# Representation-graded Bredon homology
of elementary abelian 2-groups

Markus Hausmann  and  Stefan Schwede<EMAIL_ADDRESS><EMAIL_ADDRESS>Mathematisches Institut, Universität Bonn, Germany

###### Abstract.

We calculate the representation-graded Bredon homology rings of all elementary
abelian 2-groups with coefficients in the constant mod-2 Mackey functor. We
show that these rings are domains, and we exhibit minimal presentations as
quotients of the polynomial algebra on the pre-Euler and inverse Thom classes
of all nontrivial characters, subject to an explicit finite list of relations
arising from orientability properties. Two corollaries of our presentation are
the calculation, originally due to Holler and Kriz, of the geometric fixed
point rings, and a strengthening of a calculation of Balmer and Gallauer of
the localized twisted cohomology ring.
2020 MSC: 55N91, 55P91, 55Q91

## Introduction

In this paper we calculate the representation-graded Bredon homology rings of
all elementary abelian 2-groups, with coefficients in the constant mod-2
Mackey functor $\underline{{\mathbb{F}}}_{2}$. More specifically, we determine
the ‘homological cone’ of the $RO$-graded Bredon homology ring, i.e., the
sector given by the reduced Bredon homology groups of linear representation
spheres; we denote this multigraded ring by $H(A,\star)$. The following are
our main results, to be proved as Theorems 2.2 and 2.5 below.

Theorem. Let $A$ be an elementary abelian 2-group.

  1. (i)

The representation-graded Bredon homology ring $H(A,\star)$ of $A$ with
coefficients in $\underline{{\mathbb{F}}}_{2}$ is a domain.

  2. (ii)

The ${\mathbb{F}}_{2}$-algebra $H(A,\star)$ is generated by the pre-Euler
classes $a_{\lambda}$ and the inverse Thom classes $t_{\lambda}$ for all
nontrivial $A$-characters $\lambda$.

  3. (iii)

The ideal of relations between the classes $a_{\lambda}$ and $t_{\lambda}$ is
generated by the polynomials

$\sum_{\lambda\in T}a_{\lambda}\cdot\bigg{(}\prod_{\mu\in
T\setminus\\{\lambda\\}}t_{\mu}\bigg{)}$

for all minimally dependent sets $T$ of nontrivial $A$-characters.

The relations listed in part (iii) above are also minimal, i.e., none of them
can be omitted. The origin of the relations is the fact that for a minimally
dependent set $T$ of nontrivial $A$-characters, the $A$-representation
$\bigoplus_{\lambda\in T}\lambda$ is orientable, see Proposition 1.7.

We illustrate our result for elementary abelian 2-groups of small rank. When
$A=C=\\{\pm 1\\}$ is of order 2, the calculation is classical, originally due
to Stong (unpublished), and reproved by several authors. In this case there is
only one nontrivial character, and no relations; so
$H(C,\star)={\mathbb{F}}_{2}[a,t]$ is a polynomial algebra on the pre-Euler
class and the inverse Thom class.

To the best of the authors’ knowledge, the special case $A=C^{2}$ of our
theorem is already a new result. In this case there are three nontrivial
characters $p_{1},p_{2}$ and $\mu$, and all relations are generated by the
single relation

$a_{1}t_{2}t_{\mu}+t_{1}a_{2}t_{\mu}+t_{1}t_{2}a_{\mu}\ =\ 0\ .$

So the minimal presentation of $H(C^{2},\star)$ is

$H(C^{2},\star)\ =\
{\mathbb{F}}_{2}[a_{1},a_{2},a_{\mu},t_{1},t_{2},t_{\mu}]/(a_{1}t_{2}t_{\mu}+t_{1}a_{2}t_{\mu}+t_{1}t_{2}a_{\mu})\
.$

For $A=C^{3}$ we make our presentation of $H(C^{3},\star)$ completely explicit
in Example 2.7. In this case there are 14 polynomial generators, namely the
classes $a_{\lambda}$ and $t_{\lambda}$ for each of the seven nontrivial
characters $\lambda$, and 14 minimal relations. Of these relations, seven are
cubic in the generators, and of the same general form as in the previous
example, i.e.,
$a_{\alpha}t_{\beta}t_{\gamma}+t_{\alpha}a_{\beta}t_{\gamma}+t_{\alpha}t_{\beta}a_{\gamma}=0$
for all triples of distinct nontrivial characters that satisfy
$\alpha\cdot\beta\cdot\gamma=1$. And there are seven minimal relations that
are homogeneous of degree 4 in the generators, of the form

$a_{\alpha}t_{\beta}t_{\gamma}t_{\delta}\ +\
t_{\alpha}a_{\beta}t_{\gamma}t_{\delta}\ +\
t_{\alpha}t_{\beta}a_{\gamma}t_{\delta}\ +\
t_{\alpha}t_{\beta}t_{\gamma}a_{\delta}\ =\ 0$

for quadruples of distinct nontrivial characters that satisfy
$\alpha\cdot\beta\cdot\gamma\cdot\delta=1$.

The number of minimal relations grows very quickly in the rank of the
elementary abelian 2-group, see the table in Remark 2.8. However, a basic
pattern continues as follows: when $A$ has rank $r$, then a new family of
relations appears that has no predecessor for smaller rank, given by
homogeneous polynomials of degree $r+1$ in the generators.

We use our presentation of $H(A,\star)$ to derive two interesting corollaries.
Inverting the pre-Euler classes $a_{\lambda}$ for all nontrivial
$A$-characters and restricting to integer gradings yields the $A$-geometric
fixed point ring of mod-2 Bredon homology. This ring was previously calculated
by Holler and Kriz in [4, Theorem 2], who also gave a formula for the Poincaré
series of the multigraded ring $H(A,\star)$ in [4, Theorem 5]. We explain in
Corollary 3.4 how our presentation of $H(A,\star)$ yields the Holler-Kriz
presentation upon localization. A noticeable feature is that inverting the
pre-Euler classes makes all polynomial relations of degree at least four
redundant.

More generally, we consider a subgroup $B$ of $A$ and determine the ‘mixed’
localization obtained by inverting the pre-Euler classes of all characters
that restrict nontrivially to $B$, and the inverse Thom classes of all other
characters. The resulting integer-graded ring $H(A|B)$ previously featured in
the work of Balmer and Gallauer [1] on the Balmer spectrum of the tt-category
of permutation modules. We obtain an explicit presentation of the ring
$H(A|B)$ by generators and relations in Theorem 3.2; here, too, all polynomial
relations of degree at least four in the minimal presentation of $H(A,\star)$
become redundant in the localization. Our calculation improves [1, Theorem
8.13], for the prime $2$, from a ‘presentation modulo nilpotence’ to an actual
presentation. Because $H(A,\star)$ is a domain, its localization $H(A|B)$ is a
domain, too.

Acknowledgments. The authors are members of the Hausdorff Center for
Mathematics at the University of Bonn (DFG GZ 2047/1, project ID 390685813). A
substantial part of the work for this paper was done while the second author
spent the summer term 2023 on sabbatical at Stockholm University, with
financial support from the Knut and Alice Wallenberg Foundation; the second
author would like to thank SU for the hospitality and stimulating atmosphere
during this visit.

##  1\. Representation-graded Bredon homology

In this section we review some basic features of Bredon homology in a form
adapted for our purposes. This section does not contain any new mathematics.
What is now called Bredon cohomology was introduced by Bredon in [2] for
finite groups and equivariant CW-complexes. The corresponding equivariant
homology theory was introduced by Illman in [5]. Illman develops the theory
for arbitrary topological groups, and he uses singular chains to define the
equivariant homology and cohomology groups on arbitrary equivariant spaces. In
many ways, Bredon homology and cohomology are the correct generalizations of
singular (co-)homology to the equivariant context, and of fundamental
importance in equivariant topology.

######  Construction 1.1 (Bredon homology).

We recall one construction of Bredon homology with coefficients in a constant
Mackey functor. We employ a definition that is naturally isomorphic to the
original one of Bredon and Illman, namely as the equivariant homology theory
represented by the Eilenberg-MacLane $G$-spectrum $H\underline{M}$ of the
constant Mackey functor $\underline{M}$ associated to an abelian group $M$. In
other words, we define the $m$th reduced $G$-equivariant Bredon homology group
of a based $G$-space $X$ with $\underline{M}$-coefficients as

$\tilde{H}_{m}^{G}(X;\underline{M})\ =\ \pi_{m}^{G}(H\underline{M}\wedge X)\
.$

The groups $\tilde{H}^{G}_{*}(-;\underline{M})$ form an equivariant homology
theory. In particular, they come with a suspension isomorphism, and a based
$G$-map gives rise to a long exact sequence featuring the Bredon homology
groups of source, target and the reduced mapping cone. We mostly consider
$\underline{{\mathbb{F}}}_{2}$-coefficients in this paper, and we will drop
$\underline{{\mathbb{F}}}_{2}$-coefficients from the notation.

The commutative $G$-ring spectrum structure of $H\underline{{\mathbb{F}}}_{2}$
gives rise to associative, commutative and bilinear pairings

$\cdot\ :\ \tilde{H}_{m}^{G}(X)\times\tilde{H}_{n}^{G}(Y)\ \longrightarrow\
\tilde{H}^{G}_{m+n}(X\wedge Y)$

for all based $G$-spaces $X$ and $Y$.

In this paper a representation of a finite group is a finite-dimensional
orthogonal representation. Our results are about the
$\underline{{\mathbb{F}}}_{2}$-Bredon homology groups of representation
spheres, i.e., onepoint compactifications $S^{V}$ of such representations $V$.
The following proposition collects some well-known general facts about these;
we give proofs for the readers’ convenience. Part (iii) says that
automorphisms of representation spheres are invisible to the eyes of Bredon
homology with $\underline{{\mathbb{F}}}_{2}$-coefficients. This means that we
can – and will – safely ignore the distinction between $G$-representations and
their isomorphism classes.

######  Proposition 1.2.

Let $V$ be a $d$-dimensional representation of a finite group $G$.

  1. (i)

The group $\tilde{H}_{k}^{G}(S^{V})$ is trivial for $k<0$ and for $k>d$, and
the restriction homomorphism

$\operatorname{res}^{G}_{1}\ :\ \tilde{H}_{d}^{G}(S^{V})\ \longrightarrow\
\tilde{H}_{d}(S^{V})$

is an isomorphism. Hence the ${\mathbb{F}}_{2}$-vector space
$\tilde{H}_{d}^{G}(S^{V})$ is 1-dimensional.

  2. (ii)

If $V$ is orientable, then the restriction homomorphism for constant integer
coefficients

$\operatorname{res}^{G}_{1}\ :\
\tilde{H}_{d}^{G}(S^{V};\underline{{\mathbb{Z}}})\ \longrightarrow\
\tilde{H}_{d}(S^{V};{\mathbb{Z}})$

is an isomorphism. Hence the abelian group
$\tilde{H}_{d}^{G}(S^{V};\underline{{\mathbb{Z}}})$ is free of rank 1.

  3. (iii)

For every based $G$-homotopy equivalence $\psi:S^{V}\longrightarrow S^{V}$ and
every $m\geq 0$, the map

$\tilde{H}^{G}_{m}(\psi)\ :\ \tilde{H}^{G}_{m}(S^{V})\ \longrightarrow\
\tilde{H}^{G}_{m}(S^{V})$

is the identity.

###### Proof.

(i) The groups $\tilde{H}_{*}^{G}(S^{V})$ can be calculated from the reduced
cellular chain complex of a $G$-CW-structure on $S^{V}$ by taking first
$G$-fixed points and then homology. Since the reduced cellular chain complex
is concentrated in dimensions 0 through $d$, the vanishing claims follow.
Because the underlying chain complex calculates the reduced homology of the
underlying space $S^{V}$, the kernel of the top differential
$\delta_{d}:C_{d}^{\text{cell}}(S^{V};{\mathbb{F}}_{2})\longrightarrow
C_{d-1}^{\text{cell}}(S^{V};{\mathbb{F}}_{2})$ is 1-dimensional and
necessarily with trivial $G$-action. So the kernel of
$\delta_{d}^{G}:(C_{d}^{\text{cell}}(S^{V};{\mathbb{F}}_{2}))^{G}\longrightarrow(C_{d-1}^{\text{cell}}(S^{V};{\mathbb{F}}_{2}))^{G}$
is also 1-dimensional, yielding the second claim. The argument for constant
integer coefficients in (ii) is essentially the same; the orientability
assumption is equivalent to the condition that $G$ acts trivially on the group
of $d$-cycles in the underlying chain complex.

(iii) Because $\psi:S^{V}\longrightarrow S^{V}$ is a $G$-homotopy equivalence,
its class in the $G$-equivariant 0-stem is a unit. Because Bredon homology is
represented by the commutative $G$-ring spectrum
$H\underline{{\mathbb{F}}}_{2}$, the map $\tilde{H}^{G}_{*}(\psi)$ equals
multiplication by the Hurewicz image of $\langle\psi\rangle$ in
$\pi_{0}^{G}(H\underline{{\mathbb{F}}}_{2})$. The ring
$\pi_{0}^{G}(H\underline{{\mathbb{F}}}_{2})$ is isomorphic to
${\mathbb{F}}_{2}$, so 1 is its only unit. This proves the claim. ∎

######  Construction 1.3 (The representation-graded ring $H(G,\star)$).

We introduce notation to deal with the representation-grading. For a finite
group $G$, we let $J_{G}$ be the abelian monoid, under direct sum, of
isomorphism classes of $G$-representations with trivial $G$-fixed points. So
$J_{G}$ is freely generated by the isomorphism classes of nontrivial
irreducible $G$-representations.

We choose a representative for each element of $J_{G}$. For $\rho\in J_{G}$
and $m\geq 0$, we define

$H_{m}(G,\rho)\ =\ \tilde{H}_{m}^{G}(S^{V})\ ,$

where $V$ is the chosen representative of $\rho$. Proposition 1.2 (iii)
guarantees that this definition is independent of the representative up to
preferred isomorphism, induced by any isomorphism of representations.

The pairing of Bredon homology induces an associative, commutative and
bilinear pairing

$\cdot\ :\ H_{m}(G,\rho)\times H_{n}(G,\kappa)\ \longrightarrow\
H_{m+n}(G,\rho+\kappa)$

for $\rho,\kappa\in J_{G}$ and $m,n\geq 0$. This map is defined as the
composite

$\tilde{H}_{m}^{G}(S^{V})\times\tilde{H}_{n}^{G}(S^{W})\ \xrightarrow{\ \cdot\
}\ \tilde{H}_{m+n}^{G}(S^{V}\wedge S^{W})\ \cong\ \tilde{H}_{m+n}^{G}(S^{U})\
,$

where $V$, $W$ and $U$ are the chosen representatives of $\rho$, $\kappa$ and
$\rho+\kappa$, respectively. The last isomorphism is induced by a choice of
$G$-equivariant isomorphism $V\oplus W\cong U$; it is independent of the
choice by Proposition 1.2 (iii). We emphasize that this multiplication is
strictly commutative, i.e., for $x\in H_{m}(G,\rho)$ and $y\in
H_{n}(G,\kappa)$, the classes $x\cdot y$ and $y\cdot x$ are equal in the group
$H_{m+n}(G,\rho+\kappa)=H_{n+m}(G,\kappa+\rho)$. This, one more time, uses
that automorphisms of representation spheres are invisible in
$\tilde{H}^{G}_{*}$. The multiplication maps thus make the collection of
groups $H_{m}(G,\rho)$ into a commutative $({\mathbb{N}}\times J_{G})$-graded
${\mathbb{F}}_{2}$-algebra. We denote this object by $H(G,\star)$ and refer to
it as the representation-graded Bredon homology ring of the group $G$. We will
routinely abuse notation by identifying a $G$-representation $V$ with trivial
fixed points with its class in $J_{G}$; thus we shall write $H_{m}(G,V)$ for
$H_{m}(G,[V])$.

######  Remark 1.4.

Bredon homology with coefficients in a Mackey functor is represented by a
genuine $G$-spectrum, and hence can be extended to a homology theory for
$G$-spaces that is $RO(G)$-graded. Our results are only about the ‘homological
cone’ of the $RO(G)$-graded coefficient ring, i.e., the sector given by Bredon
homology of representation spheres. The homological cone has a much nicer
algebraic structure than the rest of the $RO(G)$-graded Bredon homology, which
tends to contain many nilpotent classes and trivial products. The homological
cone contains the pre-Euler and inverse Thom classes, so it determines the
geometric fixed points (obtained by inverting all pre-Euler classes), and
various other localizations, see Construction 3.1 below.

We recall two kinds of classes that exist for every $G$-representation, the
pre-Euler class and the inverse Thom class. Our pre-Euler class is also called
‘Euler class’ by other authors.

######  Construction 1.5 (Pre-Euler and inverse Thom classes).

We let $V$ be a $G$-representation. The pre-Euler class

$a_{V}\ \in\ H_{0}(G,V)$

is the image of the multiplicative unit $1\in H_{0}(G,0)$ under the
homomorphism induced by the based $G$-map $S^{0}\longrightarrow S^{V}$ that
sends the point $0$ to the $G$-fixed point $0$ in $S^{V}$. The pre-Euler class
can be zero, for example if $V$ has nonzero $G$-fixed points.

For $d=\dim(V)$, the inverse Thom class is the unique nonzero element

$t_{V}\ \in\ H_{d}(G,V)\ ,$

compare Proposition 1.2 (i). If $W$ is another $G$-representation, then

$a_{V}\cdot a_{W}\ =\ a_{V\oplus W}\text{\qquad and\qquad}t_{V}\cdot
t_{W}=t_{V\oplus W}$

in $H_{*}(G,V\oplus W)$.

In this paper, a character of a finite group $G$ is a group homomorphism
$\lambda:G\longrightarrow C=\\{\pm 1\\}$. We shall routinely confuse a
character $\lambda$ with the 1-dimensional $G$-representation on
${\mathbb{R}}$ in which $g\in G$ acts by multiplication by $\lambda(g)$.
Elementary abelian 2-groups are characterized among finite groups by the
property that all irreducible real representations are 1-dimensional, and
hence given by characters.

We will make use of the Bockstein homomorphism

$\beta\ :\ \tilde{H}_{m}^{G}(X)\ \longrightarrow\ \tilde{H}_{m-1}^{G}(X)$

associated to the short exact sequence of constant Mackey functors
$\underline{{\mathbb{F}}}_{2}\longrightarrow\underline{{\mathbb{Z}}}/4\longrightarrow\underline{{\mathbb{F}}}_{2}$.

######  Example 1.6.

Let $\lambda:G\longrightarrow C$ be a nontrivial character with kernel $K$.
The minimal $G$-CW-structure of $S^{\lambda}$ with two fixed 0-cells and one
1-cell with isotropy $K$ shows that for every abelian group $M$, the group
$\tilde{H}_{n}^{G}(S^{\lambda};\underline{M})$ is trivial for $n\neq 0,1$. And
it yields an exact sequence of Bredon homology groups:

$\displaystyle
0\longrightarrow\tilde{H}^{G}_{1}(S^{\lambda};\underline{M})\longrightarrow
H^{G}_{0}(G/K;\underline{M})\longrightarrow
H^{G}_{0}(G/G;\underline{M})\xrightarrow{\cdot
a_{\lambda}}\tilde{H}_{0}^{G}(S^{\lambda};\underline{M})\longrightarrow 0$

The middle two groups are coefficient groups of the Mackey functor, and thus
equal to $M$. The middle homomorphism is the transfer from $K$ to $G$ in the
Mackey functor $\underline{M}$, i.e., multiplication by the index $[G:K]=2$.
So the groups $\tilde{H}_{n}^{G}(S^{\lambda};\underline{M})$ in dimension 1
and 0 are isomorphic to the kernel and cokernel, respectively, of
multiplication by $2$ on $M$. For $M={\mathbb{F}}_{2}$ we conclude that
$H_{1}(G,\lambda)$ and $H_{0}(G,\lambda)$ are 1-dimensional, generated by the
inverse Thom class $t_{\lambda}$ and the pre-Euler class $a_{\lambda}$,
respectively.

We claim that the Bockstein homomorphism takes $t_{\lambda}$ to $a_{\lambda}$.
Indeed, the short exact sequence of Mackey functors
$\underline{{\mathbb{F}}}_{2}\longrightarrow\underline{{\mathbb{Z}}}/4\longrightarrow\underline{{\mathbb{F}}}_{2}$
yields an exact sequence of Bredon homology groups:

$\displaystyle\tilde{H}^{G}_{1}(S^{\lambda};\underline{{\mathbb{F}}}_{2})\
\xrightarrow{\ \beta\
}\tilde{H}^{G}_{0}(S^{\lambda};\underline{{\mathbb{F}}}_{2})\longrightarrow\tilde{H}^{G}_{0}(S^{\lambda};\underline{{\mathbb{Z}}}/4)\longrightarrow\tilde{H}^{G}_{0}(S^{\lambda};\underline{{\mathbb{F}}}_{2})\longrightarrow
0$

Since the four nontrivial groups in this sequence are all cyclic of order 2,
exactness implies that the Bockstein homomorphism is an isomorphism. Since
source and target are spanned by $t_{\lambda}$ and $a_{\lambda}$,
respectively, the Bockstein satisfies $\beta(t_{\lambda})=a_{\lambda}$.

######  Proposition 1.7.

Let $G$ be a finite group.

  1. (i)

If $V$ is an orientable $G$-representation, then $\beta(t_{V})=0$.

  2. (ii)

Let $T$ be a set of $G$-characters whose product is 1. Then

$\sum_{\lambda\in T}a_{\lambda}\cdot\bigg{(}\prod_{\mu\in
T\setminus\\{\lambda\\}}t_{\mu}\bigg{)}\ =\ 0\ .$

###### Proof.

(i) If $V$ is orientable, then by Proposition 1.2 (ii), the inverse Thom class
$t_{V}$ lifts to a class in the Bredon homology with constant integral
coefficients. So the image of $t_{V}$ under the Bockstein homomorphism is
trivial.

(ii) We let $V=\bigoplus_{\lambda\in T}\lambda$ be the sum of the characters
in the set $T$. The determinant of $V$ is the product of the characters in
$T$, which is trivial by hypothesis. So the $G$-representation $V$ is
orientable, and hence $\beta(t_{V})=0$ by part (i). The Bockstein homomorphism
is a derivation, in the sense that $\beta(x\cdot y)=\beta(x)\cdot
y+x\cdot\beta(y)$ for all classes $x\in\tilde{H}_{m}^{G}(X)$ and
$y\in\tilde{H}_{n}^{G}(Y)$. Applying the derivation property repeatedly and
using that $\beta(t_{\lambda})=a_{\lambda}$ shows that

$0\ =\ \beta(t_{V})\ =\ \beta\big{(}{\prod}_{\lambda\in T}t_{\lambda}\big{)}\
=\ {\sum}_{\lambda\in T}\ a_{\lambda}\cdot\big{(}{\prod}_{\mu\in
T\setminus\\{\lambda\\}}t_{\mu}\big{)}\ .\vspace*{-.5cm}$

∎

##  2\. Bredon homology for elementary abelian 2-groups

In this section we specialize from general finite groups to elementary abelian
2-groups, and we prove our two main results. Theorem 2.2 shows that the
representation-graded homology ring $H(A,\star)$ is a domain. Theorem 2.5
exhibits a presentation of $H(A,\star)$ as the quotient of a polynomial
${\mathbb{F}}_{2}$-algebra on the classes $a_{\lambda}$ and $t_{\lambda}$ for
all nontrivial $A$-characters $\lambda$, by an explicit minimal set of
homogeneous polynomial relations.

Many of our arguments involve bootstrapping information about Bredon homology
of a subgroup to the ambient group. In those arguments, we need to restrict
representations to subgroups, which typically creates new fixed points. If $V$
is a $G$-representation with $V^{G}=0$, and $K$ a subgroup of $G$, we set
$k=\dim(V^{K})$, and we let $V_{K}=V-V^{K}$ be the orthogonal complement of
the $K$-fixed points in $V$. The restriction homomorphism

$\operatorname{res}^{G}_{K}\ :\ H_{m}(G,V)\ \longrightarrow\ H_{m-k}(K,V_{K})$

is the composite

$\tilde{H}^{G}_{m}(S^{V})\ \xrightarrow{\operatorname{res}^{G}_{K}}\
\tilde{H}^{K}_{m}(S^{V})\ \cong\ \tilde{H}^{K}_{m}(S^{V_{K}}\wedge S^{k})\
\xrightarrow[\cong]{(-\wedge S^{k})^{-1}}\ \tilde{H}^{K}_{m-k}(S^{V_{K}})\ .$

The first isomorphism is induced by a choice of $K$-equivariant isomorphism
$V\cong V_{K}\oplus{\mathbb{R}}^{k}$; it is independent of this choice by
Proposition 1.2 (iii). The second isomorphism is the inverse of the suspension
isomorphism. The restriction homomorphism is multiplicative, and its effect on
inverse Thom and pre-Euler classes is given by

$\operatorname{res}^{G}_{K}(t_{V})\ =\ t_{V_{K}}\text{\quad
and\quad}\operatorname{res}^{G}_{K}(a_{V})\ =\ \begin{cases}a_{V_{K}}&\text{
if $V^{K}=0$, and}\\\ \ 0&\text{ if $V^{K}\neq 0$.}\end{cases}$

######  Proposition 2.1.

Let $A$ be an elementary abelian 2-group and let $W$ be an $A$-representation
with trivial fixed points.

  1. (i)

For every subgroup $B$ of $A$, the restriction homomorphism

$\operatorname{res}^{A}_{B}\ :\ H_{m}(A,W)\ \longrightarrow\ H_{m-k}(B,W_{B})$

is surjective, where $k=\dim(W^{B})$.

  2. (ii)

Suppose that $W=V\oplus\lambda$ for an $A$-representation $V$ and a nontrivial
$A$-character $\lambda$ with kernel $K$. Then for $k=\dim(V^{K})$, the
following sequence is exact:

$0\ \longrightarrow\ H_{m}(A,V)\ \xrightarrow{\ \cdot a_{\lambda}}\
H_{m}(A,V\oplus\lambda)\ \xrightarrow{\operatorname{res}^{A}_{K}}\
H_{m-k-1}(K,V_{K})\ \longrightarrow\ 0$

  3. (iii)

The ${\mathbb{F}}_{2}$-vector space $H_{m}(A,W)$ is spanned by the classes
$a_{U}\cdot t_{V}$ for all $A$-representations $U$ and $V$ such that $U\oplus
V=W$ and $m=\dim(V)$.

###### Proof.

We prove all three statements together by induction over the rank of $A$. The
induction starts when $A$ is the trivial group, in which case there is nothing
to show. Now we let $A$ be a nontrivial elementary abelian 2-group, and we
assume that parts (i)–(iii) hold for all proper subgroups of $A$.

We start by proving (i), where we may assume that $B$ is a proper subgroup of
$A$. By part (iii) for $B$, it suffices to show that all classes of the form
$a_{U}\cdot t_{V}$ are in the image of the restriction homomorphism, whenever
$U$ and $V$ are $B$-representations such that $U\oplus V=W_{B}$ and
$m-k=\dim(V)$. Because $W$ is a sum of 1-dimensional $A$-representations, we
may choose an $A$-equivariant decomposition $W=\bar{U}\oplus\bar{V}\oplus T$
such that $\operatorname{res}^{A}_{B}(\bar{U})\cong U$,
$\operatorname{res}^{A}_{B}(\bar{V})\cong V$, and $B$ acts trivially on $T$.
Then $\operatorname{res}^{A}_{B}(a_{\bar{U}}\cdot t_{\bar{V}}\cdot
t_{T})=a_{U}\cdot t_{V}$, and we have shown part (i) for $A$.

Now we prove (ii). Smashing the cofiber sequence of based $A$-spaces

$A/K_{+}\ \xrightarrow{\quad}\ S^{0}\ \xrightarrow{\quad}\ S^{\lambda}\
\longrightarrow\ A/K_{+}\wedge S^{1}$

with $S^{V}$ and applying $A$-equivariant Bredon homology yields a long exact
sequence:

$\dots\ \longrightarrow\ H_{m}(A,V)\ \xrightarrow{\ \cdot a_{\lambda}}\
H_{m}(A,V\oplus\lambda)\ \xrightarrow{\ \partial\ }\
\tilde{H}_{m-1}^{A}(S^{V}\wedge A/K_{+})\ \longrightarrow\ \dots$

The Wirthmüller and suspension isomorphisms identify the group
$\tilde{H}_{m-1}^{A}(S^{V}\wedge A/K_{+})$ with
$\tilde{H}_{m-k-1}^{K}(S^{V_{K}})=H_{m-k-1}(K,V_{K})$. Under this
identification, the boundary map $\partial$ becomes the restriction
homomorphism
$\operatorname{res}^{A}_{K}:H_{m}(A,V\oplus\lambda)\longrightarrow
H_{m-k-1}(K,V_{K})$, which is surjective by (i). So the long exact sequence
decomposes into short exact sequences, showing (ii).

We prove (iii) by induction on the dimension of $W$. For $W=0$, the groups
$H_{*}(A,0)$ consist of a single copy of ${\mathbb{F}}_{2}$ in dimension 0,
spanned by the multiplicative unit $1=t_{0}$. If $W$ is nonzero, we write
$W=V\oplus\lambda$ for an $A$-representation $V$ and a nontrivial
$A$-character $\lambda$, with kernel $K$. By part (i), the restriction map
$\operatorname{res}^{A}_{K}:H_{m-1}(A,V)\longrightarrow H_{m-k-1}(K,V_{K})$ is
surjective. So for every class $x\in H_{m}(A,W)$, there is a class $z\in
H_{m-1}(A,V)$ such that
$\operatorname{res}^{A}_{K}(z)=\operatorname{res}^{A}_{K}(x)$. Then

$\operatorname{res}^{A}_{K}(x+z\cdot t_{\lambda})\ =\
\operatorname{res}^{A}_{K}(x)+\operatorname{res}^{A}_{K}(z)\ =\ 0\ .$

Part (ii) provides a class $y\in H_{m}(A,V)$ such that $y\cdot
a_{\lambda}=x+z\cdot t_{\lambda}$. Because the dimension of $V$ is smaller
than that of $W$, the classes $y$ and $z$ are sums of products of $a$-classes
and $t$-classes. Hence the same is true for $x=y\cdot a_{\lambda}+z\cdot
t_{\lambda}$, and we have shown (iii). ∎

Now we can prove our first main result.

######  Theorem 2.2.

For every elementary abelian 2-group $A$, the representation-graded Bredon
homology ring $H(A,\star)$ is a domain.

###### Proof.

We call an element of $H(A,\star)$ regular if multiplication by it is
injective. We will show that all nonzero homogeneous elements of $H(A,\star)$
are regular. We argue by induction over the rank of $A$. For $A=\\{1\\}$, the
ring is the field ${\mathbb{F}}_{2}$, hence a domain.

Now we suppose that $A$ is nontrivial. Proposition 2.1 (ii) shows that the
pre-Euler classes of all nontrivial $A$-characters are regular. We show next
that all the inverse Thom classes $t_{\mu}$ are regular. So we let $y\in
H_{n}(A,W)$ be a homogeneous element such that $t_{\mu}\cdot y=0$, for some
$A$-representation $W$ with trivial fixed points. We argue by induction on the
dimension of $W$. If $W=0$ there is nothing to show because the integer-graded
part of Bredon homology is spanned by the multiplicative unit, and
$t_{\mu}\neq 0$. For $W\neq 0$ we write $W=U\oplus\lambda$ for some nontrivial
$A$-character $\lambda$, with kernel $K$. Then

$\operatorname{res}^{A}_{K}(t_{\mu})\cdot\operatorname{res}^{A}_{K}(y)\ =\
\operatorname{res}^{A}_{K}(t_{\mu}\cdot y)\ =\ 0\ .$

Because $\operatorname{res}^{A}_{K}(t_{\mu})$ is either 1 (if $\lambda=\mu$),
or the inverse Thom class of the restricted character $\mu|_{K}$ (if
$\lambda\neq\mu$), and because $K$ has smaller rank than $A$, the class
$\operatorname{res}^{A}_{K}(t_{\mu})$ is nonzero and regular, so
$\operatorname{res}^{A}_{K}(y)=0$. Proposition 2.1 (ii) provides a class $u\in
H_{n}(A,U)$ such that $y=u\cdot a_{\lambda}$. Then $t_{\mu}\cdot u\cdot
a_{\lambda}=t_{\mu}\cdot y=0$, so $t_{\mu}\cdot u$ because $a_{\lambda}$ is
regular. Because $U$ has smaller dimension than $W$, we deduce that $u=0$.
Hence also $y=0$, and this concludes the special case.

Now we show that a general nonzero homogeneous element $x\in H_{m}(A,V)$ is
regular, where $V$ is an $A$-representation with trivial fixed points. The
assumption that $x$ is nonzero implies that $m\leq\dim(V)$. We argue by
induction on $\dim(V)-m$. If $m=\dim(V)$, then $x=t_{V}$, and so $x$ is a
product of inverse Thom classes of $A$-characters. Since all the factors are
regular by the special case, so is $t_{V}$.

Now we suppose that $m<\dim(V)$. We distinguish two cases. In the first case
we suppose that there is an index 2 subgroup $K$ of $A$ such that
$\operatorname{res}^{A}_{K}(x)=0$. We let $\lambda$ be the $A$-character whose
kernel is $K$. Then also $\operatorname{res}^{A}_{K}(x\cdot t_{\lambda})=0$.
Proposition 2.1 (ii) provides a class $u\in H_{m+1}(A,V)$ such that $u\cdot
a_{\lambda}=x\cdot t_{\lambda}$. Since $x\neq 0$ and $t_{\lambda}$ is regular,
the class $u$ is nonzero. Because the integer degree of $u$ is larger than
that of $x$, the induction hypothesis shows that $u$ is regular. Since $u$,
$a_{\lambda}$ and $t_{\lambda}$ are regular, so is $x$.

In the second case we suppose that for every index 2 subgroup $K$ of $A$, the
restriction $\operatorname{res}^{A}_{K}(x)$ is nonzero. We consider a
homogeneous element $y\in H_{n}(A,W)$ such that $x\cdot y=0$, where $W$ is
another $A$-representation with trivial fixed points. We perform another
induction on the dimension of $W$. If $W=0$, then $y$ lies in the integer-
graded subring, so $y=0$ or $y=1$, and we are done. For $W\neq 0$ we write
$W=U\oplus\lambda$ for some nontrivial $A$-character $\lambda$, with kernel
$K$. Then

$\operatorname{res}^{A}_{K}(x)\cdot\operatorname{res}^{A}_{K}(y)\ =\
\operatorname{res}^{A}_{K}(x\cdot y)\ =\ 0\ .$

Because $\operatorname{res}^{A}_{K}(x)\neq 0$ and $H(K,\ast)$ is a domain by
induction, this implies $\operatorname{res}^{A}_{K}(y)=0$. Proposition 2.1
(ii) provides a class $u\in H_{n}(A,U)$ such that $y=u\cdot a_{\lambda}$. Thus
$x\cdot u\cdot a_{\lambda}=x\cdot y=0$. So $x\cdot u=0$ because $a_{\lambda}$
is regular. Because the dimension of $U$ is smaller than the dimension of $W$,
we conclude that $u=0$, and hence also $y=0$. ∎

Now we move on to our second main result, the minimal presentation of the
representation-graded Bredon homology ring $H(A,\star)$. Proposition 2.1 (iii)
shows that $H(A,\star)$ is generated as an ${\mathbb{F}}_{2}$-algebra by the
classes $a_{\lambda}$ and $t_{\lambda}$ for all nontrivial $A$-characters
$\lambda$. If $A=C$, there is only one nontrivial character, and then
$H(C,\star)$ is well known to be a polynomial algebra on the classes $a$ and
$t$. If $A$ is elementary abelian of rank at least 2, however, there are
nontrivial polynomial relations between the pre-Euler classes and the inverse
Thom classes.

In the following, it will be convenient to use the notation

$A^{\circ}\ =\ \operatorname{Hom}(A,C)\setminus\\{1\\}$

for the set of nontrivial characters of an elementary abelian 2-group.

######  Definition 2.3 (Minimally dependent sets of characters).

Let $A$ be an elementary abelian 2-group. A nonempty subset of $A^{\circ}$ is
dependent if it is linearly dependent as a subset of the vector space
$\operatorname{Hom}(A,C)$. The set is minimally dependent if it is dependent,
but no proper subset is dependent.

######  Remark 2.4.

If $T$ is a minimally dependent subset of $A^{\circ}$, then the product of all
its elements must be the trivial character. Indeed, the linear dependence of
$T$ means that some subset of it has product the trivial character; if this
were the case for some proper subset of $T$, then that subset would be
dependent, contradicting minimality.

Sets of nontrivial $A$-characters that have one or two elements have a
nontrivial product. So every dependent set of $A$-characters has a least three
elements. If $A$ has rank $r$, then every set of $r+1$ nontrivial
$A$-characters is dependent, and hence has a nonempty subset whose product is
the trivial character. So a minimally dependent set of $A$-characters has at
most $r+1$ elements.

The group $C^{2}$ has precisely one minimally dependent set of nontrivial
characters, the set of all three nontrivial characters. In Example 2.7 we
enumerate all 14 minimally dependent sets of nontrivial characters of the
group $C^{3}$. In Remark 2.8 we determine the number of minimally dependent
subsets of $A^{\circ}$ as a function of the rank of $A$.

The next theorem is our main result about representation-graded Bredon
homology, providing an explicit set of homogeneous polynomial relations
between the pre-Euler and inverse Thom classes, parameterized by minimally
dependent sets of characters. Moreover, these relations form a minimal
generating set for the ideal of all relations. Holler and Kriz determined the
Poincaré series of the multigraded ring $H(A,\star)$ in [4, Theorem 5]. We
have not investigated how to derive their formula for the Poincaré series from
our presentation.

######  Theorem 2.5.

Let $A$ be an elementary abelian 2-group.

  1. (i)

The representation-graded Bredon homology ring $H(A,\star)$ is generated as an
${\mathbb{F}}_{2}$-algebra by the classes $a_{\lambda}$ and $t_{\lambda}$ for
all nontrivial $A$-characters $\lambda$.

  2. (ii)

The kernel of the surjective homomorphism of ${\mathbb{F}}_{2}$-algebras

$\epsilon_{A}\ :\ {\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in A^{\circ}]\
\longrightarrow\ H(A,\star)$

is the ideal generated by the polynomials

$r(T)\ =\ \sum_{\lambda\in T}a_{\lambda}\cdot\bigg{(}\prod_{\mu\in
T\setminus\\{\lambda\\}}\,t_{\mu}\bigg{)}$

for all minimally dependent subsets $T$ of $A^{\circ}$.

  3. (iii)

Every set of homogeneous elements that generates the kernel of $\epsilon_{A}$
contains the polynomials $r(T)$ for all minimally dependent subsets $T$ of
$A^{\circ}$.

###### Proof.

Part (i) was shown in Proposition 2.1 (iii), and is repeated here for easier
reference.

(ii) For the course of the proof we write $I(A)$ for the ideal of the
polynomial ring ${\mathbb{F}}[a_{\mu},t_{\mu}\colon\mu\in A^{\circ}]$
generated by the polynomials $r(T)$ for all minimally dependent subsets $T$ of
$A^{\circ}$. Since minimally dependent sets of characters multiply to 1,
Proposition 1.7 (ii) shows that $I(A)\subseteq\ker(\epsilon_{A})$; so it
remains to show the reverse inclusion.

We argue by induction on the rank of $A$. The induction starts with the
trivial group, where there is nothing to show. Now we let $A$ be a nontrivial
elementary abelian 2-group, and we assume part (ii) for all proper subgroups
of $A$. In the inductive step, we shall make use of the polynomials

$r(T)\ =\ \sum_{\lambda\in T}a_{\lambda}\cdot\bigg{(}\prod_{\mu\in
T\setminus\\{\lambda\\}}\,t_{\mu}\bigg{)}$

for arbitrary subsets $T$ of $A^{\circ}$, not necessarily minimally dependent;
we alert the reader that if the elements of $T$ do not multiply to the trivial
character, then the polynomial $r(T)$ will typically not map to 0 under
$\epsilon_{A}$.

We let $\lambda$ be a nontrivial $A$-character, with kernel $K$. We let
${\mathcal{J}}$ denote the homogeneous ideal in the ring
${\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in
A^{\circ}\setminus\\{\lambda\\}]$ consisting of those elements $y$ such that
$\operatorname{res}^{A}_{K}(\epsilon_{A}(y))=0$. We emphasize that elements of
${\mathcal{J}}$ are polynomials that do not involve the variables
$a_{\lambda}$ nor $t_{\lambda}$. We shall prove two properties of this ideal:

(a) The ideal ${\mathcal{J}}$ is generated by the polynomials $r(T)$ for all
subsets $T$ of $A^{\circ}\setminus\\{\lambda\\}$ such that either $T$ is
minimally dependent, or $T\cup\\{\lambda\\}$ is minimally dependent.

(b) In the graded ring ${\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in
A^{\circ}]$, the relation $t_{\lambda}\cdot{\mathcal{J}}\subset\langle
I(A),a_{\lambda}\rangle$ holds.

Proof of (a). We write $J_{A\backslash\lambda}$ for the submonoid of $J_{A}$
consisting of the classes of $A$-representations $V$ that do not involve the
character $\lambda$. Equivalently, $J_{A\backslash\lambda}$ contains the
$A$-representations with trivial $A$-fixed points such that $V^{K}=0$. The
restriction homomorphism $\operatorname{res}^{A}_{K}:{\mathbb{N}}\times
J_{A\backslash\lambda}\longrightarrow{\mathbb{N}}\times J_{K}$ lets us inflate
$({\mathbb{N}}\times J_{K})$-graded rings $R$ to $({\mathbb{N}}\times
J_{A\backslash\lambda})$-graded rings $(\operatorname{res}^{A}_{K})^{*}(R)$ by
setting

$(\operatorname{res}^{A}_{K})^{*}(R)(k,V)\ =\ R(k,V|_{K})\ .$

We alert the reader that a grading-inflated polynomial algebra is no longer a
polynomial algebra. With this grading convention, a morphism of
$({\mathbb{N}}\times J_{A\backslash\lambda})$-graded
${\mathbb{F}}_{2}$-algebras

$\rho^{A}_{K}\ :\ {\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in
A^{\circ}\setminus\\{\lambda\\}]\ \longrightarrow\
(\operatorname{res}^{A}_{K})^{*}\left({\mathbb{F}}_{2}[a_{\nu},t_{\nu}\colon\nu\in
K^{\circ}]\right)$

is given by sending $a_{\mu}$ to $a_{\mu|K}$ and $t_{\mu}$ to $t_{\mu|K}$.
Moreover, the ideal ${\mathcal{J}}$ is precisely the kernel of the composite
homomorphism

$\displaystyle{\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in
A^{\circ}\setminus\\{\lambda\\}]\ $ $\displaystyle\xrightarrow{\ \rho^{A}_{K}\
}\
(\operatorname{res}^{A}_{K})^{*}\left({\mathbb{F}}_{2}[a_{\nu},t_{\nu}\colon\nu\in
K^{\circ}]\right)$
$\displaystyle\xrightarrow{(\operatorname{res}^{A}_{K})^{*}(\epsilon_{K})}(\operatorname{res}^{A}_{K})^{*}(H(K,\star))\
.$

The kernel of $\rho^{A}_{K}$ is the ideal generated by the homogeneous
polynomials

$a_{\mu}t_{\mu\lambda}+t_{\mu}a_{\mu\lambda}\ =\ r(\\{\mu,\mu\lambda\\})$

for all $\mu\in A^{\circ}\setminus\\{\lambda\\}$. The set
$\\{\mu,\mu\lambda\\}$ is independent and
$\\{\mu,\mu\lambda\\}\cup\\{\lambda\\}$ is minimally dependent. So the
polynomials $r(\\{\mu,\mu\lambda\\})$ are among those of the second kind
specified in (a).

By the inductive hypothesis for the subgroup $K$, the kernel of
$\epsilon_{K}:{\mathbb{F}}_{2}[a_{\nu},t_{\nu}\colon\nu\in
K^{\circ}]\longrightarrow H(K,\star)$ is generated by the polynomials $r(S)$
for all minimally dependent subsets $S$ of $K^{\circ}$. So the kernel of the
homomorphism

$(\operatorname{res}^{A}_{K})^{*}(\epsilon_{K})\ :\
(\operatorname{res}^{A}_{K})^{*}\left({\mathbb{F}}_{2}[a_{\nu},t_{\nu}\colon\nu\in
K^{\circ}]\right)\ \longrightarrow\
(\operatorname{res}^{A}_{K})^{*}(H(K,\star))$

is the ideal generated by the same polynomials $r(S)$, but each occurring
multiple times in different degrees, namely for all subsets $T$ of
$A^{\circ}\setminus\\{\lambda\\}$ such that

${\bigoplus}_{\mu\in T}\ \mu|_{K}\ =\ {\bigoplus}_{\nu\in S}\ \nu\ .$

This condition means that for each character $\nu\in S$, the set $T$ contains
exactly one of the two extensions of $\nu$ to an $A$-character. Because $S$ is
a minimally dependent subset of $K^{\circ}$, the $A$-character

${\prod}_{\mu\in T}\ \mu\quad\in\ \operatorname{Hom}(A,C)$

then restricts to the trivial character on the subgroup $K$. Hence this
product is either 1 or $\lambda$. In the first case, $T$ is a minimally
dependent subset of $A^{\circ}\setminus\\{\lambda\\}$. In the second case, $T$
is an independent subset of $A^{\circ}\setminus\\{\lambda\\}$ such that
$T\cup\\{\lambda\\}$ is minimally dependent.

We have now shown that some of the polynomials $r(T)$ with $T$ as specified in
(a) generate the kernel of $\rho^{A}_{K}$, and the images of the others under
$\rho^{A}_{K}$ generate the kernel of
$(\operatorname{res}^{A}_{K})^{*}(\epsilon_{K})$. So all those polynomials
together generate the kernel of the composite, and hence the ideal
${\mathcal{J}}$. This completes the proof of (a).

Proof of (b). It suffices to show that the two types of generating polynomial
for ${\mathcal{J}}$ specified in (a) have the desired property. If $T$ is a
minimally dependent subset of $A^{\circ}\setminus\\{\lambda\\}$, then the
polynomial $r(T)$ belongs to the ideal $I(A)$. Hence also $t_{\lambda}\cdot
r(T)\in I(A)$. If $T$ is a subset of $A^{\circ}\setminus\\{\lambda\\}$ such
that $T\cup\\{\lambda\\}$ is minimally dependent, then the relation

$t_{\lambda}\cdot r(T)\ =\ r(T\cup\\{\lambda\\})\ +\
a_{\lambda}\cdot{\prod}_{\mu\in T}\ t_{\mu}$

holds in the ring ${\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in A^{\circ}]$,
and witnesses that the left hand side lies in the ideal $\langle
I(A),a_{\lambda}\rangle$. This completes the proof of (b).

Now we complete the inductive step, showing that $\ker(\epsilon_{A})\subseteq
I(A)$. We prove this for all homogeneous pieces $H_{m}(A,W)$, where $W$ is an
$A$-representation with $W^{A}=0$, by induction on the dimension of $W$. For
$W=0$ we are considering integer degrees, where source and target of
$\epsilon_{A}$ both consist only of a copy of ${\mathbb{F}}_{2}$ in degree 0.

Now we suppose that $W\neq 0$. We let $f$ be a homogeneous polynomial in
${\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in A^{\circ}]$ of degree $(k,W)$
with $\epsilon_{A}(f)=0$. We choose an $A$-character $\lambda$ and an
$A$-representation $V$ with $W=V\oplus\lambda$. We let $K$ denote the kernel
of $\lambda$. We write $f=t_{\lambda}\cdot y+a_{\lambda}\cdot z$ for some
homogeneous polynomials $y,z$ of degrees $(k-1,V)$ and $(k,V)$, respectively.
Then

$\operatorname{res}^{A}_{K}(\epsilon_{A}(y))\ =\
\operatorname{res}^{A}_{K}(t_{\lambda}\epsilon_{A}(y)+a_{\lambda}\epsilon_{A}(z))\
=\ \operatorname{res}^{A}_{K}(\epsilon_{A}(f))\ =\ 0\ .$

Case 1: The $A$-character $\lambda$ occurs in $W$ with multiplicity at least
2. Then there is an $A$-representation such that $V=U\oplus\lambda$. Because
$\operatorname{res}^{A}_{K}(\epsilon_{A}(y))=0$, there is a class $u\in
H_{k-1}(A,U)$ such that $\epsilon_{A}(y)=a_{\lambda}\cdot u$. Then

$a_{\lambda}\cdot(t_{\lambda}\cdot u+\epsilon_{A}(z))\ =\
t_{\lambda}\cdot\epsilon_{A}(y)+a_{\lambda}\cdot\epsilon_{A}(z)\ =\
\epsilon_{A}(f)\ =\ 0\ .$

Because multiplication by $a_{\lambda}$ is injective, we deduce that
$t_{\lambda}\cdot u=\epsilon_{A}(z)$. We choose a homogeneous polynomial $g$
in ${\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in A^{\circ}]$ of degree
$(k-1,U)$ such that $\epsilon_{A}(g)=u$. Then $y+a_{\lambda}g$ and
$z+t_{\lambda}g$ lie in the kernel of $\epsilon_{A}$. Since $U\oplus\lambda=V$
has smaller dimension than $W$, the classes $y+a_{\lambda}g$ and
$z+t_{\lambda}g$ are in the ideal $I(A)$, by induction. So

$f\ =\ t_{\lambda}\cdot y+a_{\lambda}\cdot z\ =\
t_{\lambda}\cdot(y+a_{\lambda}g)+a_{\lambda}\cdot(z+t_{\lambda}g)$

also lies in the ideal $I(A)$.

Case 2: The $A$-character $\lambda$ occurs in $W$ with multiplicity 1. Then
$\lambda$ does not occur in $V$, and so the polynomial $y$ does not involve
the variable $a_{\lambda}$ nor $t_{\lambda}$. Because
$\operatorname{res}^{A}_{K}(\epsilon_{A}(y))=0$, the polynomial $y$ belongs to
the ideal ${\mathcal{J}}$. By property (b) of the ideal ${\mathcal{J}}$, the
class $t_{\lambda}\cdot y$ then belongs to the ideal generated by $I(A)$ and
$a_{\lambda}$. Hence there is a homogeneous polynomial $h$ of degree $(k,V)$
such that $t_{\lambda}\cdot y$ is congruent to $a_{\lambda}\cdot h$ modulo the
ideal $I(A)$. So the polynomial $f$ in the kernel of $\epsilon_{A}$ satisfies

$f\ =\ t_{\lambda}\cdot y+a_{\lambda}\cdot z\ \equiv\
a_{\lambda}\cdot(h+z)\text{\qquad modulo $I(A)$.}$

Because $I(A)$ is contained in the kernel of $\epsilon_{A}$, we deduce the
relation

$a_{\lambda}\cdot\epsilon_{A}(h+z)\ =\ \epsilon_{A}(a_{\lambda}\cdot(h+z))\ =\
\epsilon_{A}(f)\ =\ 0\ .$

Because multiplication by $a_{\lambda}$ is injective, we conclude that
$\epsilon_{A}(h+z)=0$. Since $V$ has smaller dimension than
$W=V\oplus\lambda$, the class $h+z$ lies in the ideal $I(A)$ by induction. So
also $a_{\lambda}(h+z)$, and hence the class $f$, lies in the ideal $I(A)$.
This finishes the inductive step, and hence the proof of part (ii).

(iii) Source and target of the homomorphism $\epsilon_{A}$ are graded by the
abelian monoid of isomorphism classes of $A$-representations. This abelian
monoid is free on the classes of the $A$-characters and thus admits a
compatible partial order by declaring $V\leq W$ if $V$ is isomorphic to a
direct summand of $W$. Hence all products of a homogeneous polynomial $f$ with
other homogeneous elements of ${\mathbb{F}}_{2}[a_{\mu},t_{\mu}\colon\mu\in
A^{\circ}]$ have degrees greater or equal than that of $f$. By (ii), the
kernel of $\epsilon_{A}$ is generated in gradings that contain the sum of a
linearly dependent set of characters. So the kernel of $\epsilon_{A}$ is
nontrivial only in degrees that contain the sum of a linearly dependent set of
characters. The generating relations $r(T)$ specified in (ii) lie in degrees
that are minimal with this property, and they are the unique nontrivial
elements in the kernel of $\epsilon_{A}$ in their degrees. So each of the
relations $r(T)$ specified in (ii) is necessary to generate the kernel of
$\epsilon_{A}$. ∎

We take the time to go through the presentation of Theorem 2.5 for elementary
abelian 2-groups of rank at most 3. For the group $C$ with two elements, the
representation-graded Bredon homology ring is well studied, and a polynomial
algebra on the classes $a$ and $t$,

$H(C,\star)\ =\ {\mathbb{F}}_{2}[a,t]\ .$

This calculation is originally due to Stong (unpublished) and reproved by
several authors; the earliest published reference we know of is [6, Section
2].

######  Example 2.6 (Rank 2).

The group $C^{2}$ has precisely one dependent set of nontrivial characters,
the set $\\{p_{1},p_{2},\mu\\}$ of all three nontrivial characters, and this
set is minimally dependent. So the presentation of $H(C^{2},\star)$ given by
Theorem 2.5 has only one relation: the map from
${\mathbb{F}}_{2}[a_{1},a_{2},a_{\mu},t_{1},t_{2},t_{\mu}]$ that takes the
polynomial generators to the classes with the same names factors through an
isomorphism of ${\mathbb{F}}_{2}$-algebras

${\mathbb{F}}_{2}[a_{1},a_{2},a_{\mu},t_{1},t_{2},t_{\mu}]/(a_{1}t_{2}t_{\mu}+t_{1}a_{2}t_{\mu}+t_{1}t_{2}a_{\mu})\
\cong\ H(C^{2},\star)\ .$

To the best of the authors’ knowledge, this is already a new result about the
representation-graded Bredon homology for the group $C^{2}$.

######  Example 2.7 (Rank 3).

As a proof of concept, we make the minimal presentation given by Theorem 2.5
completely explicit for $A=C^{3}$. In this case there are 14 polynomial
generators, namely the classes $a_{\lambda}$ and $t_{\lambda}$ for each of the
seven nontrivial characters $\lambda$, and there is also a total of 14 minimal
relations. Minimally dependent subsets for $C^{3}$ have either three or four
elements. There are seven minimally dependent subsets of cardinality 3; if
$p_{i}$ denotes the projection of $C^{3}$ to the $i$th factor, then these sets
are

$\displaystyle\\{p_{1},\ p_{2},\ p_{1}p_{2}\\},\quad\\{p_{1},\ p_{3},\
p_{1}p_{3}\\},\quad\\{p_{2},\ p_{3},\ p_{2}p_{3}\\},\quad\\{p_{1},\
p_{2}p_{3},\ p_{1}p_{2}p_{3}\\},$ $\displaystyle\\{p_{2},\ p_{1}p_{3},\
p_{1}p_{2}p_{3}\\},\quad\\{p_{3}.\ p_{1}p_{2},\ p_{1}p_{2}p_{3}\\}\text{\quad
and\quad}\\{p_{1}p_{2},\ p_{1}p_{3},\ p_{2}p_{3}\\}\ .$

This gives seven minimal relations of the form

$a_{\alpha}t_{\beta}t_{\gamma}\ +\ t_{\alpha}a_{\beta}t_{\gamma}\ +\
t_{\alpha}t_{\beta}a_{\gamma}$

where $\\{\alpha,\beta,\gamma\\}$ runs over the above seven sets. And there
are also seven minimally dependent subsets with four elements, namely

$\displaystyle\\{p_{1},\ p_{2},\ p_{3},\ p_{1}p_{2}p_{3}\\},\quad\\{p_{1},\
p_{2},\ p_{1}p_{3},\ p_{2}p_{3}\\},\quad\\{p_{1},\ p_{3},\ p_{1}p_{2},\
p_{2}p_{3}\\},$ $\displaystyle\\{p_{2},\ p_{3},\ p_{1}p_{2},\
p_{1}p_{3}\\},\quad\\{p_{1},\ p_{1}p_{2},\ p_{1}p_{3},\ p_{1}p_{2}p_{3}\\},$
$\displaystyle\\{p_{2},\ p_{1}p_{2},\ p_{2}p_{3},\
p_{1}p_{2}p_{3}\\}\text{\quad and\quad}\\{p_{3},\ p_{1}p_{3},\ p_{2}p_{3},\
p_{1}p_{2}p_{3}\\}\ .$

Each such set $\\{\alpha,\beta,\gamma,\delta\\}$ gives a minimal relation of
the form

$a_{\alpha}t_{\beta}t_{\gamma}t_{\delta}\ +\
t_{\alpha}a_{\beta}t_{\gamma}t_{\delta}\ +\
t_{\alpha}t_{\beta}a_{\gamma}t_{\delta}\ +\
t_{\alpha}t_{\beta}t_{\gamma}a_{\delta}\ .$

######  Remark 2.8 (The number of minimally dependent sets).

We let $A$ be an elementary abelian 2-group of rank $r$. As we explained in
Remark 2.4, minimally dependent subsets of $A^{\circ}$ have at least 3
elements, and at most $r+1$ elements. We shall now count how many of these
there are. We consider $2\leq k\leq r$. Every minimally dependent subset with
$k+1$ elements is obtained from a linearly independent $k$-element subset $S$
of $A^{\circ}$ by adding to it the product of all its members
$\sigma=\prod_{\lambda\in S}\lambda$. The resulting minimally dependent subset
$S\cup\\{\sigma\\}$ can be obtained in $k+1$ different ways from an unordered
linearly independent $k$-element set, depending on which of the elements plays
the role of the product. An $r$-dimensional ${\mathbb{F}}_{2}$-vector space
has

$(2^{r}-1)(2^{r}-2)\cdot\ldots\cdot(2^{r}-2^{k-1})/k!$

linearly independent $k$-element subsets. Hence there are

$(2^{r}-1)(2^{r}-2)\cdot\ldots\cdot(2^{r}-2^{k-1})/(k+1)!$

minimally dependent subsets of $A^{\circ}$ with $k+1$ elements. So the total
number of relations in the minimal presentation of $H(A,\star)$ given by
Theorem 2.5 is

$\sum_{k=2}^{r}\,\prod_{i=1}^{k}\frac{2^{r}-2^{i-1}}{i+1}\ .$

The following table shows the number $2(2^{r}-1)$ of polynomial generators of
the representation-graded ring $H(A,\star)$, and the above number of minimal
relations for small ranks:

rank$(A)$ | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
no. of variables | 2 | 6 | 14 | 30 | 62 | 126 | 254
no. of relations | 0 | 1 | 14 | 308 | 20,336 | 4,994,472 | 4,610,816,280

While the number of relations grows very quickly with the rank, this is in a
sense due to the growing number of automorphisms. Indeed, the automorphism
group of $A$ acts transitively on the minimally dependent sets of nontrivial
$A$-characters of fixed cardinality. So up to automorphisms of $A$, there is
only one relation of degree $k+1$ for all $2\leq k\leq\text{rank}(A)$.

##  3\. Bredon homology with pre-Euler and inverse Thom classes inverted

In this section we study certain localizations of the representation-graded
Bredon homology ring $H(A,\star)$. We fix a subgroup $B$ of $A$ and consider
the ring obtained by inverting all pre-Euler classes that restrict
nontrivially to $B$, and all inverse Thom classes that restrict trivially to
$B$. Our presentation of $H(A,\star)$ yields a presentation of the localized
ring, see Theorem 3.2.

The localization by inverting all pre-Euler classes is also known as the
geometric fixed point ring of the Eilenberg-MacLane spectrum
$H\underline{{\mathbb{F}}}_{2}$; Holler and Kriz [4] previously obtained a
presentation of it, which we recover as the special case $B=A$. The mixed
localizations were considered by Balmer and Gallauer [1]; we improve their [1,
Theorem 8.13], for the prime $2$, from a ‘presentation modulo nilpotence’ to
an actual presentation, see Remark 3.3.

######  Construction 3.1 (Localizations).

We let $B$ be a subgroup of an elementary abelian 2-group $A$. We let $H(A|B)$
be the integer-graded part of the localization of the representation-graded
ring $H(A,\star)$ obtained by inverting the following classes:

  * •

all classes $a_{\lambda}$ for all $A$-characters $\lambda$ such that
$\lambda|_{B}$ is nontrivial, and

  * •

all classes $t_{\lambda}$ for all $A$-characters $\lambda$ such that
$\lambda|_{B}$ is trivial.

Every element of $H_{k}(A|B)$ is then of the form $x/a_{V}t_{W}$ for some
$A$-representation $V$ with $V^{B}=0$, some $A$-representation $W$ with
$W^{A}=0$ on which $B$ acts trivially, and some $x\in H_{k+\dim(W)}(A,V\oplus
W)$. These elements satisfy the relations

$x/a_{V}t_{W}\ =\ (x\cdot
a_{\bar{V}}t_{\bar{W}})/a_{V\oplus\bar{V}}t_{W\oplus\bar{W}}$

for all $A$-representations $\bar{V}$ and $\bar{W}$ with the corresponding
properties. The localizations introduce enough graded units so that the
representation-grading effectively collapses to an integer-grading. In other
words, we are not losing any information by restricting attention to the
integer-graded subrings of the localizations.

The following theorem ‘localizes’ the presentation of the representation-
graded ring $H(A,\star)$ from Theorem 2.5 to a presentation of the integer-
graded ring $H(A|B)$. A noticeable feature is that those relations in the
presentation of $H(A,\star)$ that involve minimally dependent sets of four or
more characters become redundant in the localization.

######  Theorem 3.2.

For every subgroup $B$ of an elementary abelian 2-group $A$, the graded ring
$H(A|B)$ is a domain. The ring $H(A|B)$ is generated by

  * •

the homogeneous elements $x_{\lambda}=t_{\lambda}/a_{\lambda}$ of degree $1$,
for $\lambda\in A^{\circ}$ with $\lambda|_{B}\neq 1$, and

  * •

the homogeneous elements $e_{\lambda}=a_{\lambda}/t_{\lambda}$ of degree $-1$,
for $\lambda\in A^{\circ}$ with $\lambda|_{B}=1$.

The ideal of relations between these generators is generated by the following
polynomials:

  * •

$x_{\alpha}x_{\beta}+x_{\alpha}x_{\gamma}+x_{\beta}x_{\gamma}$ for all triples
of nontrivial $A$-characters such that $\alpha\cdot\beta\cdot\gamma=1$ and
such that $\alpha$, $\beta$ and $\gamma$ are nontrivial on $B$;

  * •

$x_{\alpha}+x_{\beta}+x_{\alpha}x_{\beta}e_{\gamma}$ for all triples of
nontrivial $A$-characters such that $\alpha\cdot\beta\cdot\gamma=1$, such that
$\alpha$ and $\beta$ are nontrivial on $B$, and $\gamma|_{B}=1$;

  * •

$e_{\alpha}+e_{\beta}+e_{\gamma}$ for all triples of nontrivial $A$-characters
such that $\alpha\cdot\beta\cdot\gamma=1$ and
$\alpha|_{B}=\beta|_{B}=\gamma|_{B}=1$.

###### Proof.

The multigraded ring $H(A,\star)$ is a domain by Theorem 2.2, hence so is any
localization at a multiplicative subset of homogeneous elements. Since
$H(A|B)$ is a subring of such a localization, it is a domain, too.

We write $\chi_{+}=\\{\lambda\in A^{\circ}\colon\lambda|_{B}\neq 1\\}$ and
$\chi_{-}=\\{\lambda\in A^{\circ}\colon\lambda|_{B}=1\\}$. If we take the
presentation of $H(A,\star)$ given by Theorem 2.5, invert the relevant pre-
Euler and inverse Thom classes, and restrict to integer-gradings, we recognize
$H(A|B)$ as the quotient of the polynomial ring
${\mathbb{F}}_{2}[x_{\lambda},e_{\mu}\colon\lambda\in\chi_{+},\mu\in\chi_{-}]$
by the ideal generated by the polynomials

$\displaystyle\bar{r}(T)\ $ $\displaystyle=\ r(T)/(\prod_{\mu\in
T\cap\chi_{+}}a_{\mu}\cdot\prod_{\mu\in T\cap\chi_{-}}t_{\mu})$
$\displaystyle=\ (\sum_{\lambda\in
T\cap\chi_{+}}\prod_{\mu\in(T\cap\chi_{+})\setminus\\{\lambda\\}}x_{\mu})\ +\
(\sum_{\lambda\in T\cap\chi_{-}}e_{\lambda})\cdot\\!\prod_{\mu\in
T\cap\chi_{+}}x_{\mu}$

for all minimally dependent subsets $T$ of $A^{\circ}$. Among these are the
minimally dependent subsets $T=\\{\alpha,\beta,\gamma\\}$ that have three
elements. We note that because $\alpha\cdot\beta\cdot\gamma=1$, whenever two
of $\alpha$, $\beta$ and $\gamma$ are trivial on $B$, then so is the third. So
the relations for minimally dependent subsets with three elements are the ones
from the statement of the theorem.

In the rest of the proof we show that the polynomials $\bar{r}(T)$ for
minimally dependent subsets $T$ with more than three elements are in the ideal
generated by those with three elements. We argue by induction on the
cardinality of $T$, and we let $T$ be a minimally dependent subset of
$A^{\circ}$ with at least 4 elements. We pick two distinct elements
$\alpha\neq\beta$ from $T$; by minimality, the product
$\gamma=\alpha\cdot\beta$ then does not belong to $T$. We set
$S=T\setminus\\{\alpha,\beta\\}$. We claim that the polynomial $\bar{r}(T)$
lies in the ideal generated by $\bar{r}(S\cup\\{\gamma\\})$ and
$\bar{r}(\\{\alpha,\beta,\gamma\\})$. The sets $S\cup\\{\gamma\\}$ and
$\\{\alpha,\beta,\gamma\\}$ are both minimally dependent, and both have fewer
elements than $T$. So by induction, $\bar{r}(T)$ belongs to the ideal
generated by the ternary relations.

It remains to prove the claim. We distinguish two cases, depending on whether
the restriction of $\gamma$ to $B$ is trivial or not. If $\gamma|_{B}=1$, we
exploit the polynomial relation

$t_{\gamma}\cdot r(T)\ =\ t_{\alpha}t_{\beta}\cdot r(S\cup\\{\gamma\\})\ +\
r(\\{\alpha,\beta,\gamma\\})\cdot\prod_{\lambda\in S}t_{\lambda}$

that holds by inspection. Because the class $t_{\gamma}$ is among those being
inverted, this relation shows that the polynomial $\bar{r}(T)$ lies in the
ideal generated by $\bar{r}(S\cup\\{\gamma\\})$ and
$\bar{r}(\\{\alpha,\beta,\gamma\\})$. If $\gamma|_{B}\neq 1$, we exploit the
relation

$a_{\gamma}\cdot r(T)\ =\ (a_{\alpha}t_{\beta}+t_{\alpha}a_{\beta})\cdot
r(S\cup\\{\gamma\\})\ +\ r(\\{\alpha,\beta,\gamma\\})\cdot r(S)$

that also holds by inspection. Because the class $a_{\gamma}$ is among those
being inverted, this relation shows that the polynomial $\bar{r}(T)$ lies in
the ideal generated by $\bar{r}(S\cup\\{\gamma\\})$ and
$\bar{r}(\\{\alpha,\beta,\gamma\\})$. ∎

######  Remark 3.3 (Relation to the work of Balmer and Gallauer).

In [1], Balmer and Gallauer also study the representation-graded Bredon
homology ring $H(A,\star)$ as input for their computation of the Balmer
spectrum of the tt-category of permutation modules; this ring is called the
twisted cohomology ring of $A$ in [1, Definition 3.16], and denoted
$H^{\bullet\bullet}(A)$. The work of Balmer-Gallauer also covers elementary
abelian $p$-groups for odd primes $p$, which we do not consider. The
connection to Bredon homology comes from the fact that the homotopy category
of permutation $A$-modules is equivalent to the homotopy category of modules
in genuine $A$-spectra over the Eilenberg-MacLane spectrum
$H\underline{{\mathbb{F}}}_{2}$ for the constant Mackey functor. Under this
equivalence, the invertible object $u_{\lambda}$ introduced in [1, Definition
3.3] corresponds to the representation sphere $S^{\lambda}$. Using our
notation, Balmer-Gallauer prove the relation
$a_{\alpha}t_{\beta}t_{\gamma}+t_{\alpha}a_{\beta}t_{\gamma}+t_{\alpha}t_{\beta}a_{\gamma}=0$
in [1, Lemma 8.4], for all triples of $A$-characters whose product is 1.

The localization $H(A|B)$ is introduced as $\mathcal{O}^{\bullet}_{A}(B)$ in
[1, Definition 5.9]. In [1, Construction 8.5], Balmer-Gallauer define a ring
$\underline{\mathcal{O}}_{A}^{\bullet}(B)$ by generators and relations as in
the presentation of Theorem 3.1. Then they show in [1, Theorem 8.13] that the
morphism
$\underline{\mathcal{O}}_{A}^{\bullet}(B)\longrightarrow\mathcal{O}_{A}^{\bullet}(B)$
becomes an isomorphism after modding out the respective nilradicals. In the
notation of Balmer-Gallauer, the content of our Theorem 3.1 is that
$\underline{\mathcal{O}}_{A}^{\bullet}(B)\longrightarrow\mathcal{O}_{A}^{\bullet}(B)$
is already an isomorphism, without the need to divide out any ideal. In
particular, our theorem confirms the expectation formulated in [1, Remark
8.15].

For every genuine equivariant ring spectrum, the localization of the
representation-graded homotopy ring at the pre-Euler classes of all nontrivial
irreducible representations is isomorphic to the so-called geometric fixed
point ring, see [3, Proposition 3.20]. In particular, for every elementary
abelian 2-group $A$, the localization $H(A|A)$ is isomorphic to the geometric
fixed point ring $\Phi^{A}_{*}(H\underline{{\mathbb{F}}}_{2})$ of the
Eilenberg-MacLane ring spectrum of the constant Mackey functor
$\underline{{\mathbb{F}}}_{2}$. Holler and Kriz gave a presentation of this
ring in [4, Theorem 2]. Their result is the special case $A=B$ of Theorem 3.2:

######  Corollary 3.4.

[4, Theorem 2] Let $A$ be an elementary abelian 2-group. The graded ring
$\Phi^{A}_{*}H=H(A|A)$ is generated by the homogeneous elements
$x_{\lambda}=t_{\lambda}/a_{\lambda}$ of degree $1$, for all nontrivial
$A$-characters $\lambda$. The ideal of relations between these generators is
generated by the quadratic polynomials

$x_{\alpha}x_{\beta}+x_{\alpha}x_{\gamma}+x_{\beta}x_{\gamma}$

for all triples of nontrivial $A$-characters such that
$\alpha\cdot\beta\cdot\gamma=1$.

If we specialize Theorem 3.2 to the other extreme $B=\\{1\\}$, we see that
$H(A|\\{1\\})$ is generated by the Euler classes
$e_{\lambda}=a_{\lambda}/t_{\lambda}$ for all nontrivial $A$-characters
$\lambda$, and the ideal of relations is generated by the linear polynomials
$e_{\alpha}+e_{\beta}=e_{\gamma}$ for all triples of nontrivial $A$-characters
such that $\alpha\cdot\beta=\gamma$. So if $A$ has rank $r$, and
$\lambda_{1},\dots,\lambda_{r}$ is a basis of the vector space of
$A$-characters, then already the Euler classes
$e_{\lambda_{1}},\dots,e_{\lambda_{r}}$ generate $H(A|\\{1\\})$, and there are
no further relations between these. In other words, the ring $H(A|\\{1\\})$
obtained by inverting all inverse Thom classes is an
${\mathbb{F}}_{2}$-polynomial algebra in the classes
$e_{\lambda_{1}},\dots,e_{\lambda_{r}}$, which agrees with the group
cohomology ring $H^{*}(A;{\mathbb{F}}_{2})$.

## References

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  * [3] J P C Greenlees, J P May, Equivariant stable homotopy theory. Handbook of algebraic topology, 277–323, North-Holland, Amsterdam, 1995.
  * [4] J Holler, I Kriz, On $RO(G)$-graded equivariant ”ordinary” cohomology where $G$ is a power of ${\mathbb{Z}}/2$. Algebr. Geom. Topol. 17 (2017), no. 2, 741–763.
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