes
uncensored.

To tackle the rising phenomenon of hate speech, efforts have been made towards
data curation and analysis. When it comes to analysis of bias, previous work
has focused predominantly on race. In our work, we further investigate bias in
hate speech datasets along racial, gender and intersectional axes. We identify
strong bias against African American English (AAE), masculine and
AAE+Masculine tweets, which are annotated as disproportionately more hateful
and offensive than from other demographics. We provide evidence that BERT-
based models propagate this bias and show that balancing the training data for
these protected attributes can lead to fairer models with regards to gender,
but not race.

Analyzing Hate Speech Data along Racial, Gender and Intersectional Axes

Antonis Maronikolakis††thanks:  Equal contribution. and Philip Baader∗ and
Hinrich Schütze Center for Information and Language Processing (CIS), LMU
Munich, Germany<EMAIL_ADDRESS>

##  1 Introduction

Hate Speech. To tackle the phenomenon of online hate speech, efforts have been
made to curate datasets Davidson et al. (2017); Guest et al. (2021); Sap et
al. (2020). Since datasets in this domain are dealing with sensitive topics,
it is of upmost importance that biases are kept to a (realistic) minimum and
that data is thoroughly analyzed before use Davidson et al. (2019a); Madukwe
et al. (2020). In our work, we are contributing to this analysis by uncovering
biases along the racial, gender and intersectional axes.

Racial111While the correlation of race and African American English (AAE) is
complicated Anderson (2015), in our work we consider AAE as a proxy for race,
since it is a dialect overwhelmingly used by African Americans Spears and
Hinton (2010); Spears (2015)., Gender and Intersectional Biases. During data
collection, biases can be introduced due to–among other reasons–lack of
annotator training or divergence between annotators and user demographics. For
example, oftentimes the majority of annotators are white or male Sap et al.
(2020); Founta et al. (2018). An annotator not in the ‘in-group’ may hold
(un)conscious biases based on misconceptions about ‘in-group’ speech which may
affect their perception of speech from certain communities O’Dea et al.
(2015), leading to incorrect annotations when it comes to dialects the
annotators are not familiar with. A salient example of this is annotators
conflating African American English with hateful language Sap et al. (2019).

Figure 1: Distributions of label annotations on Davidson (neutral, offensive,
hateful) for AAE+Masculine, AAE and SAE (top-to-bottom). AAE has a higher
ratio of offensive examples than SAE, while AAE+Masculine is both highly
offensive and hateful.

Intersectionality Crenshaw (1989) is a framework for examining how different
forms of inequality (for example, racial or gender inequalities) intersect
with and reinforce each other. These new social dynamics need to be analyzed
both separately and as a whole in order to address challenges faced by the
examined communities. For example, a black woman does not face inequality
based only on race or only on gender: she faces inequality because of both
these characteristics, separately and in conjunction. In this work, we are
analyzing not only the racial or gender inequalities in hate speech datasets,
but their intersectionality as well.

With research in the area of hate speech, the NLP community aims at protecting
target groups and fostering a safer online environment. In this sensitive
area, it is pivotal that datasets and models are analyzed extensively to
ensure the biases we are protecting affected communities from do not appear in
the data itself, causing further marginalization (for example, by removing AAE
speech disproportionately more often).

Contributions. In summary, we  (i) investigate racial, gender and
intersectional bias in three hate speech datasets, Founta et al. (2018);
Davidson et al. (2017); Mathew et al. (2021),  (ii) examine classifier
predictions on existing, general-purpose African/Standard American English
(AAE/SAE) and gendered tweets,  (iii) identify model bias against AAE,
masculine and AAE+Masculine (labeled as both AAE and masculine) tweets,  (iv)
show that balancing training data for gender leads to fairer models  .

##  2 Related Work

Hate speech research has focused on dataset curation Davidson et al. (2017);
Founta et al. (2018); Sap et al. (2020); Guest et al. (2021); Hede et al.
(2021); Grimminger and Klinger (2021) and dataset analysis Madukwe et al.
(2020); Wiegand et al. (2019); Swamy et al. (2019). In our work, we further
analyze datasets to uncover latent biases.

It has been shown that data reflects social bias inherent in annotator pools
Waseem (2016); Al Kuwatly et al. (2020); Davidson et al. (2019a, b). Work has
been conducted to identify bias against AAE Sap et al. (2019); Zhou et al.
(2021); Xia et al. (2020) and gender Excell and Al Moubayed (2021).

Research has also been conducted in identifying disparities in performance
across social groups, with machine learning algorithms underperforming for
certain groups Tatman (2017); Buolamwini and Gebru (2018); Rudinger et al.
(2018).

Kim et al. (2020) investigated whether bias along the intersectional axis
exists in Founta et al. (2018). While Kim et al. (2020) focused on bias within
a single dataset, in our work we generalize to multiple hate speech datasets.
We also examine classifier behavior and methods to mitigate this bias.

Research from a sociolinguistic perspective has shown that genders exhibit
differences in online text Gefen and Ridings (2005) as well as general speech
Penelope Eckbert (2013). In Bamman et al. (2014) and Bergsma and Van Durme
(2013), gender classifiers for English tweets were developed with accuracy of
88% and 85% respectively. In our work, we develop a gender classifier of
tweets as well, focusing on precision over recall, leading to a smaller but
more accurate sample of gendered data.

Dataset | Neutral | Offensive | Hateful
---|---|---|---
Davidson | 0.92 | 0.85 | 0.53
Founta | 0.85 | 0.79 | 0.47
HateXplain | 0.69 | 0.55 | 0.70
Table 1: F1-score of BERT for each label, evaluated on Davidson, Founta and
HateXplain.

##  3 Datasets

Five English datasets were used: three hate speech datasets (Davidson, Founta
and HateXplain), one dataset of tweets labeled for race (Groenwold) and one
for gender (Volkova). We adopt the definitions of Davidson et al. (2017) for
hate speech (defined as speech that contains expressions of hatred towards a
group or individual on the basis of protected attributes like ethnicity,
gender, race and sexual orientation) and offensive speech (speech that
contains offensive language but is not hateful).

Davidson. In Davidson et al. (2017), a hate speech dataset of tweets was
collected, labeled for neutral, offensive and hateful language.

Founta. In Founta et al. (2018) a crowdsourced dataset of tweets was
presented, labeled for normal, abusive and hateful language. To unify
definitions, we rename normal to neutral language and abusive to offensive
language.

HateXplain. Mathew et al. (2021) presented a dataset from Twitter and
Gab222Gab is a social platform that has been known to host far-right groups
and rhetoric. passages. It has been labeled for normal (neutral), offensive
and hateful language.

Groenwold. In Groenwold et al. (2020) a dataset of African American English
and Standard American English tweets was introduced. The AAE tweets come from
Blodgett et al. (2016) and the SAE are direct translations of those tweets
provided by annotators.

Volkova. Volkova et al. (2013) presented a dataset of 800k English tweets from
users with an associated gender (feminine/masculine).

Dataset | Masc. | Fem. | SAE | AAE | SAE+Masc. | SAE+Fem. | AAE+Masc. | AAE+Fem.
---|---|---|---|---|---|---|---|---
Davidson | 2716 | 2338 | 3534 | 8099 | 1279 | 1240 | 3157 | 1172
Founta | 26307 | 13615 | 43330 | 4177 | 13486 | 13257 | 971 | 787
HateXplain | 4509 | 1103 | 10368 | 1103 | 4145 | 2376 | 250 | 240
Groenwold AAE | 586 | 613 | 0 | 1995 | 0 | 0 | 587 | 612
Groenwold SAE | 587 | 601 | 1980 | 0 | 587 | 601 | 0 | 0
Volkova | 41164 | 58836 | 37874 | 3755 | 16243 | 21631 | 1843 | 1912
Table 2: Protected attribute statistics for Davidson, Founta, HateXplain,
Groenwold and Volkova.

##  4 Experimental Setup

AAE Classifier. To classify tweets as AAE or SAE, we used the Blodgett et al.
(2016) classifier. We took into consideration tweets with a confidence score
over 0.5, which can be interpreted as a straightforward classifier of AAE/SAE
(whichever class has the highest score is returned).

Gender Classifier. To classify tweets as masculine or feminine, we finetuned
BERT-base333https://huggingface.co/bert-base-cased on Volkova et al. (2013),
which includes gender information as self-reported from authors. We split the
dataset into train/dev/test (50K/25K/25K) and employed a confidence score of
0.8 as the threshold for assigning gender to a tweet. For the tweets with a
confidence over the given threshold, precision was 78.4% when classifying
tweets as ‘masculine’ and 79.5% when classifying tweets as ‘feminine’.

Hate Speech Classifiers. For each of the three hate speech datasets we
finetuned BERT-base. We split each dataset into 80:10:10 (train:dev:test)
sets, used a max sequence length of 256 and trained for 3 epochs, keeping the
rest of the hyperparameters the same. Performance for the development set is
shown in Table 1444Performance on the test set is similar, omitted for
brevity.. In Davidson and Founta, BERT performs well for neutral and offensive
examples, performance drops for hateful content. In HateXplain, BERT overall
performs worse, with slightly better performance for neutral and hateful
examples over offensive ones.

Intersectionality. For our analysis, we classified tweets from all datasets
for gender and race.

##  5 Intersectionality Statistics

In Table 2, we present statistics for gender, race and their intersection as
found in the three examined hate speech datasets as well as in Groenwold and
Volkova.555Race/gender for the hate speech datasets, gender for Groenwold and
race for Volkova have been computed as described in Section 4. We show that no
dataset is balanced between AAE and SAE. In Founta and HateXplain, AAE tweets
make up approximately 1/10th of the data. In Davidson, we see stronger
representation of AAE, with the AAE tweets being almost twice as many as the
SAE tweets. Davidson is also balanced for gender. The other hate speech
datasets, while still not balanced, are more balanced for gender than they are
for race. Founta has twice as many masculine than feminine tweets and
HateXplain has four times as many.

In Table 3, we present a breakdown of protected attributes per class
(neutral/offensive/hateful) for Davidson, Founta and HateXplain. A main
takeaway for Davidson and Founta is the imbalance of AAE versus SAE. In SAE,
the neutral class makes up 52% of the data for Davidson and 81% for Founta,
while the respective numbers for AAE are 3% for Davidson and 13% for Founta.

In HateXplain, AAE and SAE are more balanced, but there is instead imbalance
between genders. For masculine and feminine speech, passages are neutral at
rates of 43% and 61% respectively. In Davidson, SAE+Feminine speech is viewed
as more offensive than SAE+Masculine (48% vs. 19%), while in HateXplain,
SAE+Masculine is more hateful than SAE+Feminine (34% vs. 16%). Finally, when
comparing genders in AAE speech, we see that while AAE+Feminine contains a
larger percentage of offensive tweets (for example, in Founta, 69% vs. 54% and
in HateXplain, 50% vs. 21%), AAE+Masculine contains disproportionately more
hateful speech (in Davidson, 7% vs. 5%, in Founta, 28% vs. 9% and in
HateXplain, 19% vs. 6%).

Overall, AAE and masculine speech is annotated as more offensive and hateful
than SAE and feminine speech. Further analyzing AAE, AAE+Masculine is viewed
as more hateful than AAE+Feminine.

Dataset | Masc. | Fem. | SAE | AAE | SAE+Masc. | SAE+Fem. | AAE+Masc. | AAE+Fem.
---|---|---|---|---|---|---|---|---
N | O | H | N | O | H | N | O | H | N | O | H | N | O | H | N | O | H | N | O | H | N | O | H
Davidson | 32.2 | 61.9 | 5.9 | 27.7 | 69.5 | 2.8 | 51.8 | 40.7 | 7.5 | 2.8 | 93.2 | 4.0 | 77.0 | 19.4 | 3.6 | 50.0 | 47.8 | 2.3 | 4.7 | 88.5 | 6.8 | 6.8 | 88.0 | 5.2
Founta | 81.2 | 12.3 | 6.4 | 71.0 | 25.0 | 4.0 | 80.5 | 14.6 | 4.9 | 13.2 | 69.2 | 17.6 | 86.9 | 7.6 | 5.5 | 86.2 | 11.4 | 2.4 | 18.3 | 53.8 | 27.9 | 21.8 | 69.4 | 8.8
HateXplain | 43.0 | 23.7 | 33.3 | 60.7 | 24.6 | 14.8 | 38.3 | 26.7 | 35.0 | 45.6 | 39.1 | 15.3 | 41.6 | 24.0 | 34.4 | 58.9 | 25.1 | 16.0 | 59.4 | 21.3 | 19.4 | 44.4 | 50.0 | 5.6
Table 3: Distribution of protected attribute annotations for neutral/offensive/hateful (N/O/H) examples. Dataset | Masc. | Fem. | SAE | AAE | SAE+Masc. | SAE+Fem. | AAE+Masc. | AAE+Fem.
---|---|---|---|---|---|---|---|---
N | O | H | N | O | H | N | O | H | N | O | H | N | O | H | N | O | H | N | O | H | N | O | H
Random | 33.8 | 63.2 | 3.0 | 27.7 | 71.2 | 1.1 | 53.1 | 40.5 | 6.4 | 4.9 | 94.1 | 1.0 | 77.3 | 19.3 | 3.4 | 45.6 | 53.2 | 1.2 | 6.4 | 91.2 | 2.4 | 3.0 | 94.3 | 2.7
Balanced | 25.3 | 71.5 | 3.2 | 25.4 | 71.1 | 3.5 | 54.3 | 39.2 | 6.5 | 4.3 | 95.1 | 1.6 | 71.0 | 22.8 | 6.2 | 52.3 | 46.4 | 2.3 | 5.8 | 92.1 | 2.1 | 6.2 | 93.1 | 0.7
Table 4: Distribution of predictions for protected attributes on random and balanced datasets based on Davidson. The balanced set is balanced on race (equal number of AAE and SAE tweets) and gender (equal number of feminine and masculine tweets). Shown are percentages for neutral/offensive/hateful (N/O/H) predictions. Dataset |  All |  AAE
---|---|---
Davidson |  n*ggerize, subhuman, bastards, border, pigfucking, feminist, wetbacks, savages, wetback, jumpers |  queer, n*gros, n*ggaz, racial, shittiest, wet, savage, skinned, darky, f*gs
Founta |  moron, insult, muslims, aggression, puritan, haters, arabs, coloured, ousted, pedophiles |  white, killing, pathetic, n*gga, slave, n*ggas, sells, hell, children, violent
HateXplain |  towelhead, muzzrat, muscum, n*gresses, n*ggerette, n*glets, musloid, n*ggerish, n*ggery, gorilla |  spic, fuck, f*ggots, gorilla, towel, sandn*gger, zhid, c*ons, rag, fowl
Table 5: Top 10 most contributing words for Davidson, Founta and HateXplain as
computed with LIME for hateful predictions.

##  6 Bias in BERT

We investigate to what extent data bias is learned by BERT. We compare our
findings against a dataset balanced for race and gender, to examine whether
balanced data leads to fairer models. Namely, we compare a randomly sampled
with a balanced set the Davidson dataset.666Founta and HateXplain were not
considered for this study as they do not contain enough AAE examples to make
confident inferences. In the balanced set we sample the same number of AAE and
SAE tweets (3000) and the same number of masculine and feminine tweets (1750).
We also include 8000 neutral tweets without race or gender labels. For the
randomly sampled set, for a fair comparison, we sampled the same number of
tweets as the balanced set.777Experiments were conducted with the entirety of
the original dataset with similar results. They are omitted for brevity. All
sampling was stratified to preserve the original label distributions. Results
are shown in Table 4.

In the randomly sampled set, there is an imbalance both for gender and race.
For gender, while masculine tweets are more hateful (3% vs. 1%), feminine
tweets are more offensive (71% vs. 63%). For race, AAE is marked almost
entirely as offensive (94%), while SAE is split in neutral and offensive (53%
and 41%). In the SAE subset of tweets, there is an imbalance between genders,
with SAE+Feminine being marked disproportionately more often as offensive than
SAE+Masculine (54% vs. 19%).

###  6.1 Balanced Training

In Table 4, before balancing, 34% of masculine and 28% of feminine tweets are
marked as neutral. After balancing, these rates are both at 25%. There is an
improvement in the intersection of AAE and gender, with the distributions of
AAE+Masculine and AAE+Feminine tweets converging. For SAE, SAE+Masculine and
SAE+Feminine distributions converge too, although still far apart. Overall,
balanced data improves fairness for gender but not for race, which potentially
stems from bias in annotation.

###  6.2 Interpretability with LIME

In Table 5, we show the top contributing words for offensive and hateful
predictions in Davidson, Founta and HateXplain. We see that for AAE, terms
such as ‘n****z’ and ‘n***a’ contribute in classifying text as non-neutral
even though the terms are part of African American vernacular Rahman (2012),
showing that this dialect is more likely to be flagged. In non-AAE speech
(which includes–but is not exclusive to–SAE), we see the n-word variant with
the ‘-er’ spelling appearing more often in various forms, which is correctly
picked up by the model as an offensive and hateful term. On both sets, we also
see other slurs, such as ‘f*ggots’, ‘moron’ and ‘wetback’ (a slur against
foreigners residing in the United States, especially Mexicans) being picked
up, showing the model does recognize certain slurs and offensive terms.

##  7 Conclusion

In our work, we analyze racial, gender and intersectional bias in hate speech
datasets. We show that tweets from AAE and AAE+Masculine users (as classified
automatically) are labeled disproportionately more often as offensive. We
further show that BERT learns this bias, flagging AAE speech as significantly
more offensive than SAE. We perform interpretability analysis using LIME,
showing that the inability of BERT to differentiate between variations of the
n-word across dialects is a contributing factor to biased predictions.
Finally, we investigate whether training on a dataset balanced for race and
gender mitigates bias. This method shows mixed results, with gender bias being
mitigated more than racial bias. With our work we want to motivate further
investigation in model bias not only for the usual gender and racial
attributes, but also for their intersection.

##  8 Bias Statement

Research in the sphere of hate speech has produced annotated data that can be
used to train classifiers to detect hate speech as found in online media. It
is known that these datasets contain biases that models will potentially
propagate. The representational harm that can be triggered is certain target
groups getting their speech inadvertently censored/deleted due to existing
biases that marginalize certain groups. In our work we investigate this
possibility along intersectional axes (gender and race). We find that tweets
written by female users are seen as disproportionately more offensive, while
male users write tweets that appear more hateful.

##  9 Ethical Considerations

In our work we are dealing with data that can catalyze harm against
marginalized groups. We do not advocate for the propagation or adoption of
this hateful rhetoric. With our work we wish to motivate further analysis and
documentation of sensitive data that is to be used for the training of models
(for example, using templates from Mitchell et al. (2019); Bender and Friedman
(2018)).

Further, while classifying protected attributes such as race or gender is
important in analyzing and identifying bias, care should be taken for the race
and gender classifiers to not be misused or abused, in order to protect the
identity of users, especially those from marginalized demographics who are
more vulnerable to hateful attacks and further marginalization. In our work we
only predict these protected attributes for investigative purposes and do not
motivate the direct application of such classifiers. Further, in our work we
are using dialect (AAE) associated with African Americans as a proxy to race
due to a lack of available annotated data. It should be noted that not all
African Americans make use of AAE and not all AAE users are African Americans.

Finally, in our work we only focused on English and a specific set of
attributes. Namely, we considered race (African American) and gender. This is
a non-exhaustive list of biases and more work needs to be done for greater
coverage of languages and attributes.

##  10 Acknowledgments

Antonis Maronikolakis was partly supported by the European Research Council
(#740516).

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