# Computing Equivariant Homotopy with a Splitting Method

Yutao Liu
University of Chicago
Email<EMAIL_ADDRESS>
#### Abstract:

We develop a new method in the computation of equivariant homotopy, which is
based on the splitting of cofiber sequences associated to universal spaces in
the category of equivariant spectra. In particular, we use this method to
compute the homotopy of $H\underline{\mathbb{Z}}$ for $G=D_{2p}$ and $A_{5}$.

###### Contents

  1. 1 Introduction
  2. 2 Universal spaces and homological properties
    1. 2.1 Universal $G$-spaces
    2. 2.2 $\mathbb{Z}$-graded homology with coefficients in $A_{G}$
    3. 2.3 Homology of $E\mathscr{F}$ in degree $0$
    4. 2.4 Homology of $E\mathscr{F}$ in positive degrees
  3. 3 Splitting cofiber sequences
    1. 3.1 A categorical splitting
    2. 3.2 Splitting criterion
  4. 4 Splitting method
    1. 4.1 Decomposition of $E\mathscr{F}_{+}\wedge X$
    2. 4.2 Input of the computation
    3. 4.3 Computability of $H\underline{\mathbb{Z}}$
    4. 4.4 Computibility of $MU_{G}$
    5. 4.5 Some Examples
    6. 4.6 An algebraic point of view
  5. 5 Computation of $\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$
    1. 5.1 Decomposition into $C_{2}$ and $C_{p}$
    2. 5.2 A complete expression
  6. 6 Computation of $\pi_{\bigstar}^{A_{5}}(H\underline{\mathbb{Z}})$
  7. 7 Mackey functor valued homology
    1. 7.1 One example
    2. 7.2 Another example
  8. A Homology of a join
  9. B Computations on $K_{4}$-homology
    1. B.1 The positive cone
    2. B.2 The negative cone

##  1 Introduction

Let $G$ be a compact Lie group. In [LMM81], Lewis, May and McClure defined
Eilenberg-Maclane $RO(G)$-graded cohomology with Mackey functor coefficients,
which gave rise to a fundamental but hard question in equivariant stable
homotopy: computing $RO(G)$-graded cohomology (or homology) of a point, which
is equivalent to the equivariant homotopy of Eilenberg-Maclane spectra. We
make this question more precise:

######  Question 1.1.

Let $M$ be an arbitrary $G$-Mackey functor. Compute

$\pi_{V}^{G}(HM):=[S^{V},HM]^{G}$

for any virtual $G$-representation $V$. Use $\bigstar$ to denote the
$RO(G)$-grading. Then we want to compute $\pi_{\bigstar}^{G}(HM)$ as an
$RO(G)$-graded abelian group.

Moreover, we have the Mackey functor valued homotopy
$\underline{\pi}_{\bigstar}^{G}(HM)$ given by

$\underline{\pi}_{V}^{G}(HM)(G/H):=[G/H_{+}\wedge
S^{V},HM]^{G}\cong[S^{V},HM]^{H}.$

A more general question is to compute $\underline{\pi}_{\bigstar}^{G}(HM)$ as
an $RO(G)$-graded Mackey functor.

In addition, when $M$ is chosen as a Green functor, which is a monoid in the
category of Mackey functors, there will be extra multiplicative structures on
the homotopy groups of $HM$. A final question is to compute
$\pi_{\bigstar}^{G}(HM)$ or $\underline{\pi}_{\bigstar}^{G}(HM)$ as an
$RO(G)$-graded ring or Green functor.

Common choices of $M$ include constant Mackey functors
$\underline{\mathbb{Z}}$, $\underline{\mathbb{F}_{p}}$,
$\underline{\mathbb{Q}}$ and the Burnside ring Mackey functor $A_{G}$, which
is the unit object in the category of Mackey functors. All these choices admit
natural Green functor structures.

When $G$ is a cyclic group with prime order, the question is completely solved
in multiple ways. Partial computations have been done for larger groups like
$C_{p^{2}}$, $C_{2}^{n}$, dihedral group $D_{2p}$, and quaternion group
$Q_{8}$, for which we refer to [Ell20], [Geo21], [HK17], [KL20], [Lu21],
[Zen18], and [Zou18]. In this paper, we will develop a new method which
computes equivariant stable homotopy by decomposition, which works for general
equivariant spectra:

######  Theorem 1.2.

(Main Theorem) Assume that $G$ is a finite group whose order contains multiple
prime factors. For any $G$-spectrum $X$ and $G$-virtual representation $V$,
$\pi_{V}^{G}(X)$ can be expressed as the limit of a diagram with objects
$\pi_{V|_{H}}^{H}(X)$ and $\pi_{(V|_{L})^{K}}^{L/K}(\Phi^{K}X)$ for proper
$H\subset G$ and $K\lhd L\subset G$ (with proper localizations) such that
$|H|,|L/K|$ are prime powers, and maps induced by restriction, conjugacy and
localization.

In other words, the whole equivariant stable homotopy theory with finite
acting groups is encoded in the cases of $p$-groups.

A more detailed statement is given in Theorem 4.6.

The idea of our method is motivated by the study of rational $G$-spectra for
finite $G$, which is first discussed by Greenlees and May:

######  Theorem 1.3.

[GM95] There is an orthogonal basis $\\{e_{H}:H\subset G\\}$ of the rational
Burnside ring, which only contains idempotent elements. For any rational
$G$-spectrum $X$, define $e_{H}X$ as

$e_{H}X:=colim(X\xrightarrow{e_{H}}X\xrightarrow{e_{H}}X\xrightarrow{e_{H}}...)$

Then we have

$X\simeq\bigvee_{H}e_{H}X$

and

$[X,Y]^{G}\cong\prod_{H}[e_{H}X,e_{H}Y]^{G}$

with one $H$ chosen from each conjugacy class of subgroups.

This theorem is further studied by Barnes in [Bar08], which reproves the
splitting above by some topological constructions and sets up an algebraic
model which completely describes the behavior of rational $G$-spectra.

One important idea in Barnes’ proof is to use the universal $G$-space
$E\mathscr{F}_{H}$, which is characterized by the fixed point subspaces:

$(E\mathscr{F}_{H})^{K}\simeq\begin{cases}S^{0},\text{ if }K\text{ is
conjugate to }H,\\\ *,\text{ otherwise.}\end{cases}$

######  Theorem 1.4.

[Bar08] For rational $G$-spectra $X,Y$, we have an isomorphism

$[X,Y]^{G}\cong\prod_{H}[E\mathscr{F}_{H}\wedge X,E\mathscr{F}_{H}\wedge
Y]^{G}$

with one $H$ chosen from each conjugacy class. Moreover, the functor

$X\mapsto\bigvee_{H}E\mathscr{F}_{H}\wedge X$

is symmetric monoidal.

The computation in the rational world is not hard since the category of
rational $G$-spectra splits into small and simple pieces (with one piece
corresponding to each conjugacy class of subgroups). In fact, in order to
obtain this full splitting, it suffices to invert all prime factors of $|G|$
instead of applying rationalization.

Our idea comes from a weaker splitting. If we just invert some prime factors
of $|G|$, sometimes the category of $G$-spectra will still split, but into
some less simple pieces. If we choose the inverted primes properly, the
homotopy of the split pieces may still remain computable. Thus we can compute
the localized homotopy by collecting all the pieces.

Each localization loses torsion information at the inverted primes, but that
information is retained at other localizations and no torsion information is
lost when we glue together different localizations. That is, when $|G|$
contains multiple prime factors, we can apply the idea above multiple times
with different prime factors inverted each time. The unlocalized homotopy can
be recovered by collecting different localized data.

For example, let $G=D_{2p}$. When inverting $2$ or $p$, we get two different
splittings in the category of $G$-spectra. Both make the equivariant homotopy
computable. For any $G$-spectrum $X$, we can compute
$\pi_{\bigstar}^{G}(X)[1/2]$ and $\pi_{\bigstar}^{G}(X)[1/p]$. The unlocalized
$\pi_{\bigstar}^{G}(X)$ can be computed by the following pullback diagram:

$\textstyle{\pi_{\bigstar}^{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\bigstar}^{G}(X)[1/p]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\bigstar}^{G}(X)[1/2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{\bigstar}^{G}(X)[1/2,1/p]}$

The key element in our splitting method is the universal space $E\mathscr{F}$,
where $\mathscr{F}$ is a family of subgroups of $G$. There is a natural
cofiber sequence

$E\mathscr{F}_{+}\rightarrow S^{0}\rightarrow\widetilde{E\mathscr{F}},$

where $\widetilde{E\mathscr{F}}$ is the unreduced suspension of
$E\mathscr{F}$. When certain prime factors of $|G|$ are inverted, the cofiber
sequence above will split after taking $\Sigma^{\infty}$. Thus we have

$X\simeq(E\mathscr{F}_{+}\wedge X)\vee(\widetilde{E\mathscr{F}}\wedge X)$

for any $G$-spectrum $X$.

The inverted prime factors depend on the choice of $\mathscr{F}$. When we
invert all prime factors of $|G|$, we can choose all families of subgroups
together, which will result in a full splitting of $G$-spectra and recover
Theorem 1.4. In our computation, we invert all but one prime factor and make
special choices of families. Details will be provided in sections 3 and 4.

#### Structure of the paper:

In section 2, we will introduce the universal space $E\mathscr{F}$. We are
especially interested in the equivariant homology of $E\mathscr{F}$ with
coefficients in the Burnside ring Mackey functor $A_{G}$, which is the
starting point of our splitting method. To be precise, we will prove:

######  Proposition 1.5.

When $|G|$ is inverted, the Mackey functor valued homology

$\underline{H}_{*}^{G}(E\mathscr{F};A_{G})$

concentrates in degree $0$. Moreover, the trivial map
$E\mathscr{F}\rightarrow{*}$ induces an inclusion

$\underline{H}_{0}^{G}(E\mathscr{F};A_{G})\hookrightarrow\underline{H}_{0}^{G}(*;A_{G})\cong
A_{G}$

which makes $\underline{H}_{0}^{G}(E\mathscr{F};A_{G})$ into a direct summand
of $A_{G}$.

In section 3, we will study the cofiber sequence

$E\mathscr{F}_{+}\rightarrow S^{0}\rightarrow\widetilde{E\mathscr{F}}.$

We will give a necessary and sufficient criterion about which primes we need
to invert in order to split the cofiber sequence above after applying
$\Sigma^{\infty}$.

Section 4 is the core of the paper, in which our splitting method is fully
explained. We will explain the choice of families and how to decompose the
split pieces into the data of smaller subgroups of $G$. The splitting method
itself is not a computation, but a machine which uses the computations on
smaller groups to obtain the data on larger groups. The input of the machine
consists of equivariant homotopy on $p$-subgroups of $G$ with restriction and
conjugacy maps. We will also discuss the computability of
$H\underline{\mathbb{Z}}$ and $MU_{G}$ with our splitting method.

In sections 5 and 6, we will apply our splitting method to compute the
homotopy of $H\underline{\mathbb{Z}}$ for $G=D_{2p}$ and $A_{5}$.

In section 7, we will explain how to compute the Mackey functor valued
homotopy of $H\underline{\mathbb{Z}}$. The input of our machine becomes the
Mackey functor valued
$\underline{\pi}_{\bigstar}^{H}(H\underline{\mathbb{Z}})$ for proper subgroups
$H$ of $G$. The structure maps in
$\underline{\pi}_{\bigstar}^{G}(H\underline{\mathbb{Z}})$ can be expressed by
the structure maps in
$\underline{\pi}_{\bigstar}^{H}(H\underline{\mathbb{Z}})$. We will explicitly
compute the Mackey functor valued
$\underline{\pi}_{V}^{A_{5}}(H\underline{\mathbb{Z}})$ for some special
choices of $V$.

#### List of computations:

$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$: Theorem 5.7.

$\pi_{\bigstar}^{A_{5}}(H\underline{\mathbb{Z}})$: Theorems 6.2, 6.3, 6.4,
B.7, and B.15.

General computations on $D_{2p},A_{5}$-spectra: Theorems 4.16, 4.17, and 4.18.

#### Notations:

In this paper, we use $*$ when the homotopy or homology is graded over
$\mathbb{Z}$, and use $\bigstar$ when graded over $RO(G)$. We add an underline
to express the Mackey functor valued homotopy or homology: $\underline{H}$,
$\underline{\pi}$.

To be more precise, let $X$ be an unbased $G$-space and $M$ be a Mackey
functor. Define the Mackey functor valued equivariant homology of $X$ with
coefficients in $M$ as

$\underline{H}_{*}^{G}(X;M):=\underline{\pi}_{*}^{G}(X_{+}\wedge HM).$

So we have

$\underline{H}_{*}^{G}(X;M)(G/H)=[\Sigma^{*}G/H_{+},HM\wedge X_{+}]^{G},$
$H_{*}^{G}(X;M)=\underline{H}_{*}^{G}(X;M)(G/G).$

Here $HM$ is the equivariant Eilenberg-Maclane spectrum corresponding to $M$.

In this paper, $G$ will always be a finite group.

#### Acknowledgment:

The author would like to thank his PhD advisor, Peter May, for numerous
suggestions on the organization and editing of this paper.

##  2 Universal spaces and homological properties

We introduce the universal space $E\mathscr{F}$ and study its equivariant
homology in this section. The most important properties are given in
Proposition 2.13 and Theorem 2.16.

###  2.1 Universal $G$-spaces

######  Definition 2.1.

A family $\mathscr{F}$ is a collection of subgroups of $G$ which is closed
under conjugation and taking subgroups. The corresponding universal space
$E\mathscr{F}$ is an unbased $G$-space such that $(E\mathscr{F})^{H}$ is
contractible for all $H\in\mathscr{F}$ and empty otherwise.

######  Remark 2.2.

When $\mathscr{F}$ only contains the trivial subgroup, $E\mathscr{F}$ becomes
$EG$, which is a contractible space with free $G$-action.

One construction of $E\mathscr{F}$ is given in [Dieck72] as a join of spaces.
We will explain this idea later and use it to prove some important properties.
An alternative construction is given by [Elm83] as a categorical bar
construction.

We can fully characterize $E\mathscr{F}$ by fixed point subspaces:

######  Lemma 2.3.

The universal space $E\mathscr{F}$ is unique up to weak equivalence.

#### Proof:

Assume that $X,Y$ are $G$-spaces with the same conditions on fixed point
subspaces as $E\mathscr{F}$.

If we have a $G$-map $X\rightarrow Y$, its restriction on each fixed point
subspace is a weak equivalence since $X,Y$ have the same types of fixed point
subspaces as either $*$ or $\emptyset$. Thus this map is a weak
$G$-equivalence.

If we do not have a map between $X,Y$, consider $X\times Y$, which is another
$G$-space with the same conditions on fixed point subspaces. The above
argument shows that the projections $X\times Y\rightarrow X$ and $X\times
Y\rightarrow Y$ are weak $G$-equivalences. Therefore, $E\mathscr{F}$ is unique
up to weak equivalence. $\Box$

We assume $E\mathscr{F}$ to be a $G$-CW complex by applying the $G$-CW
approximation.

######  Lemma 2.4.

Any $G$-self map of $E\mathscr{F}$ is $G$-homotopic to the identity.

We need the following definition first:

######  Definition 2.5.

Let $\mathcal{O}_{G}$ be the category of $G$-orbits. Define an
$\mathcal{O}_{G}$-space to be a contravariant functor from $\mathcal{O}_{G}$
to topological spaces.

######  Theorem 2.6.

[May96, VI.6] The following pair of functors

$\Phi:(G-spaces)\rightleftarrows(\mathcal{O}_{G}-spaces):\Psi$

defined by $\Phi(X)(G/H):=X^{H}$ and $\Psi(T):=T(G/\\{e\\})$ form a Quillen
equivalence.

#### Proof of Lemma 2.4:

Construct an $\mathcal{O}_{G}$-space $T$ by setting

$T(G/H)=\begin{cases}*,\text{ if }H\in\mathscr{F}\\\ \emptyset,\text{
otherwise}\end{cases}$

Let $CT$ be a cofibrant approximation of $T$. Then

$\Psi(CT)^{H}=\Phi\Psi(CT)(G/H)\cong CT(G/H)\cong\begin{cases}*,\text{ if
}H\in\mathscr{F}\\\ \emptyset,\text{ otherwise}\end{cases}$

Recall that

$\Phi E\mathscr{F}(G/H)=(E\mathscr{F})^{H}\simeq\begin{cases}*,\text{ if
}H\in\mathscr{F}\\\ \emptyset,\text{ otherwise}\end{cases}$

The unique map $\Phi E\mathscr{F}\rightarrow T$ induces $\Phi
E\mathscr{F}\rightarrow CT$, whose left adjoint
$E\mathscr{F}\rightarrow\Psi(CT)$ becomes a $G$-equivalence.

So we have the following equivalences of homotopy classes on unbased maps:

$[E\mathscr{F},E\mathscr{F}]^{G}\cong[E\mathscr{F},\Psi(CT)]^{G}\cong[\Phi
E\mathscr{F},CT]^{\mathcal{O}_{G}}\cong[\Phi
E\mathscr{F},T]^{\mathcal{O}_{G}},$

which contains a single element.

Thus the identity map is the only self-map of $E\mathscr{F}$ up to
$G$-homotopy. $\Box$

Notice that $E\mathscr{F}\times E\mathscr{F}$ also appears as a universal
space for $\mathscr{F}$. Thus the above two lemmas imply:

######  Lemma 2.7.

The universal space $E\mathscr{F}$ is a $G$-topological semigroup up to
homotopy.

Usually $E\mathscr{F}$ is not a Hopf space since there is no unit. But under
proper localizations, $\Sigma^{\infty}E\mathscr{F}_{+}$ will appear as a
homotopy ring spectrum. We will explain this idea in section 3.

Notice that the based spaces $E\mathscr{F}_{+}$ and $\widetilde{E\mathscr{F}}$
are also characterized by their fixed point subspaces:

$(E\mathscr{F}_{+})^{H}\simeq\begin{cases}S^{0},\text{ if }H\in\mathscr{F},\\\
*,\text{ otherwise.}\end{cases}$
$(\widetilde{E\mathscr{F}})^{H}\simeq\begin{cases}*,\text{ if
}H\in\mathscr{F},\\\ S^{0},\text{ otherwise.}\end{cases}$

Thus we can apply the same argument as above, but in a based version:

######  Proposition 2.8.

Any self $G$-map of $E\mathscr{F}_{+}$ or $\widetilde{E\mathscr{F}}$ which is
a $G$-equivalence is $G$-homotopic to the identity.

The based spaces $E\mathscr{F}_{+}$ and $\widetilde{E\mathscr{F}}$ are based
$G$-topological semigroups up to $G$-homotopy. In addition, the natural map
$S^{0}\rightarrow\widetilde{E\mathscr{F}}$ commutes with the following
products up to $G$-homotopy:

$S^{0}\wedge S^{0}\xrightarrow{\simeq}S^{0},$
$\widetilde{E\mathscr{F}}\wedge\widetilde{E\mathscr{F}}\xrightarrow{\simeq}\widetilde{E\mathscr{F}}.$

###  2.2 $\mathbb{Z}$-graded homology with coefficients in $A_{G}$

We recall the definition of the Burnside ring and the corresponding Mackey
functor:

######  Definition 2.9.

For any finite group $G$, the collection of isomorphism classes of finite
$G$-sets forms a commutative monoid, with addition induced by disjoint unions.
The Burnside ring $A(G)$ is defined to be the group completion of this monoid.
We can express $A(G)$ as a free $\mathbb{Z}$-module, whose basis corresponds
to isomorphism classes of $G$-orbits. We use $\\{G/H\\}$ to denote the basis
element of the orbit $G/H$.

The Burnside ring Mackey functor $A_{G}$ is defined by $A_{G}(G/H):=A(H)$. The
Mackey functor structure is given by the following maps:

For any $L\subset H$, the transfer map

$T_{L}^{H}:A(L)\rightarrow A(H)$

sends $\\{L/K\\}$ to $\\{H/K\\}$ for any $K\subset L$.

The restriction map

$R_{L}^{H}:A(H)\rightarrow A(L)$

sends $\\{H/N\\}$ to itself, but viewed as an $L$-space.

When $L=g^{-1}Hg$, we also have an isomorphism

$C_{L}^{H}:A(L)\rightarrow A(H)$

sending $\\{L/K\\}$ to $\\{H/gKg^{-1}\\}$ for any $K\subset L$.

The Cartesian product of finite $G$-sets makes $A(G)$ into a ring. This
multiplicative structure also makes $A_{G}$ into a Green functor, which is a
monoid under the box product.

######  Proposition 2.10.

For a $G$-CW complex $X$, the integer graded homology of $X$ with coefficients
in $A_{G}$ can be computed as

$H_{*}^{G}(X;A_{G})\cong\bigoplus_{H}H_{*}(X^{H}/W_{G}H;\mathbb{Z})$

with one $H$ chosen from each conjugacy class of subgroups of $G$. Here
$W_{G}H$ is the Weyl group of $H$ in $G$.

As a Mackey functor,

$\underline{H}_{*}^{G}(X;A_{G})(G/L)\cong
H_{*}^{L}(X;A_{L})\cong\bigoplus_{K}H_{*}(X^{K}/W_{L}K;\mathbb{Z})$

with one $K$ chosen from each conjugacy class of subgroups of $L$.

#### Proof:

It suffices to prove the first equation. The second one can be proved by the
same argument.

When we view $A_{G}$ as a covariant coefficient system, the restriction maps
are removed. For any $G$-map $G/L_{1}\rightarrow G/L_{2}$ induced by the
multiplication of $g\in G$ (where we require $g^{-1}L_{1}g\subset L_{2}$), we
have an induced map

$A(L_{1})\rightarrow A(L_{2})$

sending each $\\{L_{1}/H\\}$ to $\\{L_{2}/g^{-1}Hg\\}$. Therefore, as a
coefficient system, $A_{G}$ can be decomposed as

$A_{G}=\bigoplus_{H}A_{G}^{H}$

with one $H\subset G$ chosen from each conjugacy class. Here $A_{G}^{H}$ is
the sub-coefficient system of $A_{G}$ such that each $A_{G}^{H}(G/L)\subset
A(L)$ is generated by all $\\{L/g^{-1}Hg\\}$ with $g^{-1}Hg\subset L$.

Now we have

$H_{*}^{G}(X;A_{G})\cong\bigoplus_{H}H_{*}^{G}(X;A_{G}^{H}).$

It suffices to prove

$H_{*}^{G}(X;A_{G}^{H})\cong H_{*}(X^{H}/W_{G}H;\mathbb{Z})$

for any $H\subset G$.

As described in [Wil75], the equivariant homology can be computed in a
cellular way:

Let $\underline{C}_{*}(X)$ be the chain complex of contravariant coefficient
systems such that

$\underline{C}_{*}(X)(G/H):=C_{*}(X^{H}),$

where $C_{*}(X^{H})$ is the cellular chain complex of $X^{H}$ and the orbit
maps are sent to conjugacies and inclusions of subcomplexes. For any covariant
coefficient system $M$, $H_{*}^{G}(X;M)$ is the homology of the chain complex:

$C_{*}^{G}(X;M):=\underline{C}_{*}(X)\otimes_{\mathcal{O}_{G}}M,$

where the tensor product is taken over the category $\mathcal{O}_{G}$ of
$G$-orbits. To be more precise, we have

$C_{*}^{G}(X;M):=\left(\bigoplus_{L\subset G}C_{*}(X^{L})\otimes
M(G/L)\right)/\sim.$

For any $G$-map $f:G/L_{1}\rightarrow G/L_{2}$, the equivalence relation
identifies $f^{*}a\otimes b$ and $a\otimes f_{*}b$ for all $a\in
C_{*}(X^{L_{2}})$ and $b\in M(G/L_{1})$.

Now we choose $M=A_{G}^{H}$. Since each $M(G/L)$ is freely generated by
$\\{L/g^{-1}Hg\\}$ and the structure maps send these basis elements to each
other, $C_{*}^{G}(X;M)$ is the free $\mathbb{Z}$-module generated by the
equivalence classes of $e\otimes\\{L/g^{-1}Hg\\}$, for all cells $e$ in
$X^{L}$ and $g^{-1}Hg\subset L$.

We can simplify the expression of $C_{*}^{G}(X;A_{G})$ by the following four
facts:

(1) The free $\mathbb{Z}$-module $A_{G}^{H}(G/H)$ is generated by a single
element $\\{H/H\\}$.

(2) Each $e\otimes\\{L/g^{-1}Hg\\}$ is identified with $ge\otimes\\{H/H\\}$ by
the equivalence relation.

(3) If there exists another $g^{\prime}\in G$ such that

$ge\otimes\\{H/H\\}\sim e\otimes\\{L/g^{-1}Hg\\}\sim
g^{\prime}e\otimes\\{H/H\\},$

then $\\{L/g^{-1}Hg\\}=\\{L/(g^{\prime})^{-1}Hg^{\prime}\\}$. Thus

$g^{-1}Hg=l^{-1}((g^{\prime})^{-1}Hg^{\prime})l$

for some $l\in L$. So $g^{\prime}lg^{-1}\in W_{G}H$. Since $e$ is $L$-fixed,
we have $(g^{\prime}lg^{-1})(ge)=g^{\prime}e$. Thus $ge$ and $g^{\prime}e$ are
in the same $W_{G}H$-orbit.

(4) The converse of (3) is also true: $e_{1}\otimes\\{H/H\\}$ and
$e_{2}\otimes\\{H/H\\}$ are identified if $e_{1},e_{2}$ are in the same
$W_{G}H$-orbit.

Now we have a 1-1 correspondence between the basis of $C_{*}^{G}(X;A_{G}^{H})$
and the cells in $X^{H}/W_{G}H$. Therefore, we get

$C_{*}^{G}(X;A_{G}^{H})\cong C_{*}(X^{H}/W_{G}H).$

Taking the homology on both sides gives us the required equation. $\Box$

######  Remark 2.11.

Proposition 2.10 follows from the fact that the underlying coefficient system
of $A_{G}$ splits into $A_{G}^{H}$. However, such splitting cannot be lifted
to the Mackey functor level. Thus the decomposition does not work for
cohomology with coefficients in $A_{G}$.

In fact, since the orbit spectra $(-)/G$ of genuine $G$-spectra can only be
defined after passing from a complete universe to the trivial universe,

$X\mapsto H_{*}(X^{H}/W_{G}H)$

is a homology theory only for $G$-spaces, and hence cannot be represented by
any $G$-spectrum.

###  2.3 Homology of $E\mathscr{F}$ in degree $0$

Since $E\mathscr{F}$ only has empty and contractible fixed point subspaces,
Proposition 2.10 helps us to compute the 0th degree equivariant homology of
$E\mathscr{F}$ explicitly:

######  Proposition 2.12.

The trivial map $E\mathscr{F}\rightarrow\\{*\\}$ implies an inclusion

$\underline{H}_{0}^{G}(E\mathscr{F};A_{G})\hookrightarrow\underline{H}_{0}^{G}(*;A_{G})\cong
A_{G}.$

The image of $\underline{H}_{0}^{G}(E\mathscr{F},A_{G})(G/L)$ in $A(L)$ is
generated by all $\\{L/K\\}\in A(L)$ with $K\in\mathscr{F}$.

We already have a small splitting here:

######  Proposition 2.13.

When $|G|$ is inverted, $\underline{H}_{0}^{G}(E\mathscr{F};A_{G})$ is a
direct summand of $A_{G}$ as Mackey functors.

######  Notation 2.14.

We use $M_{\mathscr{F}}$ to denote the Mackey functor
$\underline{H}_{0}^{G}(E\mathscr{F};A_{G})$ and use the same notation for its
localizations.

When $M_{\mathscr{F}}$ appears to be a direct summand of $A_{G}$, denote its
complement as $N_{\mathscr{F}}$. A more concrete description of
$N_{\mathscr{F}}$ is given in the proof of Proposition 2.13 below.

#### Proof of Proposition 2.13:

Assume that $|G|$ is inverted everywhere. For any $H\subset G$, define a
linear map

$\chi_{H}:A(G)\rightarrow\mathbb{Z}$

which sends each $G$-set $S$ to $|S^{H}|$.

Let $s_{(K,H)}$ denote $\chi_{K}(\\{G/H\\})=|G/H|^{K}$, which can be computed
as the product between $|W_{G}H|$ and the number of subgroups of $G$
containing $K$ and in the conjugacy class of $H$. Thus $s_{(K,H)}\neq 0$ if
and only if $K$ is sub-conjugate to $H$.

When $|G|$ is inverted, all $|W_{G}H|$ and non-zero $s_{(K,H)}$ become
invertible. We choose elements $e_{H}\in A(G)$ inductively by defining

$e_{H}=|W_{G}H|^{-1}\left(\\{G/H\\}-\sum_{K}s_{(K,H)}^{-1}e_{K}\right)$

with one $K$ chosen from each conjugacy class that contains a proper subgroup
of $H$.

By induction, we have

$\chi_{K}(e_{H})=\begin{cases}1,\text{ if }K\text{ is conjugate to }H\\\
0,\text{ otherwise}\end{cases}$

Since the dimension of $A(G)$ agrees with the number of conjugacy classes, the
collection of $e_{H}$, with one $H$ chosen from each conjugacy class, forms a
basis of $A(G)$.

In general, for any $H\subset L\subset G$, define a linear map

$\chi_{H}^{L}:A(L)\rightarrow\mathbb{Z}$

which sends each $L$-set to the size of its $H$-fixed subset. We can get a
similar basis $\\{e_{H}^{L}\\}$, with one $H$ chosen from each conjugacy class
of subgroups of $L$, such that

$\chi_{K}^{L}(e_{H}^{L})=\begin{cases}1,\text{ if }K\text{ is conjugate to
}H\text{ in }L\\\ 0,\text{ otherwise}\end{cases}$

Now we discuss how these maps interact with transfer and restriction maps:

For any $K\subset L_{1}\subset L_{2}\subset G$, the restriction map
$R_{L_{1}}^{L_{2}}$ keeps each $L_{2}$-set but views it as an $L_{1}$-set,
hence does not change its $K$-fixed subset. So we have

$\chi_{K}^{L_{2}}=\chi_{K}^{L_{1}}\circ R_{L_{2}}^{L_{1}}.$

On the other hand, for any $K\subset L_{1}$, the elements in $\\{L_{1}/K\\}$
have isotropy groups conjugate to $K$ inside $L_{1}$. The elements in
$T_{L_{1}}^{L_{2}}(\\{L_{1}/K\\})=\\{L_{2}/K\\}$ have isotropy groups
conjugate to $K$ inside $L_{2}$. There may be more isotropy groups. But these
additional isotropy groups are still chosen from the conjugacy class of $K$
inside $G$. According to the construction of the basis element $e_{H}^{L}$, we
have

$\chi_{H}^{L_{2}}(T_{L_{1}}^{L_{2}}e_{K}^{L_{1}})\neq 0\text{ only if }H\text{
is conjugate to a subgroup of }K\text{ in }G$

Let

$N(L):=\bigcap_{H\in\mathscr{F},H\subset L}\ker\chi_{H}^{L}$

In other words, $N(L)$ is generated by $e_{H}^{L}$ for all $H\subset L$ and
$H\notin\mathscr{F}$. The above discussion tells us that $N(L)$ is closed
under transfer and restriction maps. Thus we get a sub-Mackey functor whose
value at $G/L$ agrees with $N(L)$. Denote that as $N_{\mathscr{F}}$.

Notice that for each $L\subset G$, $e_{H}^{L}\in N(L)=N_{\mathscr{F}}(G/L)$ if
$H\notin\mathscr{F}$, $e_{H}^{L}\in M_{\mathscr{F}}(G/L)$ if
$H\in\mathscr{F}$, according to Proposition 2.12. Moreover, consider any
nontrivial element

$a_{1}\\{L/K_{1}\\}+a_{2}\\{L/K_{2}\\}+...+a_{n}\\{L/K_{n}\\}\in
M_{\mathscr{F}}(G/L)$

with $a_{1},a_{2},...,a_{n}\neq 0$, $K_{1},K_{2},...,K_{n}$ in different
conjugacy classes in $\mathscr{F}$, and
$|K_{1}|\leq|K_{2}|\leq...\leq|K_{n}|$. The map $\chi_{K_{n}}^{L}$ sends
$\\{L/K_{1}\\},...,\\{L/K_{n-1}\\}$ to zero but $\\{L/K_{n}\\}$ to a positive
value. Thus this element is not in $\ker\chi_{K_{n}}^{L}$, and hence
$M_{\mathscr{F}}(G/L)\cap N(L)=\emptyset$.

In conclusion, we have $A_{G}=M_{\mathscr{F}}\oplus N_{\mathscr{F}}$ and
$M_{\mathscr{F}}$ appears as a direct summand.. $\Box$

######  Remark 2.15.

When the multiplicative structure is added into consideration, the maps
$\chi_{H}^{L}$ appear as the components of the ring isomorphism from $A(L)$ to
several copies of $\mathbb{Z}[|G|^{-1}]$, with one $H$ chosen from each
conjugacy class inside $L$. The generators $e_{H}^{L}$ become idempotent
elements. So we can view $M_{\mathscr{F}}$ as a direct summand of $A_{G}$ as a
Green functor.

###  2.4 Homology of $E\mathscr{F}$ in positive degrees

In positive degrees, we have:

######  Theorem 2.16.

For any family $\mathscr{F}$, the $\mathbb{Z}$-graded, Mackey functor valued
homology

$\underline{H}_{*}^{G}(E\mathscr{F};A_{G})$

contains only torsion when the degree is positive. Moreover, the torsion only
has prime factors which divide $|G|$.

This is the most important property we want about $E\mathscr{F}$. We will use
the rest of this section to prove it.

The main idea is an induction on the size of $\mathscr{F}$. For the base case,
we have

######  Lemma 2.17.

Let $BG=EG/G$ be the classifying space of principal $G$-bundles. Then
$H_{*}(BG;\mathbb{Z})$ contains only torsion when the degree is positive.
Moreover, the torsion only has prime factors which divide $|G|$.

This is a standard result about $EG$. We give one possible proof below.

#### Proof:

Give $EG$ the standard $G$-CW structure which only contains $G$-free cells.
Consider the map between cellular chain complexes induced by the projection
$EG\rightarrow EG/G=BG$:

$C_{*}(EG)\rightarrow C_{*}(BG).$

Define another map between chain complexes

$C_{*}(BG)\rightarrow C_{*}(EG)$

which sends each cell $e$ in $BG$ to the sum of all cells in $EG$ which are
sent to $e$ under $EG\rightarrow BG$. Then the composition

$C_{*}(BG)\rightarrow C_{*}(EG)\rightarrow C_{*}(BG)$

is the multiplication by $|G|$ since all cells in $EG$ are $G$-free. The same
$|G|$-multiplication passes to homology:

$H_{*}(BG)\rightarrow H_{*}(EG)\rightarrow H_{*}(BG).$

Since $EG$ is contractible, $H_{*}(EG)=\mathbb{Z}$, which is concentrated in
degree $0$. Thus when the degree is positive, $H_{*}(BG)$ contains only
torsion which divides $|G|$. $\Box$

In the case when $\mathscr{F}$ only contains the trivial subgroup,
$E\mathscr{F}$ becomes $EG$ and only has orbit spaces homotopic to $BH$ for
$H\subset G$. Lemma 2.17 and Proposition 2.10 imply that it satisfies Theorem
2.16.

For general $\mathscr{F}$, we can construct $E\mathscr{F}$ as a join. More
details about joins and their properties are given in Appendix A.

######  Proposition 2.18.

The join of $G\times_{N_{G}H}EW_{G}H$, with one $H$ chosen from each conjugacy
class in $\mathscr{F}$, is a valid construction for $E\mathscr{F}$. Here
$N_{G}H$ is the normalizer of $H$ in $G$. We view $EW_{G}H$ as a left
$N_{G}H$-space and $G$ as a right $N_{G}H$-set.

#### Proof:

Notice that $E\mathscr{F}$ is characterized by its fixed point subspaces:

$(E\mathscr{F})^{K}\simeq\begin{cases}*,\text{ if }K\in\mathscr{F},\\\
0,\text{ if }K\notin\mathscr{F}\end{cases}$

According to Lemma A.2, it suffices to show:

(a) If $K\in\mathscr{F}$, $(G\times_{N_{G}H}EW_{G}H)^{K}$ is contractible for
some $H\in\mathscr{F}$.

(b) If $K\notin\mathscr{F}$, $(G\times_{N_{G}H}EW_{G}H)^{K}$ is empty for all
$H\in\mathscr{F}$.

Consider any point in $G\times_{N_{G}H}EW_{G}H$ expressed by $a\times x$ for
$a\in G$ and $x\in EW_{G}H$. For any $g\in G$, if the action of $g$ fixes the
point $a\times x$:

$g(a\times x)=ga\times x=a\times x,$

then there is $h\in N_{G}H$, such that $hx=x$, $gah^{-1}=a$. Since all points
in $EW_{G}H$ have isotropy group $H$, we have $h\in H$. Then $g=aha^{-1}\in
aHa^{-1}$.

Therefore, the isotropy group of $a\times x$ is $aHa^{-1}$. Thus the $K$-fixed
point subspace of $G\times_{N_{G}H}EW_{G}H$ is non-empty if and only if $K$ is
sub-conjugate to $H$. Moreover, when $K=gHg^{-1}$, we have

$(G\times_{N_{G}H}EW_{G}H)^{K}=\\{g\\}\times EW_{G}H,$

which is contractible. Thus both conditions (a), (b) are satisfied. $\Box$

######  Remark 2.19.

Since all points in $G\times_{N_{G}H}EW_{G}H$ have isotropy groups conjugate
to $H$, according to Definition A.1, the collection of isotropy groups of all
points in $E\mathscr{F}$ agrees with $\mathscr{F}$.

#### Proof of Theorem 2.16:

Assume that Theorem 2.16 is true for all families smaller than $\mathscr{F}$.
Let $\mathscr{F}^{\prime}$ be a smaller family which is obtained by removing
the conjugacy class of one largest subgroup $H\in\mathscr{F}$. According to
Proposition 2.18, we have

$E\mathscr{F}\simeq E\mathscr{F}^{\prime}\ast(G\times_{N_{G}H}EW_{G}H).$

For any $K\lhd L\subset G$, the discussion in the proof of Proposition 2.18
shows that the space $(G\times_{N_{G}H}EW_{G}H)^{K}$ is either empty or
several copies of $EW_{G}H$. Since all points in each copy of $EW_{G}H$ have
the same isotropy group, $(G\times_{N_{G}H}W_{G}H)^{K}/L$ is either empty or
the disjoint union of $BN$ for some subgroups $N\subset N_{G}H\subset G$.
Lemma 2.17 shows that its positive degree homology only contains torsion
dividing $|G|$. Thus Proposition 2.10 shows that $G\times_{N_{G}H}EW_{G}H$
satisfies Theorem 2.16.

Since $E\mathscr{F}^{\prime}$ satisfies Theorem 2.16, according to Lemma A.3,
now it suffices to prove the theorem for
$E\mathscr{F}^{\prime}\times(G\times_{N_{G}H}EW_{G}H)$. Using $X(H)$ to denote
$G\times_{N_{G}H}EW_{G}H$, the homology of $E\mathscr{F}^{\prime}\times X(H)$
can be computed by the equivariant Künneth spectral sequence [LM04]:

$E_{p,q}^{2}=\underline{Tor}^{A_{G}}_{p,q}(\underline{H}_{*}^{G}(E\mathscr{F}^{\prime};A_{G}),\underline{H}_{*}^{G}(X(H);A_{G}))\Rightarrow\underline{H}_{*}^{G}(E\mathscr{F}^{\prime}\times
X(H);A_{G}).$

When we invert $|G|$, the homology of both $E\mathscr{F}^{\prime}$ and $X(H)$
is concentrated in degree 0. Moreover,
$\underline{H}_{0}^{G}(E\mathscr{F}^{\prime};A_{G})=M_{\mathscr{F}^{\prime}}$
appears as a direct summand of $A_{G}$ by Proposition 2.13. Thus the
$E_{2}$-page collapses into a single box product:

$E^{2}=E^{2}_{0,0}=M_{\mathscr{F}}\text{ }\Box\text{
}\underline{H}_{*}^{G}(X(H);A_{G}).$

Therefore, $E\mathscr{F}^{\prime}\times X(H)$ has trivial homology in positive
degrees when $|G|$ is inverted. $\Box$

For special choices of $\mathscr{F}$, the torsion in
$\underline{H}^{G}_{*}(E\mathscr{F};A_{G})$ may not see all prime factors of
$|G|$. We emphasize the following criterion which is implied by Proposition
2.10:

######  Proposition 2.20.

The torsion types in $\underline{H}_{*}^{G}(E\mathscr{F},A_{G})$ come from
$H_{*}(E\mathscr{F}^{K}/L)$ for all $K\lhd L\subset G$.

##  3 Splitting cofiber sequences

Let $\mathscr{F}$ be a family of subgroups of $G$. Consider the cofiber
sequence

$E\mathscr{F}_{+}\rightarrow S^{0}\rightarrow\widetilde{E\mathscr{F}}$

where the map $E\mathscr{F}_{+}\rightarrow S^{0}$ sends $E\mathscr{F}$ into
the non-basepoint of $S^{0}$.

###  3.1 A categorical splitting

######  Theorem 3.1.

When certain prime factors of $|G|$ are inverted, the cofiber sequence above
splits the category of $G$-spectra, in the sense that

$X\simeq(E\mathscr{F}_{+}\wedge X)\vee(\widetilde{E\mathscr{F}}\wedge X),$
$[E\mathscr{F}_{+}\wedge X,\widetilde{E\mathscr{F}}\wedge
Y]^{G}=[\widetilde{E\mathscr{F}}\wedge Y,E\mathscr{F}_{+}\wedge X]^{G}=0$

for any $G$-spectra $X,Y$. Moreover, the functors

$X\mapsto E\mathscr{F}_{+}\wedge X,\text{
}X\mapsto\widetilde{E\mathscr{F}}\wedge X$

are symmetric monoidal up to homotopy.

The proof will give information about which primes must be inverted; see
Remark 3.3.

Consider the suspension of the cofiber sequence:

$\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow
S\rightarrow\Sigma^{\infty}\widetilde{E\mathscr{F}}.$

######  Theorem 3.2.

When $|G|$ is inverted, there exists a left inverse of
$\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow S$.

#### Proof:

According to Proposition 2.12, the map
$\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow S$ induces an inclusion in
homology:

$\underline{HA_{G}}_{0}(\Sigma^{\infty}E\mathscr{F}_{+})=:M_{\mathscr{F}}\hookrightarrow\underline{HA_{G}}_{0}S=A_{G}.$

When $|G|$ is inverted, Theorem 2.16 shows that the $HA_{G}$-homology is
concentrated in degree 0. Thus it suffices to find a map
$S\rightarrow\Sigma^{\infty}E\mathscr{F}_{+}$ which induces a projection from
$A_{G}$ to $M_{\mathscr{F}}$ on their $HA_{G}$-homology. After composition
with $\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow S$, we get a self-map of
$\Sigma^{\infty}E\mathscr{F}_{+}$ that induces an isomorphism on
$HA_{G}$-homology. Since $\Sigma^{\infty}E\mathscr{F}_{+}$ is a connective
spectrum, this self map must be a weak equivalence.

For each $H\in\mathscr{F}$, the transfer and restriction maps in Mackey
functors between images of $G/G$ and $G/H$ are induced by stable maps

$T:S=\Sigma^{\infty}S^{0}=\Sigma^{\infty}G/G_{+}\rightarrow\Sigma^{\infty}G/H_{+}$
$R:\Sigma^{\infty}G/H_{+}\rightarrow\Sigma^{\infty}G/G_{+}=S.$

$R$ is induced by the space-level map $G/H_{+}\rightarrow S^{0}$, which sends
$G/H$ into the non-basepoint of $S^{0}$. According to Remark 2.19,
$E\mathscr{F}$ contains points with any $H\in\mathscr{F}$ as isotropy groups.
Thus we can find a map from $G/H$ to $E\mathscr{F}$ and factorize $R$ as

$\Sigma^{\infty}G/H_{+}\rightarrow\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow
S.$

Let $\iota_{H}$ be the composition of the first map above and $T$:

$\iota_{H}:S\xrightarrow{T}\Sigma^{\infty}G/H_{+}\rightarrow\Sigma^{\infty}E\mathscr{F}_{+}.$

The composition

$S\xrightarrow{\iota_{H}}\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow S$

agrees with $R\circ T$, which is the equivariant Euler characteristic of
$G/H$. We can fully describe the self-map on

$\underline{HA_{G}}_{0}S\cong A_{G}\cong\underline{\pi}_{0}S$

induced by the Euler characteristic as follows:

For each $L\subset G$, the induced self-map on $A_{G}(G/L)=A(L)$ is the
multiplication by the underlying $L$-set of $\\{G/H\\}$ (which is defined in
Definition 2.9). We refer to [LMS86], V.1 and V.2] for more details.

Now we choose one $H$ from each conjugacy class in $\mathscr{F}$. Assign a
random number $c_{H}\in\mathbb{Z}[|G|^{-1}]$ for each such $H$. Consider the
map

$\sum_{H}c_{H}\iota_{H}:S\rightarrow\Sigma^{\infty}E\mathscr{F}_{+}.$

The main strategy is to choose $c_{H}$ properly such that the map above
induces an isomorphism on equivariant homology with coefficient in $A_{G}$
after composing with $\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow S$, hence
appears to be the required left inverse.

The composition

$S\xrightarrow{\sum_{H}c_{H}\iota_{H}}\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow
S$

induces a self-map on $A_{G}(G/L)=A(L)$ as the multiplication by
$\sum_{H}c_{H}\\{G/H\\}$.

Recall that
$\underline{HA_{G}}_{0}(\Sigma^{\infty}E\mathscr{F}_{+})(G/L)=M_{\mathscr{F}}(G/L)\subset
A(L)$ is generated by all $\\{L/J\\}$ with $J\in\mathscr{F}$. The statement
that $\sum_{H}c_{H}\iota_{H}$ induces the projection $A_{G}\rightarrow
M_{\mathscr{F}}$ is equivalent to

$\\{L/J\\}\cdot\sum_{H}c_{H}\\{G/H\\}=\\{L/J\\}$

for any $J\in\mathscr{F}$, $J\subset L$.

Recall the ring map $\chi_{K}^{L}:A(L)\rightarrow\mathbb{Z}$ for any $K\subset
L$ which sends each $L$-set to the number of $K$-fixed points. According to
Remark 2.15, $\prod_{K\subset L}\chi_{K}^{L}$ is an injective ring
homomorphism from $A(L)$ to copies of $\mathbb{Z}$. Thus it suffices to check
the equation above on the images under each $\chi_{K}^{L}$. The equation above
under $\chi_{K}^{L}$ becomes

$|(L/J)^{K}|\cdot\sum_{H}c_{H}|(G/H)^{K}|=|(L/J)^{K}|.$

Notice that $|(L/J)^{K}|=0$ for any $K\notin\mathscr{F}$ (since $K$ is not
sub-conjugate to $J\in\mathscr{F}$). Thus it suffices to choose $c_{H}$ such
that

$\sum_{H}c_{H}|(G/H)^{K}|=\sum_{H}c_{H}s_{(K,H)}=1,\text{ }\forall
K\in\mathscr{F}$

with one $H$ chosen from each conjugacy class in $\mathscr{F}$. (Recall that
$s_{(K,H)}:=|(G/H)^{K}|$ is defined in the proof of Proposition 2.13.)

Notice that $s_{(K,H)}$ is non-zero if and only if $K$ is sub-congujate to
$H$. Denote this relation by $[K]\leq[H]$. We have

$\sum_{[K]\leq[H]\subset\mathscr{F}}c_{H}s_{(K,H)}=1.$

Thus

$c_{K}=s_{(K,K)}^{-1}\left(1-\sum_{[K]\lneqq[H]\subset\mathscr{F}}c_{H}s_{(K,H)}\right).$

Since $s_{(K,K)}=|W_{G}K|$ is invertible when $|G|$ is inverted, $c_{K}$ can
be chosen inductively from larger subgroups to smaller ones. Moreover, it’s
clear that the denominator of any $c_{K}$ only contains prime factors dividing
$|G|$.

Now the map $\sum_{H}c_{H}\iota_{H}$ gives us the required projection on the
0th homology, hence becomes the left inverse of
$\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow S$. $\Box$

#### Proof of Theorem 3.1:

Assume that $|G|$ is inverted. Theorem 3.2 implies that the cofiber sequence

$\Sigma^{\infty}E\mathscr{F}_{+}\rightarrow
S\rightarrow\Sigma^{\infty}\widetilde{E\mathscr{F}}$

splits. Thus we have

$X\simeq(E\mathscr{F}_{+}\wedge X)\vee(\widetilde{E\mathscr{F}}\wedge X).$

In fact, both $\Sigma^{\infty}E\mathscr{F}_{+}$ and
$\Sigma^{\infty}\widetilde{E\mathscr{F}}$ become ring spectra (up to
homotopy). The unit map on $\Sigma^{\infty}E\mathscr{F}_{+}$ is given by the
left inverse in Theorem 3.2, and the product is induced by the space level map
$E\mathscr{F}_{+}\wedge E\mathscr{F}_{+}\rightarrow E\mathscr{F}_{+}$. For
$\Sigma^{\infty}\widetilde{E\mathscr{F}}$, the unit and product maps are both
induced by the space level maps $S^{0}\rightarrow\widetilde{E\mathscr{F}}$ and
$\widetilde{E\mathscr{F}}\wedge\widetilde{E\mathscr{F}}\rightarrow\widetilde{E\mathscr{F}}$.
Proposition 2.8 and the fact that

$E\mathscr{F}_{+}\wedge E\mathscr{F}_{+}\simeq E\mathscr{F}_{+}\text{ ,
}\widetilde{E\mathscr{F}}\wedge\widetilde{E\mathscr{F}}\simeq\widetilde{E\mathscr{F}}$

guarantee all commutative diagrams in the definition of a ring spectrum.

Since both

$S\rightarrow\Sigma^{\infty}E\mathscr{F}_{+}\text{ ,
}S\rightarrow\Sigma^{\infty}\widetilde{E\mathscr{F}}$

are maps between ring spectra, the functors

$X\mapsto E\mathscr{F}_{+}\wedge X,\text{
}X\mapsto\widetilde{E\mathscr{F}}\wedge X$

are symmetric monoidal.

Moreover, $E\mathscr{F}_{+}\wedge\widetilde{E\mathscr{F}}\simeq*$ since all
its fixed point subspaces are contractible. Use $A,B$ to denote the
suspensions of $E\mathscr{F}_{+}$ and $\widetilde{E\mathscr{F}}$ (in either
order). The universal coefficient spectral sequence [LM04] tells us that for
any $G$-spectra $X,Y$, we have

$\underline{Ext}^{*,*}_{\underline{B}_{*}}(\underline{B}_{*}(A\wedge
X),\underline{(B\wedge Y)}_{*})\Rightarrow\underline{(B\wedge Y)}^{*}(A\wedge
X)=[A\wedge X,\Sigma^{*}B\wedge Y]^{G}.$

Since

$\underline{B}_{*}(A\wedge X)=\underline{\pi}_{*}(B\wedge A\wedge X)=0,$

the spectral sequence has trivial $E_{2}$-page. Thus we have

$[E\mathscr{F}_{+}\wedge X,\widetilde{E\mathscr{F}}\wedge
Y]^{G}=[\widetilde{E\mathscr{F}}\wedge Y,E\mathscr{F}_{+}\wedge X]^{G}=0.$

$\Box$

Now we can get a criterion about which prime factors of $|G|$ need to be
inverted according to the proofs above:

######  Remark 3.3.

It suffices to invert the prime factors of the denominators of all $c_{H}$,
and the torsion in $\underline{H}_{*}^{G}(E\mathscr{F};A_{G})$.

The numbers $c_{H}$ for all $H\in\mathscr{F}$ are defined in an inductive way
by the equations

$\sum_{[K]\leq[H]\subset\mathscr{F}}c_{H}s_{(K,H)}=1,\text{ for any fixed
}K\in\mathscr{F}.$

According to Proposition 2.20, the torsion in
$\underline{H}_{*}^{G}(E\mathscr{F};A_{G})$ comes from the torsion in
$H_{*}(E\mathscr{F}^{K}/L)$ for all $K\lhd L\subset G$.

###  3.2 Splitting criterion

In this subsection, we will improve Remark 3.3 and make a better criterion,
which plays the key role in our splitting method:

######  Theorem 3.4.

The cofiber sequence in Theorem 3.1 splits after taking $\Sigma^{\infty}$ if
and only if we invert all prime $p$ such that there exists $H\lhd J\subset G$
with $H\in\mathscr{F}$, $J\notin\mathscr{F}$, and $|J/H|$ is a power of $p$.

We prove the easier half first:

#### Proof of “only if” part:

Consider any prime $p$ such that the corresponding $H,J$ in the theorem exist.
We can assume that $J/H\cong C_{p}$, otherwise we replace $H$ by larger proper
normal subgroups of $J$. If the cofiber sequence

$E\mathscr{F}_{+}\rightarrow S^{0}\rightarrow\widetilde{E\mathscr{F}}$

splits after $\Sigma^{\infty}$, we have

$S\simeq\Sigma^{\infty}E\mathscr{F}_{+}\vee\Sigma^{\infty}\widetilde{E\mathscr{F}}.$

Take equivariant homology with coefficients in $A_{G}$, we have

$\widetilde{H}_{*}^{G}(S^{0};A_{G})=\widetilde{H}_{*}^{G}(E\mathscr{F}_{+};A_{G})\oplus\widetilde{H}_{*}^{G}(\widetilde{E\mathscr{F}};A_{G}).$

Since $\widetilde{H}_{*}^{G}(S^{0};A_{G})\cong A_{G}$ concentrates in degree
$0$, $\widetilde{H}_{*}^{G}(E\mathscr{F}_{+};A_{G})$ must concentrate in
degree $0$. According to Proposition 2.10,
$H_{*}(E\mathscr{F}^{H}/W_{J}H;\mathbb{Z})$ is a direct summand of
$\widetilde{H}_{*}^{G}(E\mathscr{F}_{+};A_{G})$, hence also concentrates in
degree $0$.

Since $H\in\mathscr{F}$, $J\notin\mathscr{F}$, $J/H\simeq C_{p}$, we have
$H_{*}(E\mathscr{F}^{H}/W_{J}H;\mathbb{Z})\simeq H_{*}(BC_{p};\mathbb{Z})$,
which concentrates in degree $0$ only if $p$ is inverted. $\Box$

Now we consider the “if” part. Assume that all primes described in Theorem 3.4
are inverted. According to Remark 3.3, it suffices to prove:

######  Proposition 3.5.

For any $K\lhd L\subset G$ and $m>0$, $H_{m}(E\mathscr{F}^{K}/L)=0$.

######  Proposition 3.6.

Consider numbers $c_{H}$ for all $H\in\mathscr{F}$ determined by the following
equations: For any $K\in\mathscr{F}$, we have

$\sum_{H}c_{H}\cdot|(G/H)^{K}|=1.$

Here we choose one $H$ from each conjugacy class inside $\mathscr{F}$. Then
the denominators of all $c_{H}$’s become invertible after we invert all prime
factors of $|G|$ not in $T$.

#### Proof of Proposition 3.5:

We divide the proof into three steps:

#### Step 1:

Let $p$ be a prime factor of $|G|$ which is not inverted and $P$ be any Sylow
$p$-subgroup of $G$. Then $|G/P|\cdot H_{m}(E\mathscr{F}/G)=0$ for all $m>0$.

Fix any $G$-CW structure on $E\mathscr{F}$, which induces CW structures on
both $E\mathscr{F}/P$ and $E\mathscr{F}/G$. The natural map
$E\mathscr{F}/P\rightarrow E\mathscr{F}/G$ induces a map between cellular
chain complexes:

$\phi:C_{*}(E\mathscr{F}/P)\rightarrow C_{*}(E\mathscr{F}/G).$

Define a map between abelian groups in the opposite direction

$\psi:C_{*}(E\mathscr{F}/G)\rightarrow C_{*}(E\mathscr{F}/P)$

as follows:

For any $G$-cell $G/H_{+}\wedge e$ of $E\mathscr{F}$, use $e_{P}$ and $e_{G}$
to denote the corresponding cells in $E\mathscr{F}/P$ and $E\mathscr{F}/G$.
Let $Pg_{1},Pg_{2},...,Pg_{k}$ be all right cosets of $P$ in $G$. Define
$\psi$ by sending each $e_{G}\in C_{*}(E\mathscr{F}/G)$ to
$\sum_{i}(g_{i}e)_{P}\in C_{*}(E\mathscr{F}/P)$.

We can show that $\psi$ is well-defined by the following two facts:

(1) If $g,g^{\prime}$ are in the same right coset of $P$, then
$(ge)_{P}=(g^{\prime}e)_{P}$. Thus $\psi$ does not depend on the choice of
$g_{1},g_{2},...,g_{k}$.

(2) For any $e,e^{\prime}$, $e_{G}=e^{\prime}_{G}$ if and only if
$e^{\prime}=ge$ for some $g\in G$. Since $g_{1}g,g_{2}g,...,g_{k}g$ also cover
all right cosets of $P$, we have
$\sum_{i}(g_{i}(ge))_{P}=\sum_{i}((g_{i}g)e)_{P}=\sum_{i}(g_{i}e)_{P}$. Thus
$\psi$ is uniquely defined on each element of $C_{*}(E\mathscr{F}/G)$.

Moreover, $\psi$ commutes with the boundary maps:

We use $d,d_{P},d_{G}$ to denote the cellular boundary maps for
$E\mathscr{F},E\mathscr{F}/P,E\mathscr{F}/G$ respectively. For any $G$-cell
$G/H_{+}\wedge e$ of $E\mathscr{F}$, write

$de=\sum_{t}\left(\sum_{g\in
G}n_{g}g\right)e_{t}=\sum_{t}\left[\sum_{i}\left(\sum_{a\in
P}n_{ag_{i}}a\right)g_{i}\right]e_{t}$

where the cells $e_{t}$ are in different $G$-orbits and $n_{g}\in\mathbb{Z}$.
Then we have

$d_{G}e_{G}=\sum_{t}\left(\sum_{g\in P}n_{g}\right)(e_{t})_{G}$
$d_{P}e_{P}=\sum_{t,i}\left(\sum_{a\in P}n_{ag_{i}}\right)(g_{i}e_{t})_{P}$

Notice that

$d(\sum_{g\in G}ge)=\sum_{g\in G}g(de)=\sum_{t}\left(\sum_{g\in
G}n_{g}\right)\left(\sum_{g\in
G}g\right)e_{t}=\sum_{t}\left[\sum_{i}\left(\sum_{a\in P}(\sum_{g\in
G}n_{g})a\right)g_{i}\right]e_{t}.$

Thus we have

$d_{P}(\sum_{g\in G}(ge)_{P})=|P|\sum_{t,i}(\sum_{g\in
G}n_{g})(g_{i}e_{t})_{P}.$

Since $\sum_{g\in G}(ge)_{P}=|P|\sum_{i}(g_{i}e)_{P}$, we have

$d_{P}(\sum_{i}(g_{i}e)_{P})=\frac{1}{|P|}d_{P}(\sum_{g\in
G}(ge)_{P})=\sum_{t,i}(\sum_{g\in G}n_{g})(g_{i}e_{t})_{P}.$

Therefore, we get

$d_{P}(\psi e_{G})=d_{P}(\sum_{i}(g_{i}e)_{P})=\sum_{t,i}(\sum_{g\in
G}n_{g})(g_{i}e_{t})_{P}$ $=\sum_{t}(\sum_{g\in
G}n_{g})(\sum_{i}(g_{i}e_{t})_{P})=\sum_{t}(\sum_{g\in
G}n_{g})\psi((e_{t})_{G})=\psi(d_{G}e_{G}).$

Now $\psi$ becomes a morphism between chain complexes. Notice that
$\phi\circ\psi$ is multiplication by $|G/P|$. Thus the induced map on homology

$H_{*}(E\mathscr{F}/G)\xrightarrow{\psi_{*}}H_{*}(E\mathscr{F}/P)\xrightarrow{\phi_{*}}H_{*}(E\mathscr{F}/G)$

is also multiplication by $|G/P|$.

Notice that since $p$ is not inverted, $\mathscr{F}$ must contain all
$p$-subgroups of $G$; otherwise we can choose $H$ to be the trivial subgroup
and $J$ to be the $p$-subgroup not in $\mathscr{F}$, which does not satisfy
the condition in Theorem 3.4. Thus $E\mathscr{F}$ is contractible as a
$P$-space, and hence $H_{m}(E\mathscr{F}/P)=0$ for all $m>0$. The maps above
imply that $|G/P|\cdot H_{m}(E\mathscr{F}/G)=0$ for all $m>0$.

#### Step 2:

Now we list all prime factors of $|G|$ which are not inverted:
$p_{1},p_{2},...,p_{n}$.

For any $i\in\\{1,2,...,n\\}$, let $P_{i}$ be a Sylow-$p_{i}$ subgroup of $G$.
By step 1, we have $|G/P_{i}|\cdot H_{m}(E\mathscr{F}/G)=0$ for all $m>0$.

Letting $i$ vary among $1,2,...,n$, we get

$(|G/P_{1}|,|G/P_{2}|,...,|G/P_{n}|)\cdot H_{m}(E\mathscr{F}/G)=0$

for $m>0$. Here $(|G/P_{1}|,|G/P_{2}|,...,|G/P_{n}|)$ is the greatest common
divisor of all $|G/P_{i}|$. Since all prime factors of $G$ except
$p_{1},p_{2},...,p_{n}$ are inverted, $(|G/P_{1}|,|G/P_{2}|,...,|G/P_{n}|)$ is
inverted. Thus $H_{m}(E\mathscr{F}/G)=0$ for any $m>0$.

#### Step 3:

Consider the general case of any $K\lhd L\subset G$. By comparing the fixed
point subspaces, $E\mathscr{F}^{K}$ is $L/K$-equivalent to $E\mathscr{G}$,
where $\mathscr{G}$ is the following family of subgroups of $L/K$:

$\mathscr{G}:=\\{H/K\text{ }|\text{ }K\subset H\subset L,H\in\mathscr{F}\\}.$

Consider any $H_{1},H_{2}\subset G$ such that $H_{1}/K\in\mathscr{G}$,
$H_{2}/K\notin\mathscr{G}$, $(H_{1}/K)\lhd(H_{2}/K)$ and
$|(H_{2}/K)/(H_{1}/K)|$ is a power of prime $p$. We must have
$H_{1}\in\mathscr{F}$, $H_{2}\notin\mathscr{F}$, $H_{1}\lhd H_{2}$ and
$|H_{2}/H_{1}|$ is a power of $p$. Thus $p$ must be inverted. By induction on
the group size and the previous two steps, we have
$H_{m}(E\mathscr{F}^{K}/L)\cong H_{m}(E\mathscr{G}/(L/K))=0$ for all $m>0$.
$\Box$

We now prove Proposition 3.6. Let $T=\\{p_{1},...,p_{n}\\}$ be the set of all
prime factors of $|G|$ which are not inverted.

For any $H\subset G$, recall the ring map $\chi_{H}:A(G)\rightarrow\mathbb{Z}$
defined in the proof of Proposition 2.13 which sends each $G$-set $S$ to
$|S^{H}|$. Let $C(G):=\prod_{[H]}\mathbb{Z}$. Then we have a ring map

$\chi=\prod_{[H]}\chi_{H}:A(G)\rightarrow C(G).$

The proof of Proposition 2.13 implies that $\chi$ is a ring isomorphism when
$|G|$ is inverted. If we only invert the prime factors not in $T$, $\chi$
becomes a monomorphism and $C(G)/Im(\chi)$ only contains torsion of
$p_{1},...,p_{n}$.

We want to show that the denominators of all $c_{H}$’s are invertible, which
is equivalent to the existence of the element $\sum_{H}c_{H}\\{G/H\\}$ in
$A(G)$, when all prime factors of $|G|$ not in $T$ are inverted. Since $\chi$
is a monomorphism, this existence can be detected by the image of $\chi$:

$\chi_{K}(\sum_{H}c_{H}\\{G/H\\})=\sum_{H}c_{H}|(G/H)^{K}|=\begin{cases}1,\text{
if }K\in\mathscr{F}\\\ 0,\text{ otherwise}\end{cases}$

In other words, it suffices to show that the element
$\alpha=(\alpha_{H})_{[H]}\in C(G)$, which is defined as

$\alpha_{H}=\begin{cases}1,\text{ if }K\in\mathscr{F},\\\ 0,\text{
otherwise,}\end{cases}$

is inside the image of $\chi$.

We will need the following lemma:

######  Lemma 3.7.

For any $H\in\mathscr{F}$ and $p\in T=\\{p_{1},p_{2},...,p_{n}\\}$, there
exists $L\in\mathscr{F}$, such that $\chi_{H}(G/L)$ is not divided by $p$.

#### Proof:

Let $M$ be the subgroup of $H$ generated by all elements whose order is not
divided by $p$.

List all prime factors of $|G|$ other than $p$ as $q_{1},q_{2},...,q_{m}$.
Notice that $M$ is generated by elements $h\in G$ whose order has the form
$|h|=q_{1}^{a_{1}}q_{2}^{a_{2}}...q_{m}^{a_{m}}$. Since $h$ is contained in
the group generated by $h^{|h|/q_{i}^{a_{i}}}$ for all $i=1,2,...,m$ and each
$h^{|h|/q_{i}^{a_{i}}}$ has a $q_{i}$-power order, $M$ is also generated by
its Sylow $q_{i}$-subgroups, for $i=1,2,...,m$.

We observe that $M$ is a normal subgroup of $H$ since conjugations do not
change the order of an element. Thus $H\subset N_{G}M$.

Write $H=Q\ltimes M$ and $N_{G}M=W_{G}M\ltimes M$ with $Q\subset W_{G}M$.

For any $h\in H$, write the order of $h$ as $|h|=p^{a}c$ with $c$ not divided
by $p$. Then $|h^{p^{a}}|=c$ and hence $h^{p^{a}}\in M$. Therefore, $H/M=Q$ is
a $p$-group.

Choose $P\in W_{G}M$ to be a Sylow-$p$ group containing $Q$. Let $L=P\ltimes
M\supset H$. Since $M\in\mathscr{F}$ as a subgroup of $H$ and $L/M\simeq P$
has a $p$-power order, we have $L\in\mathscr{F}$. We will show that
$\chi_{H}(G/L)$ is not divided by $p_{1}$.

Recall that $\chi_{H}(G/L)=|(G/L)^{H}|$. For any $g\in G$, $gL\in G/L$ is
fixed by $H$ if and only if $g^{-1}Hg\subset L$. If this happens, then
$g^{-1}Mg\subset L=P\ltimes M$.

Let $N$ be any Sylow $q_{i}$-subgroup of $M$ for some $i\in\\{1,2,...,m\\}$.
Then $g^{-1}Ng\subset g^{-1}Mg\subset L$. Since $|L|=|P|\cdot|M|$ and $|P|$ is
a $p$-power, both $N$ and $g^{-1}Ng$ are Sylow-$q_{i}$ subgroups of $L$, hence
are conjugate to each other inside $L$. Since $M$ is a normal subgroup of $L$
and $N\subset M$, we get $g^{-1}Ng\subset M$. Since $M$ is generated by its
Sylow-$q_{1},...,q_{m}$ subgroups, we have $g^{-1}Mg=M$, and hence $g\in
N_{G}M$.

When $gL\in(G/L)^{H}$, we have $g\in N_{G}M$ and $gL\in(N_{G}M/L)^{H}$. Thus

$\chi_{H}(G/L)=|(G/L)^{H}|=|(N_{G}M/L)^{H}|=|(W_{G}M/P)^{Q}|.$

Recall that $P$ is a Sylow-$p$ group of $W_{G}M$ containing $Q$. Thus
$|W_{G}M/P|$ is not divisible by $p$. Since the sizes of all $Q$-orbits except
$Q/Q$ are divided by $p$, the number of $Q/Q$ orbits in $W_{G}M/P$ is not
divided by $p$. Thus $|(W_{G}M/P)^{Q}|$ is not divided by $p$.

In conclusion, $\chi_{H}(G/L)$ is not divided by $p$. $\Box$

#### Proof of Proposition 3.6:

Since $C(G)$ has finite rank and $C(G)/Im(\chi)$ only contains torsion of
$p_{1},...,p_{n}$, there exist $k_{1},k_{2},...,k_{n}\in\mathbb{Z}^{+}$, such
that $p_{1}^{k_{1}}...p_{n}^{k_{n}}C(G)\subset Im(\chi)$.

We divide our proof into three steps:

Step 1: For any $H\in\mathscr{F}$ and $i\in\\{1,...,n\\}$, there exists an
element $\beta(H,i)\in Im(\chi)$, such that, modulo $p_{i}^{k_{i}}$, we have

$\beta(H,i)_{K}\equiv\begin{cases}1,\text{ if }K\in[H]\\\ 1\text{ or }0,\text{
if }K\in\mathscr{F}\\\ 0,\text{ if }K\notin\mathscr{F}\end{cases}$

According to Lemma 3.7, there exists $L\in\mathscr{F}$ such that
$\chi_{H}(G/L)$ is not divided by $p_{i}$. Notice that $\chi_{K}(G/L)=0$ for
any $K\notin\mathscr{F}$. Since
$|(\mathbb{Z}/p_{i}^{k_{i}})^{\times}|=p_{i}^{k_{i}-1}(p_{i}-1)$, the image of
$\\{G/L\\}^{p_{i}^{k_{i}-1}(p_{i}-1)}$ is a valid choice for $\beta(H,i)$.

Step 2: For any $H\in\mathscr{F}$, there exists an element $\beta(H)\in
Im(\chi)$, such that, modulo $p_{1}^{k_{1}}...p_{n}^{k_{n}}$, we have

$\beta(H)_{K}\equiv\begin{cases}1,\text{ if }K\in[H]\\\ 1\text{ or }0,\text{
if }K\in\mathscr{F}\\\ 0,\text{ if }K\notin\mathscr{F}\end{cases}$

For $i=1,2,...,n$, choose $c_{i}\in\mathbb{Z}$ such that $c_{1}\equiv 1$ (mod
$p_{i}^{k_{i}}$) and $c_{1}\equiv 0$ (mod $p_{j}^{k_{j}}$) for all $j\neq i$.
We only need to choose

$\beta(H)=c_{1}\beta(H,1)+...+c_{n}\beta(H,n).$

Step 3: Define $a+_{*}b=a+b-ab$ for any $a,b$ in the same ring. Notice that
for any $a,b,c\in\mathbb{Z}$ such that $a,b\equiv 0$ or $1$ (mod $c$), we have

$a+_{*}b\equiv\begin{cases}0,\text{ if }a\equiv b\equiv 0\\\ 1,\text{
otherwise}\end{cases}$

mod $c$.

Choose one subgroup from each conjugacy class inside $\mathscr{F}$ and list
them as $H_{1},H_{2},...,H_{m}$. Let

$\beta=\beta(H_{1})+_{*}\beta(H_{2})+_{*}...+_{*}\beta(H_{m})$

Then modulo $p_{1}^{k_{1}}...p_{n}^{k_{n}}$, we have

$\beta_{K}\equiv\begin{cases}1,\textbf{ if }K\in\mathscr{F}\\\ 0,\textbf{ if
}K\notin\mathscr{F}\end{cases}$

Since $\beta\in Im(\chi)$ and $p_{1}^{k_{1}}...p_{n}^{k_{n}}\cdot C(G)\subset
Im(\chi)$, the special element $\alpha=(\alpha_{H})_{[H]}$, which is defined
as

$\alpha_{H}=\begin{cases}1,\text{ if }H\in\mathscr{F},\\\ 0,\text{ if
}H\notin\mathscr{F},\end{cases}$

must be contained in $Im(\chi)$. $\Box$

##  4 Splitting method

In this section, we will set up our splitting method in the computation for
any equivariant spectrum $X$.

The procedure of the splitting method can be explained as follows:

(I) Fix a prime factor $p$ of $|G|$. Invert all other prime factors.

(II) Choose a specific family $\mathscr{F}$ such that the criterion in Theorem
3.4 is satisfied.

(III) Apply Theorem 3.1 to split $X$ into multiple pieces. Compute the
homotopy of each piece and glue them together. Now we have computed the
localized $\pi_{\bigstar}^{G}(X)$ when all prime factors of $|G|$ except $p$
are inverted.

(IV) List all prime factors of $|G|$ as $p_{1},p_{2},...,p_{n}$. For
$i=1,2,...,n$, we can compute

$R_{i}:=\pi_{\bigstar}^{G}(X)[p_{1}^{-1},...,\widehat{p_{i}^{-1}},...,p_{n}^{-1}]$

by the three steps above with $p=p_{i}$.

Consider the diagram with objects
$R_{1},R_{2},...,R_{n},\pi_{\bigstar}^{G}(HM)[|G|^{-1}]$ and morphisms

$R_{i}\rightarrow\pi_{\bigstar}^{G}(X)[|G|^{-1}],\text{ }i=1,2,...,n$

as localizations. This is a diagram inside the category of $RO(G)$-graded
rings. The limit of the diagram is exactly $\pi_{\bigstar}^{G}(X)$.

The hardest part is the computation of each piece in step (III), which will be
discussed in Theorem 4.3.

First we explain the choice of family in step (II):

Let $H$ be the subgroup of $G$ generated by all elements whose orders are not
divisible by $p$. Choose $\mathscr{F}$ to be the family of all subgroups of
$G$ not containing $H$. The criterion in Theorem 3.4 is satisfied because of
the following lemma:

######  Lemma 4.1.

$H$ is a normal subgroup of $G$ with $p$-power index and $H$ is generated by
Sylow $q$-subgroups of $G$ for all $q\neq p$. Moreover, $H$ itself does not
have any proper normal subgroups with $p$-power index.

#### Proof:

The first half can be proved in a similar way as in the proof of Lemma 3.7.

For the second half, assume that $M$ is a normal subgroup of $H$ with
$p$-power index. Then for any $q\neq p$, the Sylow $q$-subgroup of $M$ is also
a Sylow $q$-subgroup of $H$. Since $M$ is normal in $H$, it contains all Sylow
$q$-subgroups of $H$ for any $q\neq p$. Thus $M=H$ since $H$ is generated by
these Sylow subgroups. $\Box$

We have $X\simeq(E\mathscr{F}_{+}\wedge X)\vee(\widetilde{E\mathscr{F}}\wedge
X)$ when Theorem 3.1 holds. With this specific choice of $\mathscr{F}$, we
have

$\pi_{V}^{G}(\widetilde{E\mathscr{F}}\wedge X)\cong\pi_{V^{H}}^{H}(\Phi^{H}X)$
(4.1)

where $\Phi^{H}X$ is the $H$-geometric fixed point spectrum of $X$.

###  4.1 Decomposition of $E\mathscr{F}_{+}\wedge X$

When the splitting happens, we can compute the homotopy of
$E\mathscr{F}_{+}\wedge X$ in a systematic way. Recall that $M_{\mathscr{F}}$
is defined in Notation 2.14 as the homology of $E\mathscr{F}_{+}$ in degree
$0$.

######  Proposition 4.2.

Let $\mathscr{F}$ be an arbitrary family. Invert proper prime factors of $|G|$
such that Theorem 3.1 holds. Then for any $G$-spectrum $X$, we have

$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)\cong
Map(M_{\mathscr{F}},\underline{\pi}_{V}^{G}(X)).$

Here $\underline{\pi}_{V}^{G}(X)$ is the Mackey functor valued homotopy which
sends each $G/H$ to $\pi_{V|_{H}}^{H}(X)$.

Together with the expression of $M_{\mathscr{F}}$ in Proposition 2.13, we
have:

######  Theorem 4.3.

Let $\mathscr{F}$ be an arbitrary family of subgroups of $G$. Invert proper
prime factors of $|G|$ such that Theorem 3.1 holds.

For any virtual $G$-representation $V$ and $G$-spectrum $X$,
$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)$ is isomorphic to the limit of the
diagram with objects

$\pi_{V|_{H}}^{H}(X),\text{ }H\in\mathscr{F}$

and morphisms

$res_{H_{1}}^{H_{2}}:\pi_{V|_{H_{2}}}^{H_{2}}(X)\rightarrow\pi_{V|_{H_{1}}}^{H_{1}}(X),\text{
}H_{1}\subset H_{2}\in\mathscr{F},$
$c_{g}:\pi_{V|_{H}}^{H}(X)\rightarrow\pi_{V|_{gHg^{-1}}}^{gHg^{-1}}(X),\text{
}H\in\mathscr{F},g\in G.$

The map from $\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)$ to each object
$\pi_{V|_{H}}^{H}(X)$ can be factorized as

$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge
X)\rightarrow\pi_{V}^{G}(X)\xrightarrow{res_{H}^{G}}\pi_{V|_{H}}^{H}(X).$

Moreover, when $X$ is a ring spectrum, since taking limits preserves
multiplicative structures, this decomposition can also be used to compute
$\pi_{\bigstar}^{G}(E\mathscr{F}_{+}\wedge X)$ as a graded ring.

######  Remark 4.4.

The image of $res_{P}^{G}$ in $\pi_{V|_{P}}^{P}(X)$ does not depend only on
$V|_{P}$ since the choice of $g$ such that $g^{-1}Hg\subset P$ may not be
contained in $P$, not even in $N_{G}P$. Thus it’s possible that for two
different $G$-representations $V,W$, we have $V|_{P}\cong W|_{P}$ and yet
$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)$ and
$\pi_{W}^{G}(E\mathscr{F}_{+}\wedge X)$ are different.

#### Proof of Theorem 4.3:

According to Proposition 4.2, we have

$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)\cong
Map(M_{\mathscr{F}},\underline{\pi}_{V}^{G}X).$

According to Proposition 2.13, as a sub-Mackey functor of $A_{G}$,
$M_{\mathscr{F}}$ is generated by $\\{H/H\\}$ for all $H\in\mathscr{F}$, with
relations generated by

$res_{H_{1}}^{H_{2}}(\\{H_{2}/H_{2}\\})=\\{H_{1}/H_{1}\\},\text{ }H_{1}\subset
H_{2}\in\mathscr{F},$ $c_{g}(\\{H/H\\})=\\{gHg^{-1}/gHg^{-1}\\},\text{
}H\in\mathscr{F},g\in G.$

Each map from $M_{\mathscr{F}}$ to $\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)$ is
determined by the images of $\\{H/H\\}$ in

$\underline{\pi}_{V}^{G}(X)(G/H)=\pi_{V|_{H}}^{H}(X)$

for all $H\in\mathscr{F}$ which are compatible with maps $res_{H_{1}}^{H_{2}}$
and $c_{g}$. Thus $\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)$ is expressed as the
limit described in the theorem. $\Box$

In order to prove Proposition 4.2, we need a small lemma on the homotopy of
$E\mathscr{F}$ first. Recall that $N_{\mathscr{F}}$ is defined in Notation
2.14 as the complement of $M_{\mathscr{F}}$ in $A_{G}$.

######  Lemma 4.5.

Assume that proper primes are inverted such that Theorem 3.1 holds. Then

$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\cong\underline{\pi}_{\bigstar}^{G}(S)\text{
}\Box\text{ }M_{\mathscr{F}},$
$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\cong\underline{\pi}_{\bigstar}^{G}(S)\text{
}\Box\text{ }N_{\mathscr{F}}.$

Here $S=\Sigma^{\infty}S^{0}$ is the sphere spectrum.

#### Proof:

According to Theorem 3.1, we have an equivalence of ring spectra:

$S\simeq\Sigma^{\infty}E\mathscr{F}_{+}\vee\Sigma^{\infty}\widetilde{E\mathscr{F}}.$

Thus the product in $\underline{\pi}_{\bigstar}^{G}(S)$ is induced by the
products in $\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})$
and $\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})$.

In particular, the map

$\underline{\pi}_{\bigstar}^{G}(S)\text{ }\Box\text{
}\underline{\pi}_{0}^{G}(S)\rightarrow\underline{\pi}_{\bigstar}^{G}(S)$

is given by

$\underline{\pi}_{\bigstar}^{G}(S)\text{ }\Box\text{
}\underline{\pi}_{0}^{G}(S)\cong\left(\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\right)\oplus\left(\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\right)$
$\oplus\left(\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\right)\oplus\left(\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\right)\xrightarrow{f}$
$\left(\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\right)\oplus\left(\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\right)\rightarrow$
$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\oplus\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\cong\underline{\pi}_{\bigstar}^{G}(S).$

Here the map $f$ is the projection onto the first two summands.

On the other hand, the map

$\underline{\pi}_{\bigstar}^{G}(S)\text{ }\Box\text{
}\underline{\pi}_{0}^{G}(S)\rightarrow\underline{\pi}_{\bigstar}^{G}(S)$

is an isomorphism since $\underline{\pi}_{0}^{G}(S)\cong A_{G}$ is the unit
Mackey functor. Thus the projection $f$ must be an isomorphism and hence

$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\text{
}\Box\text{
}\underline{\pi}_{0}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})=\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\text{
}\Box\text{ }\underline{\pi}_{0}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})=0.$

Since $\Sigma^{\infty}E\mathscr{F}_{+}$ and
$\Sigma^{\infty}\widetilde{E\mathscr{F}}$ are connective, we have

$\underline{\pi}_{0}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\cong\widetilde{\underline{H}}_{0}^{G}(E\mathscr{F}_{+};A_{G})=M_{\mathscr{F}},$
$\underline{\pi}_{0}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\cong\widetilde{\underline{H}}_{0}^{G}(\widetilde{E\mathscr{F}};A_{G})=N_{\mathscr{F}}.$

Thus

$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\text{
}\Box\text{
}N_{\mathscr{F}}=\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\text{
}\Box\text{ }M_{\mathscr{F}}=0$

and hence

$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}E\mathscr{F}_{+})\cong\underline{\pi}_{\bigstar}^{G}(S)\text{
}\Box\text{ }M_{\mathscr{F}},$
$\underline{\pi}_{\bigstar}^{G}(\Sigma^{\infty}\widetilde{E\mathscr{F}})\cong\underline{\pi}_{\bigstar}^{G}(S)\text{
}\Box\text{ }N_{\mathscr{F}}.$

$\Box$

#### Proof of Proposition 4.2:

According to Theorem 3.1, we have

$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)=[S^{V},E\mathscr{F}_{+}\wedge
X]^{G}\cong[E\mathscr{F}_{+},\Sigma^{-V}X]^{G}=X^{-V}E\mathscr{F}_{+}.$

Here we express the homotopy of $E\mathscr{F}_{+}\wedge X$ as the
$X$-cohomology of the space $E\mathscr{F}_{+}$.

View $X$ as an $S$-module and apply the equivariant universal coefficient
spectral sequence in [LM04]:

$Ext^{*,*}_{\underline{\pi}_{*}^{G}(S)}(\underline{\pi}_{*}^{G}(\Sigma^{\infty}E\mathscr{F}_{+}),\underline{X}^{*-V})\Rightarrow
X^{*-V}E\mathscr{F}_{+}.$

According to Lemma 4.5 and Proposition 2.13, we have

$Ext^{*,*}_{\underline{\pi}_{*}^{G}(S)}(\underline{\pi}_{*}^{G}(\Sigma^{\infty}E\mathscr{F}_{+}),\underline{X}^{*-V})\cong
Ext^{*,*}_{\underline{\pi}_{*}^{G}(S)}(\underline{\pi}_{*}^{G}(S)\text{
}\Box\text{ }M_{\mathscr{F}},\underline{X}^{*-V})$ $\cong
Ext^{*,*}_{A_{G}}(M_{\mathscr{F}},\underline{X}^{*-V})\cong
Map(M_{\mathscr{F}},\underline{\pi}_{V}^{G}X)$

and the $E_{2}$-page is trivial except on the bottom line. Thus we have

$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge X)=[S^{V},E\mathscr{F}_{+}\wedge
X]^{G}\cong\vskip 3.0pt plus 1.0pt minus
1.0ptMap(M_{\mathscr{F}},\underline{\pi}_{V}^{G}(X)).$

$\Box$

###  4.2 Input of the computation

When we apply the decomposition into smaller subgroups by Theorem 4.3 and
(missing) 4.1, if the subgroups still contain multiple prime factors, we can
apply the splitting method again. Finally everything will be decomposed into
equivariant homotopy for subgroups with prime power orders. We will give a
precise list of input of this computational machine. We need the following
notation:

For any finite group $H$, let $H_{p}$ be the subgroup of $H$ generated by all
elements whose orders are not divisible by $p$. For any $K\subset H$, let
$\mathscr{F}_{H}[K]$ be the family of subgroups of $H$ not containing $K$.
Then our splitting method uses the family $\mathscr{F}_{H}[H_{p}]$ to split
$H$-spectra when localized at $p$.

######  Theorem 4.6.

Let $X$ be a $G$-spectrum and $V$ be a virtual $G$-representation. Assume that
all prime factors of $|G|$ except $p$ are inverted. Then $\pi_{V}^{G}(X)$ is
the limit of the diagram consisting of:

(i) $\pi_{V|_{H}}^{H}(X)$ for any $p$-subgroup $H\subset G$;

(ii) $\pi_{(V|_{K})^{K_{p}}}^{K/K_{p}}(\Phi^{K_{p}}X)$ for any
$K\in\mathscr{F}_{G}[G_{p}]\cup\\{G\\}$ such that $K$ is not a $p$-group;

(iii) Restriction map
$\pi_{V|_{H_{2}}}^{H_{2}}(X)\rightarrow\pi_{V|_{H_{1}}}^{H_{1}}(X)$ for any
$H_{1}\subset H_{2}$;

(iv) Restriction map
$\pi_{(V|_{L})^{L_{p}}}^{L/L_{p}}(\Phi^{L_{p}}X)\rightarrow\pi_{(V|_{K})^{K_{p}}}^{K/K_{p}}(\Phi^{K_{p}}X)$
for any $K\subset L$ such that $K_{p}=L_{p}$;

(v) Conjugacy map on $\pi_{V|_{H}}^{H}(X)$ by any $g\in W_{G}H$;

(vi) Conjugacy map on $\pi_{(V|_{K})^{K_{p}}}^{K/K_{p}}(\Phi^{K_{p}}X)$ by any
$g\in W_{G}K$ (notice that $g^{-1}K_{p}g=K_{p}$ according to the definition of
$K_{p}$).

We will prove this theorem in the rest of this subsection. Notice that Theorem
1.2 is a consequence of this theorem.

Before we start the proof, we need a more precise explanation about how to
repeat the splitting method. After we apply the splitting method on the
$G$-spectrum $X$ with family $\mathscr{F}_{G}[G_{p}]$, we obtain a diagram
with objects $\pi_{V|_{H}}^{H}(X)$, for all $H$ in $\mathscr{F}_{G}[G_{p}]$
and $\pi_{V^{G_{p}}}^{G/G_{p}}(\Phi^{G_{p}}X)$. We don’t have to further split
the geometric fixed point spectrum since it is acted on by the $p$-group
$G/G_{p}$. But we can apply the splitting method again on each
$\pi_{V|_{H}}^{H}(X)$ with family $\mathscr{F}_{H}[H_{p}]$ to make it into the
limit of another diagram.

The conjugacy map on $\pi_{V|_{H}}^{H}(X)$ splits in the same way since the
splitting in Theorem 3.1 is categorical. The following lemma implies that the
restriction maps are also compatible with the splittings. Thus repeating the
splitting method not only expresses each object as a limit, but also enlarge
the whole diagram.

By applying Theorem 4.3 and tracing the restriction maps, we have

######  Lemma 4.7.

Consider any $H\subset L\subset G$. Assume that all prime factors of $|G|$
except $p$ are inverted. The restriction map
$res_{H}^{L}:\pi_{V|_{L}}^{L}(X)\rightarrow\pi_{V|_{H}}^{H}(X)$ can be
expressed as follows:

(1) If $H\in\mathscr{F}_{L}[L_{p}]$, or equivalently, $H$ does not contain
$L_{p}$, then after applying the splitting on $\pi_{V|_{L}}^{L}(X)$ with
family $\mathscr{F}_{L}[L_{p}]$, $res_{H}^{L}$ can be factorized as

$\pi_{V|_{L}}^{L}(X)\rightarrow\pi_{V|_{L}}^{L}(E\mathscr{F}_{L}[L_{p}]_{+}\wedge
X)\rightarrow\pi_{V|_{H}}^{H}(X)$

since $\pi_{V|_{H}}^{H}(X)$ is an object in the diagram computing
$\pi_{V|_{L}}^{L}(E\mathscr{F}_{L}[L_{p}]_{+}\wedge X)$ in Theorem 4.3.

(2) If $H\notin\mathscr{F}_{L}[L_{p}]$, or equivalently, $H$ contains $L_{p}$
and hence $H_{p}=L_{p}$, we can apply the splitting on both
$\pi_{V|_{L}}^{L}(X)$ and $\pi_{V|_{H}}^{H}(X)$ with families
$\mathscr{F}_{L}[L_{p}]$ and $\mathscr{F}_{H}[H_{p}]$. Then $res_{H}^{L}$ is
split into two pieces:

$\pi_{V|_{L}}^{L}(E\mathscr{F}_{L}[L_{p}]_{+}\wedge
X)\rightarrow\pi_{V|_{H}}^{H}(E\mathscr{F}_{H}[H_{p}]_{+}\wedge X)$
$\pi_{(V|_{L})^{L_{p}}}^{L/L_{p}}(\Phi^{L_{p}}X)\rightarrow\pi_{(V|_{H})^{H_{p}}}^{H/H_{p}}(\Phi^{H_{p}}X)$

The second map is obtained by taking the underlying $H$-spectrum. The first
map is obtained by the fact that the diagram computing
$\pi_{V|_{H}}^{H}(E\mathscr{F}_{H}[H_{p}]_{+}\wedge X)$ in Theorem 4.3 is a
sub-diagram of the diagram computing
$\pi_{V|_{L}}^{L}(E\mathscr{F}_{L}[L_{p}]_{+}\wedge X)$.

######  Remark 4.8.

The splittings on the underlying spectra can be applied in the following way
to keep enlarging the diagram:

Choose an object $\pi_{V|_{H}}^{H}(X)$ such that $H_{p}$ has the largest size
(if the largest size is $1$, no more splittings are required). Apply the
splitting on all $\pi_{V|_{K}}^{K}(X)$ such that $K_{p}=H_{p}$. Lemma 4.7
guarantees that all restriction maps are transferred into the new diagram.
Thus the diagram is enlarged in each such round of splitting.

Now we can repeat the spiltting method to enlarge the diagram as much as
possible. Notice that when $H$ is a $p$-group, $H_{p}$ is trivial and hence
the splitting does not change the $H$-homotopy. In other words, the diagram
will finally stabilize when each summand is acted on by a $p$-group.

#### Proof of Theorem 4.6:

After the diagram stabilizes, the argument above and Lemma 4.7 shows that all
objects and maps will have the forms in (i)-(vi). On the other hand, all
objects and maps in (i)-(vi) appear after the splitting is applied once on all
underlying spectra where the acting group is not a $p$-group, and don’t change
in further splittings, hence survive in the stabilized diagram. $\Box$

###  4.3 Computability of $H\underline{\mathbb{Z}}$

In this subsection, we will discuss the computability of Eilenberg-Maclane
spectra with constant Mackey functors, with $H\underline{\mathbb{Z}}$ the
universal choice among them. Recall that the splitting method decomposes
$G$-spectra into underlying and geometric fixed point spectra.
$H\underline{\mathbb{Z}}$ is quite special because the geometric fixed point
spectra will be trivial when the splitting happens:

######  Proposition 4.9.

If Theorem 3.1 holds, we have $\widetilde{E\mathscr{F}}\wedge
H\underline{\mathbb{Z}}\simeq*$. Thus

$H\underline{\mathbb{Z}}\simeq E\mathscr{F}_{+}\wedge
H\underline{\mathbb{Z}}.$

#### Proof:

Consider the equivariant universal coefficient spectral sequence [LM04]:

$E^{2}_{*,*}=\underline{Tor}_{*,*}^{A_{G}}(\underline{HA_{G}}_{*}\widetilde{E\mathscr{F}},\underline{\mathbb{Z}})\Rightarrow\underline{H\underline{\mathbb{Z}}}_{*}\widetilde{E\mathscr{F}}.$

According to the proof of Proposition 2.13,
$\underline{HA_{G}}_{*}\widetilde{E\mathscr{F}}$ is concentrated in degree $0$
and appears as the direct summand $N_{\mathscr{F}}$ of $A_{G}$. Thus the
$E^{2}$-page is trivial except

$E^{2}_{0,0}\cong\underline{\mathbb{Z}}\text{ }\Box\text{ }N_{\mathscr{F}}.$

Since

$\underline{\mathbb{Z}}=\underline{\mathbb{Z}}\text{ }\Box\text{
}A_{G}=\underline{\mathbb{Z}}\text{ }\Box\text{
}M_{\mathscr{F}}\oplus\underline{\mathbb{Z}}\text{ }\Box\text{
}N_{\mathscr{F}},$

there must be one $\underline{\mathbb{Z}}$ and one $0$ in
$\\{\underline{\mathbb{Z}}\text{ }\Box\text{
}M_{\mathscr{F}},\underline{\mathbb{Z}}\text{ }\Box\text{
}N_{\mathscr{F}}\\}$.

Notice that $M_{\mathscr{F}}(G/e)=H_{0}(E\mathscr{F};\mathbb{Z})=\mathbb{Z}$.
Thus

$\underline{\mathbb{Z}}\text{ }\Box\text{
}M_{\mathscr{F}}(G/e)=\underline{\mathbb{Z}}(G/e)\otimes
M_{\mathscr{F}}(G/e)=\mathbb{Z}.$

Now we get

$\underline{\mathbb{Z}}\text{ }\Box\text{
}M_{\mathscr{F}}=\underline{\mathbb{Z}},\text{ }\underline{\mathbb{Z}}\text{
}\Box\text{ }N_{\mathscr{F}}=0.$

Therefore, the $E^{2}$-page of the spectral sequence is trivial, hence we have
$\widetilde{E\mathscr{F}}\wedge H\underline{\mathbb{Z}}\simeq*$. $\Box$

Proposition 4.9 still works when $\underline{\mathbb{Z}}$ is replaced by other
constant Mackey functors, as can be proved by considering the induced maps
from $\underline{\mathbb{Z}}$.

We now list the currently known computations on $p$-groups that can be fed in
as inputs in Theorem 4.6:

(i) $\pi_{\bigstar}^{C_{p}}(H\underline{\mathbb{Z}})$, which is computed by
multiple people with different methods.

(ii) $\pi_{\bigstar}^{C_{p^{2}}}(H\underline{\mathbb{Z}})$ by Zeng [Zen18] for
odd $p$ and Georgakopoulos [Geo21] for $p=2$.

(iii) $\pi_{\bigstar}^{C_{2}^{2}}(H\underline{\mathbb{F}_{2}})$ and partial
information about $\pi_{\bigstar}^{C_{2}^{2}}(H\underline{\mathbb{Z}})$ by
Ellis-Bloor [Ell20].

(iv) $\pi_{\bigstar}^{Q_{8}}(H\underline{\mathbb{Z}})$ by Lu [Lu21], only with
additive structure.

The computation which works for $H\underline{\mathbb{Z}}$ usually works for
other constant Mackey functors as well.

Most of the computations above are done in a cellular way. Thus we can obtain
the restriction and conjugacy maps by treating the homotopy of
$H\underline{\mathbb{Z}}$ as the equivariant homology of the representation
spheres.

In conclusion, now we have a computability list with our splitting method:

######  Theorem 4.10.

For a finite group $G$, $\pi_{\bigstar}^{G}(H\underline{\mathbb{Z}})$ is
computable if all Sylow subgroups $P\subset G$ satisfy one of the following
conditions:

(i) $P=C_{p}$ or $C_{p^{2}}$;

(ii) $P=C_{2}^{2}$ and $P\subset A_{4}\subset W_{G}P$ (this guarantees that we
only need the partial information in [Ell20]);

(iii) $P=Q_{8}$ (we can only compute the additive structure in this case).

######  Remark 4.11.

When $\underline{\mathbb{Z}}$ is replaced by a general Mackey functor $M$, the
geometric fixed point part is still computable since we have

$\widetilde{E\mathscr{F}}\wedge HM\simeq H(M\text{ }\Box\text{
}N_{\mathscr{F}}).$

However, we don’t know how to compute the homotopy of general $HM$ for
$p$-groups other than $C_{p}$. Thus $\pi_{\bigstar}^{G}(HM)$ is only
computable when all Sylow subgroups of $G$ has prime order.

###  4.4 Computibility of $MU_{G}$

We will discuss the computibility of $MU_{G}$ in this section. According to
the special structure of Thom spectra, $MU_{G}$ is more computable than
$H\underline{\mathbb{Z}}$ in some sense. We will prove the following theorem:

######  Theorem 4.12.

The homotopy of $MU_{G}$ is computable if all Sylow subgroups of $G$ are
abelian.

We just need to check the list of input in Theorem 4.6:

The homotopy of $MU_{H}$ for abelian $H$ is computed by [AK15], which uses the
Tate diagram repeatedly with families corresponding to all sequences of
subgroups.

The conjugacy maps become isomorphisms of diagrams. For any $H_{1}\subset
H_{2}$, $res_{H_{1}}^{H_{2}}$ can be obtained by the fact that the diagram
computing $MU_{H_{1}}$ is isomorphism to a sub-diagram (after taking
underlying $H_{1}$-spectra) of the diagram computing $MU_{H_{2}}$.

The study of geometric fixed point spectrum $\Phi^{K}MU_{G}$ traces back to
[Dieck70]. That paper only discusses the case when $G$ is abelian, but some of
the results still work for the non-abelian case. We refer to [Gre05] for a
more modern discussion:

######  Theorem 4.13.

Let $J(K)$ be the set of all complex virtual $K$-representations with
dimension $0$, and $t(K)$ be the number of isomorphism classes of non-trivial
irreducible complex representations of $K$. Then as a $G/K$-spectrum, we have

$\Phi^{K}MU_{G}\simeq\bigvee_{j\in
J(K)}\Sigma^{|j^{K}|}MU_{G/K}\wedge(BU_{G/K})_{+}^{\wedge t(K)}$

When $G/K$ is abelian, the homotopy of each summand is computed in [CGK02]:

######  Theorem 4.14.

Let $H$ be an abelian group. Then

$\pi_{*}^{H}(MU_{H}\wedge BU_{H}^{\wedge
n})=(\pi_{*}^{H}MU_{H})[\gamma_{1},\gamma_{2},...]^{\otimes n}.$

The conjugacy maps fix the polynomial generators $\gamma_{i}$. For any
$H_{1}\subset H_{2}$ such that $(H_{1})_{p}=(H_{2})_{p}$ (written as $H_{p}$),
the $H_{p}$-geometric fixed point spectra of $MU_{H_{1}}$ and $MU_{H_{2}}$
have the same number of $BU$ factors in the expression in Theorem 4.13. The
restriction map $res_{H_{1}}^{H_{2}}$ can be obtained by the restriction map

$\pi_{*}^{H_{2}/H_{p}}MU_{H_{2}/H_{p}}\rightarrow\pi_{*}^{H_{1}/H_{p}}MU_{H_{1}/H_{p}}$

and identity maps on the polynomial generators.

Therefore, Theorem 4.12 is proved since all inputs in Theorem 4.6 are
computable.

Theorem 4.6 also implies the following approach on the evenness conjecture of
$MU_{G}$:

######  Theorem 4.15.

If all Sylow subgroups of $G$ are abelian, then the homotopy of $MU_{G}$
concentrates in even degrees.

In general, if for any $p$-subquotient $P$ of $G$ and $n\geq 0$, the homotopy
of $MU_{P}\wedge(BU_{P})_{+}^{\wedge n}$ is even, then the homotopy of
$MU_{G}$ is even.

###  4.5 Some Examples

In this subsection, we will apply Theorem 4.6 to compute general $G$-spectra
with $G=D_{2p}$ and $A_{5}$. These results will be used in later computations.

For $G=D_{2p}$, the subgroups of $G$ (up to conjugacy) are listed as
$e,C_{2},C_{p},G$.

######  Theorem 4.16.

Let $G=D_{2p}$ and $V$ be a $G$-virtual representation. For any $G$-spectrum
$X$, $\pi_{V}^{G}(X)$ is isomorphic to:

(i) The direct sum of $\pi_{V^{G}}(\Phi^{G}X)$,
$\pi_{(V|_{C_{2}})^{C_{2}}}^{C_{2}}(\Phi^{C_{2}}X)$ and
$\pi_{V|_{C_{p}}}^{C_{p}}(X)^{G/C_{p}}$ when $2$ is inverted;

(ii) The direct sum of $\pi_{V^{C_{p}}}^{G/C_{p}}(\Phi^{C_{p}}X)$ and the
pullback of

$\pi_{V|_{C_{2}}}^{C_{2}}(X)\xrightarrow{res_{e}^{C_{2}}}\pi_{|V|}(X)\hookleftarrow\pi_{|V|}(X)^{G}$

when $p$ is inverted.

#### Proof:

When $2$ is inverted, according to Theorem 4.6, $\pi_{V}^{G}(X)$ is the limit
of the diagram with objects $\pi_{|V|}(X)$ (with $G$-action),
$\pi_{V|_{C_{p}}}^{C_{p}}(X)$ (with $G/C_{p}$-action),
$\pi_{V^{G}}(\Phi^{G}X)$, $\pi_{(V|_{C_{2}})^{C_{2}}}^{C_{2}}(\Phi^{C_{2}}X)$,
and the map

$res_{e}^{C_{p}}:\pi_{V|_{C_{p}}}^{C_{p}}(X)\rightarrow\pi_{|V|}(X).$

Since the image of $res_{e}^{C_{p}}$ is fixed by $C_{p}$, the image of
$\pi_{V|_{C_{p}}}^{C_{p}}(X)^{G/C_{p}}$ under $res_{e}^{C_{p}}$ is fixed by
$G$. Thus all information related to $\pi_{|V|}(X)$ and
$\pi_{V|_{C_{p}}}^{C_{p}}(X)$ is encoded in
$\pi_{V|_{C_{p}}}^{C_{p}}(X)^{G/C_{p}}$.

When $p$ is inverted, part (ii) is a direct consequence of Theorem 4.6. $\Box$

For $G=A_{5}$, the subgroups of $G$ (up to conjugacy) are listed as
$e,C_{2},C_{3},K_{4},C_{5},D_{6},D_{10},A_{4},G$. Here $K_{4}=C_{2}\times
C_{2}$ is the Sylow $2$-subgroup of $G$.

The case of $G=A_{5}$ is a little special. Notice that $G$ has no proper
normal subgroups and we have $G=G_{p}$ for all prime $p$. When we apply the
splitting method described at the begining of section 4, the family
$\mathscr{F}$ will always be the family of all proper subgroups no matter
which primes we invert. Thus we can apply Theorem 4.3 and (missing) 4.1
directly without any localizations:

######  Theorem 4.17.

Let $G=A_{5}$ and $V$ be a $G$-virtual representation. For any $G$-spectrum
$X$, $\pi_{V}^{G}(X)$ is isomorphic to the direct sum of
$\pi_{V^{G}}(\Phi^{G}X)$ and the limit of the following diagram:

$\textstyle{\pi_{V|_{A_{4}}}^{A_{4}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{res_{e}^{A_{4}}}$$\textstyle{\pi_{V|_{D_{6}}}^{A_{4}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{res_{e}^{D_{6}}}$$\textstyle{\pi_{|V|}(X)}$$\textstyle{\pi_{V|_{D_{10}}}^{D_{10}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{res_{e}^{D_{10}}}$

The diagram above comes from Theorem 4.3 and is simplified by similar
arguments as in the proof of Theorem 4.16 part (i).

On the other hand, we can also apply Theorem 4.6 after inverting proper primes
to get a complete decomposition. For simplicity, we will use the same notation
to denote the underlying representations of any $G$-virtual representation.

######  Theorem 4.18.

Let $G=A_{5}$ and $V$ be a $G$-virtual representation. For any $G$-spectrum
$X$, $\pi_{V}^{G}(X)$ is isomorphic to:

(i) The direct sum of $\pi_{V^{G}}(\Phi^{G}X)$,
$\pi_{V^{D_{6}}}(\Phi^{D_{6}}X)$, $\pi_{V^{A_{4}}}(\Phi^{A_{4}}X)$,
$\pi_{V^{D_{10}}}(\Phi^{D_{10}}X)$,
$\pi_{V^{K_{4}}}(\Phi^{K_{4}}X)^{A_{4}/K_{4}}$,
$\pi_{V^{C_{3}}}(\Phi^{C_{3}}X)^{D_{6}/C_{3}}$,
$\pi_{V^{C_{2}}}(\Phi^{C_{2}}X)^{K_{4}/C_{2}}$ and the limit of

$\pi_{V}^{C_{5}}(X)^{D_{10}/C_{5}}\xrightarrow{res_{e}^{C_{5}}}\pi_{V}(X)\hookleftarrow\pi_{V}(X)^{G}$

when $2,3$ are inverted.

(ii) The direct sum of $\pi_{V^{G}}(\Phi^{G}X)$,
$\pi_{V^{D_{6}}}(\Phi^{D_{6}}X)$,
$\pi_{V^{K_{4}}}^{A_{4}/K_{4}}(\Phi^{K_{4}}X)$,
$\pi_{V^{D_{10}}}(\Phi^{D_{10}}X)$,
$\pi_{V^{K_{4}}}(\Phi^{K_{4}}X)^{A_{4}/K_{4}}$,
$\pi_{V^{C_{5}}}(\Phi^{C_{5}}X)^{D_{10}/C_{5}}$,
$\pi_{V^{C_{2}}}(\Phi^{C_{2}}X)^{K_{4}/C_{2}}$, and the limit of

$\pi_{V}^{C_{3}}(X)^{D_{6}/C_{3}}\xrightarrow{res_{e}^{C_{3}}}\pi_{V}(X)\hookleftarrow\pi_{V}(X)^{G}$

when $2,5$ are inverted.

(iii) The direct sum of $\pi_{V^{G}}(\Phi^{G}X)$,
$\pi_{V^{C_{3}}}^{D_{6}/C_{3}}(\Phi^{C_{3}}X)$,
$\pi_{V^{A_{4}}}(\Phi^{A_{4}}X)$,
$\pi_{V^{C_{5}}}^{D_{10}/C_{5}}(\Phi^{C_{5}}X)$,
$\pi_{V^{C_{3}}}(\Phi^{C_{3}}X)^{D_{6}/C_{3}}$,
$\pi_{V^{C_{5}}}(\Phi^{C_{5}}X)^{D_{10}/C_{5}}$, and the limit of

$\pi_{V|_{K_{4}}}^{K_{4}}(X)^{A_{4}/K_{4}}\xrightarrow{res_{e}^{K_{4}}}\pi_{|V|}(X)\hookleftarrow\pi_{|V|}(X)^{G}$

when $3,5$ are inverted.

###  4.6 An algebraic point of view

We want to mention a special case of Theorem 4.3 and its relation with a
recent paper by Angeltveit [Ang22]:

######  Theorem 4.19.

Let $\mathscr{F}=\mathscr{F}_{p}$ be the family of all $p$-subgroups of $G$.
Let $P\subset G$ be a Sylow $p$-subgroup. Assume that all prime factors of
$|G|$ except $p$ are inverted. Then for any virtual $G$-representation $V$,
The composition

$\pi_{V}^{G}(E\mathscr{F}_{+}\wedge
X)\rightarrow\pi_{V}^{G}(X)\xrightarrow{res_{P}^{G}}\pi_{V|_{P}}^{P}(X)$

is injective. Its image consists of all elements $x\in\pi_{V|_{P}}^{P}(X)$
such that

$res^{P}_{H}(x)=c_{g}(res^{P}_{g^{-1}Hg}(x))$

for any $H\subset P$ and $g\in G$ such that $g^{-1}Hg\subset P$. Here $c_{g}$
is the conjugation map induced by $g\in G$.

#### Proof:

It suffices to check the criterion in Theorem 3.4: For any $H\in\mathscr{F}$,
$H\lhd J$ such that $|J/H|$ is a power of $p$, since $H$ itself is a
$p$-group, $J$ is also a $p$-group. Thus $J$ cannot be outside $\mathscr{F}$
according to the construction of $\mathscr{F}$. $\Box$

When we choose $X=HM$ for some cohomological $M$, the same decomposition
problem is studied in a more algebraic way:

######  Definition 4.20.

A Mackey functor $M$ is called cohomological if $tr_{K}^{H}\circ res_{K}^{H}$
is the multiplication by $|H/K|$ for any $K\subset H\subset G$.

######  Theorem 4.21.

[Ang22, Theorem 3.1] Let $P_{1},...,P_{n}$ be Sylow subgroups of $G$, with one
for each prime factor. There is an isomorphism

$\oplus_{i}tr_{P_{i}}^{G}:\bigoplus_{i}M(G/P_{i})/\sim\rightarrow M(G/G),$

where $\sim$ is generated by

$tr_{H}^{P_{i}}(y)\sim tr_{g^{-1}Hg}^{P_{j}}(c_{g^{-1}}(y))$

for any $y\in M(G/H)$ and $g\in G$ such that $H\subset P_{i}$,
$g^{-1}Hg\subset P_{j}$.

This theorem can be used to compute equivariant homology with coefficients in
a cohomological Mackey functor (like $\underline{\mathbb{Z}}$) since

######  Proposition 4.22.

[Ang22, Proposition 2.5] Let $X$ be a $G$-CW spectrum. If $M$ is a
cohomological Mackey functor, then $\underline{\pi}_{\bigstar}^{G}(X\wedge
HM)$ is also cohomological.

Notice that, in Theorem 4.21, the equivalence relation is generated by the
following two types:

(1) The equivalence relation on one single summand $M(G/P_{i})$:

$tr_{H}^{P_{i}}(y)\sim tr_{g^{-1}Hg}^{P_{i}}(c_{g^{-1}}(y))$

for $y\in M(G/H)$ such that $H,g^{-1}Hg\subset P_{i}$;

(2) For $i\neq j$, we have

$tr_{e}^{P_{i}}(y)\sim tr_{e}^{P_{j}}(y)$

for any $y\in M(G/e)$. This is equivalent to $tr_{P_{i}}^{G}\circ
tr_{e}^{P_{i}}=tr_{P_{j}}^{G}\circ tr_{e}^{P_{j}}$ as part of the Mackey
functor structure.

Since $tr_{P_{i}}^{G}\circ res_{G}^{P_{i}}$ and $tr_{e}^{P_{i}}\circ
res_{e}^{P_{i}}$ are multiplications by $|G/P_{i}|$ and $|P_{i}|$, it’s not
hard to check that Theorem 4.21 is implied by (in fact equivalent to) its
localized version:

######  Proposition 4.23.

Assume that all prime factors of $|G|$ except $p$ are inverted. Let $P$ be a
Sylow $p$-subgroup of $G$. For any cohomological Mackey functor $M$, we have
an isomorphism

$tr_{P}^{G}:M(G/P)/\sim\rightarrow M(G/G),$

where $\sim$ is generated by

$tr_{H}^{P}(y)\sim tr_{g^{-1}Hg}^{P}(c_{g^{-1}}(y))$

for any $y\in M(G/H)$ and $g\in G$ such that $H,g^{-1}Hg\subset P$.

This proposition is related to Theorem 4.19 by the following two lemmas, which
we will prove later:

######  Lemma 4.24.

With the same assumptions as in Proposition 4.23, let $R$ be the subgroup of
$M(G/P)$ which consists of all elements $x$ such that

$res_{H}^{P}(x)=c_{g}(res_{g^{-1}Hg}^{P}(x))$

for any $H\subset P$ and $g\in G$ such that $g^{-1}Hg\subset P$. Then the
composition

$R\hookrightarrow M(G/P)\xrightarrow{tr_{P}^{G}}M(G/G)$

is an isomorphism.

######  Lemma 4.25.

Assume that all prime factors of $|G|$ except $p$ are inverted. Let
$\mathscr{F}=\mathscr{F}_{p}$. Then for any cohomological Mackey functor $M$,
we have

$\widetilde{E\mathscr{F}}\wedge HM\simeq*.$

######  Remark 4.26.

When we are computing equivariant homology with coefficients in a
cohomological Mackey functor, Lemma 4.25 shows that smashing with
$E\mathscr{F}_{+}$ makes no changes. Then Lemma 4.24 shows that the methods in
[Ang22] and our paper provide the same result.

In fact, our method also works for some non-cohomological $M$. We only need to
guarantee

$\widetilde{E\mathscr{F}}\wedge HM\simeq*.$

This condition holds if and only if $M$ is an $M_{\mathscr{F}}$-module.

######  Remark 4.27.

From an algebraic point of view, both our method and the method in [Ang22]
share the same idea: Expressing the top level $M(G/G)$ of the Mackey functor
$M$ by the lower levels.

The only difference is whether we use transfer maps or restriction maps, each
of which has its own advantage: The expression with transfer maps does not
require any localizations, while the expression with restriction maps
preserves the multiplicative structure.

#### Proof of Lemma 4.24:

Since $M$ is cohomological, $tr_{P}^{G}\circ res_{P}^{G}$ is multiplication by
$|G/P|$, hence is an isomorphism from $M(G/G)$ to itself. Thus $res_{P}^{G}$
is injective and $tr_{P}^{G}$ is surjective.

Fix $z\in M(G/G)$. For any $H\subset P$ and $g\in G$ such that
$g^{-1}Hg\subset P$, we have

$c_{g}(res_{g^{-1}Hg}^{P}(res_{P}^{G}(z)))=c_{g}(res_{g^{-1}Hg}^{G}(z))=res_{H}^{G}(c_{g}(z))=res_{H}^{G}(z)=res_{H}^{P}(res_{P}^{G}(z)).$

Since $H,g$ are arbitrary, $res_{P}^{G}(z)$ is contained in $R$. Thus
$res_{P}^{G}$ can be viewed as an injective map from $M(G/G)$ to $R$, which is
the right inverse of the composition

$R\hookrightarrow M(G/P)\xrightarrow{tr_{P}^{G}}M(G/G).$

On the other hand, for any $x\in R$, choose $H=P$ and $g\in N_{G}P$. We get
$x=c_{g}(x)$. Thus

$res_{P}^{G}(tr_{P}^{G}(x))=\sum_{gP\in N_{G}P/P}c_{g}(x)=|W_{G}P|x.$

Since $|W_{G}P|$ is inverted, $res_{P}^{G}\circ tr_{P}^{G}$ becomes a self-
isomorphism of $R$. Therefore, the composition

$R\hookrightarrow M(G/P)\xrightarrow{tr_{P}^{G}}M(G/G)$

is an isomorphism. $\Box$

#### Proof of Lemma 4.25:

With the same argument as in the proof of Proposition 4.9, it suffices to
prove $M=M\text{ }\Box\text{ }M_{\mathscr{F}}$ for any cohomological Mackey
functor $M$.

For any $H\subset G$, let $P$ be a Sylow $p$-subgroup of $H$. Then
$tr_{P}^{H}\circ res_{P}^{H}$ is multiplication by $|H/P|$, hence is an
isomorphism. Thus $res_{P}^{H}$ is an injection.

Notice that for any $p$-subgroup $K$ of $G$, according to the definition of
$M_{\mathscr{F}}$, we have $M_{\mathscr{F}}(G/K)=A(K)$. Thus $M\text{
}\Box\text{ }M_{\mathscr{F}}(G/P)\cong M(G/P)$.

Consider the following commutative diagram.

$\textstyle{M(G/H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{res_{P}^{H}}$$\textstyle{M\text{
}\Box\text{
}M_{\mathscr{F}}(G/H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{res_{P}^{H}}$$\textstyle{M(G/P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\text{
}\Box\text{ }M_{\mathscr{F}}(G/P)}$

We have shown that the left vertical map and the bottom horizonal map are
isomorphisms. Moreover, the top horizontal map is surjective, since

$M=(M\text{ }\Box\text{ }M_{\mathscr{F}})\oplus(M\text{ }\Box\text{
}N_{\mathscr{F}}).$

Thus all maps in the diagram above are isomorphisms. Since $H\subset G$ is
arbitrary, we have $M=M\text{ }\Box\text{ }M_{\mathscr{F}}$. $\Box$

##  5 Computation of $\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$

Starting from this section, we will apply our splitting method to compute
$\pi_{\bigstar}^{G}(H\underline{\mathbb{Z}})$ for several choices of $G$ as
$RO(G)$-graded rings.

We will use the same way to define generators as in [HHR17] Definition 3.4:

######  Definition 5.1.

For any non-virtual $G$-representation $V$ with $V^{G}=0$, let
$a_{V}\in\pi_{-V}(S^{0})$ be the map $S^{0}\rightarrow S^{V}$ embedding
$S^{0}$ to $0$ and $\infty$. We also use $a_{V}$ to denote the Hurewicz image
of $a_{V}\in\pi_{-V}(S^{0})$ in $\pi_{-V}(H\underline{\mathbb{Z}})$.

For any non-virtual orientable representation $V$ of dimension $n$, let
$u_{V}$ be the generator of
$\pi_{n-V}(H\underline{\mathbb{Z}})=H_{n}^{G}(S^{V};\underline{\mathbb{Z}})$
which restricts to the choice of orientation in

$\underline{H}_{n}^{G}(S^{V};\underline{\mathbb{Z}})(G/e)\cong
H_{n}(S^{n};\mathbb{Z}).$

Some important relations on these generators are given below:

######  Proposition 5.2.

(a) For any $V_{1},V_{2}$,

$a_{V_{1}+V_{2}}=a_{V_{1}}a_{V_{2}},\text{
}u_{V_{1}+V_{2}}=u_{V_{1}}u_{V_{2}}.$

(b) Let $G_{V}$ be the isotropy subgroup of $V$. Then $|G/G_{V}|a_{V}=0$.

(c) For $V,W$ both oriented with dimension $2$, with $G_{V}\subset G_{W}$, we
have

$a_{W}u_{V}=|G_{W}/G_{V}|a_{V}u_{W}.$

#### Convention:

For any $H\subset G$ and $H$-representations $V_{1},V_{2}$, we will identify
them in our computation if

$S^{V_{1}}\wedge HA_{H}\simeq S^{V_{2}}\wedge HA_{H}.$

We may not have $V_{1}\simeq V_{2}$. But in the computation of homology, it’s
not necessary to distinguish these two representations.

For the rest of this section, let $G=D_{2p}$.

Up to conjugacy, the nontrivial subgroups of $G$ consist of $C_{2}$, $C_{p}$
and $G$ itself. According to the splitting method in section 4.1, the
computation of $D_{2p}$-homotopy can be decomposed into computations of
$C_{2}$ and $C_{p}$-homotopy.

We first list all irreducible representations for these subgroups except the
trivial representation:

For $C_{2}$, we have the 1-dimensional sign representation $\sigma$.

For $C_{p}$, we have $p-1$ different 2-dimensional rotation representations,
which are identified according to the convention above. We denote them by
$\lambda$.

For $D_{2p}$, we have the sign representation $\sigma$ for $D_{2p}/C_{p}$, and
the 2-dimensional dihedral representation $\gamma$, where all elements with
order $p$ act as rotations and all elements with order $2$ act as reflections.

###  5.1 Decomposition into $C_{2}$ and $C_{p}$

We apply the splitting method and invert $2$ and $p$ separately. Notice that
$\pi_{*}(H\underline{\mathbb{Z}})=\mathbb{Z}$ which is $G$-fixed. According to
Theorem 4.16 and Proposition 4.9, we get:

(i)
$\pi_{V}^{D_{2p}}(H\underline{\mathbb{Z}})[1/p]\cong\pi^{C_{2}}_{V|_{C_{2}}}(H\underline{\mathbb{Z}})[1/p]$.

(ii)
$\pi_{V}^{D_{2p}}(H\underline{\mathbb{Z}})[1/2]\cong\pi^{C_{p}}_{V|_{C_{p}}}(H\underline{\mathbb{Z}})^{D_{2p}/C_{p}}[1/2]$.

It suffices to compute the action of $D_{2p}/C_{p}$ on
$\pi_{V|_{C_{p}}}^{C_{p}}(H\underline{\mathbb{Z}})$. Express the generators of
$D_{2p}$ by $\zeta$ and $\tau$ such that $\zeta^{p}=\tau^{2}=1$,
$\zeta\tau=\tau\zeta^{-1}$.

First we consider the case that $V$ has no copies of $\sigma$:

######  Proposition 5.3.

Let $V=k+n\gamma$ for some integers $k,n$. Then the action of $\tau$ on
$[S^{V},\Sigma^{t}H\underline{\mathbb{Z}}]^{C_{p}}$ is multiplication by $-1$
if $|k-t|\equiv 2$ or $3$ (mod $4$). Otherwise, the action is multiplication
by $1$.

#### Proof:

Notice that $[S^{V},\Sigma^{t}H\underline{\mathbb{Z}}]^{C_{p}}$ can be viewed
as the equivariant cohomology or homology of $S^{|n|\gamma}$ with coefficients
in $\underline{\mathbb{Z}}$. The computation can be done in a cellular way. We
first give $S^{|n|\gamma}$ a $C_{p}$-CW structure.

We view $S^{\gamma}$ as $\gamma$ compactified at $\infty$. The action of
$\zeta$ is the counter-clockwise rotation by $2\pi/p$ and the action of $\tau$
is the reflection by the $x$-axis. The standard $CW$ structure of $S^{\gamma}$
can be described as follows:

As a based non-equivariant space, $S^{\gamma}\cong S^{2}$ can be constructed
by one single $2$-cell and the base point at the origin. Denote that $2$-cell
by $a$.

As a based $C_{p}$-space, $S^{\gamma}\cong S^{\lambda}$ can be constructed by:

One $C_{p}$-free $2$-cell denoted by $\\{b_{1},b_{2},...,b_{p}\\}$, such that
each $b_{i}$ has image

$\left\\{(r\cos\theta,r\sin\theta):\text{ }0\leq
r\leq\infty,\frac{2\pi(i-1)}{p}\leq\theta\leq\frac{2\pi i}{p}\right\\};$

One $C_{p}$-free $1$-cell denoted by $\\{c_{1},c_{2},...,c_{p}\\}$, such that
each $c_{i}$ has image

$\left\\{\left(r\cos\frac{2\pi i}{p},r\sin\frac{2\pi i}{p}\right):\text{
}0\leq r\leq\infty\right\\};$

One $C_{p}$-fixed $0$-cell $d$ at $\infty$ and the base point at $0$.

Here the subscripts of $b_{i}$ and $c_{i}$ are defined modulo $p$.

For both $CW$ structures, the action of $\tau$ on $S^{\gamma}$ can be made
into a cellular map:

The action of $\tau$ on the whole $S^{\gamma}$ is a reflection. Thus $\tau
a=-a$.

The action of $\tau$ gives a permutation among the images of
$b_{1},b_{2},...,b_{p}$ and reverses the orientations of these $2$-cells. Thus
$\tau b_{i}=-b_{p+1-i}$.

The action of $\tau$ gives a permutation among the images of
$c_{1},c_{2},...,c_{p}$ and keeps the orientations of these $1$-cells. Thus
$\tau c_{i}=c_{p+1-i}$.

Finally, $\tau d=d$ since the point $\infty$ is fixed.

Since $S^{|n|\gamma}=S^{\gamma}\wedge S^{\gamma}\wedge...\wedge S^{\gamma}$,
we can construct a CW structure as follows:

For $i\in\mathbb{Z}$ and $0\leq j\leq|n|-1$, let

$e_{2j+2,i}:=(\zeta^{i}a,...,\zeta^{i}a,b_{i},d,...,d)$
$e_{2j+1,i}:=(\zeta^{i}a,...,\zeta^{i}a,c_{i},d,...,d)$

with $j$ copies of $\zeta^{i}a$ and $|n|-1-j$ copies of $d$.

We also have one $0$-cell $e_{0}:=(d,...,d)$ and the base point.

Notice that $\zeta b_{i}=b_{i+1}$, $\zeta c_{i}=c_{i+1}$. This construction
makes $S^{|n|\gamma}$ into a $C_{p}$-CW complex with one $C_{p}$-cell (except
the base point) in each degree between $0$ and $2|n|$. The action of $\tau$ on
$S^{|n|\gamma}$ is induced by the action on each copy of $S^{\gamma}$, hence
is also made cellular:

$\tau e_{2j+2,i}=(-1)^{j+1}e_{2j+2,p+1-i},\text{ }\tau
e_{2j+1,i}=(-1)^{j}e_{2j+1,p+1-i},\text{ }\tau e_{0}=e_{0}.$

Recall that $H_{*}^{C_{p}}(S^{|n|\gamma};\underline{\mathbb{Z}})$ and
$H^{*}_{C_{p}}(S^{|n|\gamma};\underline{\mathbb{Z}})$ can be computed by the
following chain and cochain complexes:

$C_{*}^{C_{p}}(S^{|n|\gamma};\underline{\mathbb{Z}}):=\underline{C}_{*}(S^{|n|\gamma})\otimes_{\mathcal{O}_{C_{p}}}\underline{\mathbb{Z}};$
$C^{*}_{C_{p}}(S^{|n|\gamma};\underline{\mathbb{Z}}):=Hom_{\mathcal{O}_{C_{p}}}(\underline{C}_{*}(S^{|n|\gamma});\underline{\mathbb{Z}}),$

where we view $\underline{\mathbb{Z}}$ as a covariant coefficient system in
the first equation and a contravariant one in the second equation.

The actions of $\tau$ on the homology and cohomology are obtained by applying
$\tau$ on both $\underline{C}_{*}(S^{|n|\gamma})$ and
$\underline{\mathbb{Z}}$.

Since $S^{|n|\gamma}$ has only $C_{p}$-free cells in positive degrees and only
$C_{p}$-fixed cells in degree $0$, and $\underline{\mathbb{Z}}$ is the
constant coefficient system, we have

$C_{*}^{C_{p}}(S^{|n|\gamma};\underline{\mathbb{Z}}):=\underline{C}_{*}(S^{|n|\gamma})\otimes_{\mathcal{O}_{C_{p}}}\underline{\mathbb{Z}}\cong
C_{*}(S^{|n|\gamma})/C_{p};$
$C^{*}_{C_{p}}(S^{|n|\gamma};\underline{\mathbb{Z}}):=Hom_{\mathcal{O}_{C_{p}}}(\underline{C}_{*}(S^{|n|\gamma});\underline{\mathbb{Z}})\cong
Hom_{\mathbb{Z}}(C_{*}(S^{|n|\gamma})/C_{p};\mathbb{Z}).$

Let $e_{2j}$ and $e_{2j-1}$ be the orbits of cells $e_{2j,i}$ and
$e_{2j-1,i}$. The induced $\tau$-action on $C_{*}(S^{|n|\gamma})/C_{p}$ is
expressed as

$\tau e_{2j}=(-1)^{j}e_{2j},\text{ }\tau e_{2j-1}=(-1)^{j-1}e_{2j-1}.$

Therefore, the $\tau$-action on the homology and cohomology is multiplication
by $-1$ when the degree is congruent to $2$ or $3$ mod $4$. Otherwise the
$\tau$-action is multiplication by $1$. $\Box$

Now we consider the case that $V$ also contains copies of $\sigma$. For any
$D_{2p}$-spectra $X,Y$, let $\tau$ act on $[X,Y]^{C_{p}}$ by conjugation.
Since $S^{\sigma}\cong S^{1}$ as $C_{p}$-spaces, we have

$[X,Y]^{C_{p}}\cong[X,S^{m(\sigma-1)}\wedge Y]^{C_{p}}.$

Since $\tau$ acts on $S^{\sigma}$ as a reflection, the composition

$[X,Y]^{C_{p}}\cong[X,S^{m(\sigma-1)}\wedge
Y]^{C_{p}}\xrightarrow{\tau}[X,S^{m(\sigma-1)}\wedge
Y]^{C_{p}}\cong[X,Y]^{C_{p}}$

agrees with the $\tau$-action on $[X,Y]^{C_{p}}$ multiplied by $(-1)^{m}$.

Together with Proposition 5.3, we get

######  Proposition 5.4.

Let $V=k+m\sigma+n\gamma$. The action of $\tau$ on
$[S^{V},\Sigma^{t}H\underline{\mathbb{Z}}]^{C_{p}}$ is multiplication by $-1$
if $[|k+m-t|/2]+m$ is odd. Otherwise, the action is multiplication by $1$.

In conclusion, we have

######  Proposition 5.5.

For any integers $k,m,n$,

$[S^{k+m\sigma+n\gamma},H\underline{\mathbb{Z}}]^{G}[1/p]\cong[S^{(k+n)+(m+n)\sigma},H\underline{\mathbb{Z}}]^{C_{2}}[1/p],$
$[S^{k+m\sigma+n\gamma},H\underline{\mathbb{Z}}]^{G}[1/2]\cong\begin{cases}0,\text{
if }[|k+m|/2]+m\text{ is odd,}\\\
[S^{(k+m)+n\lambda},H\underline{\mathbb{Z}}]^{C_{p}}[1/2],\text{ if
}[|k+m|/2]+m\text{ is even.}\end{cases}$

###  5.2 A complete expression

Now we combine Proposition 5.5 with the computations of
$\pi_{\bigstar}^{C_{2}}(H\underline{\mathbb{Z}})$ and
$\pi_{\bigstar}^{C_{p}}(H\underline{\mathbb{Z}})$ to get an explicit
expression for $\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$. We refer to
[Zen18], which provides a modern method to compute $C_{2}$ and
$C_{p}$-homotopy of $H\underline{\mathbb{Z}}$ and gives the following
expression:

######  Theorem 5.6.

The $C_{2}$ and $C_{p}$-homology of a point are given below:

$\pi^{C_{2}}_{\bigstar}(H\underline{\mathbb{Z}})=\mathbb{Z}[u_{2\sigma},a_{\sigma}]/(2a_{\sigma})\oplus\left(\bigoplus_{i>0}2\mathbb{Z}\langle
u_{2\sigma}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/2\langle\Sigma^{-1}u_{2\sigma}^{-j}a_{\sigma}^{-k}\rangle\right).$
$\pi^{C_{p}}_{\bigstar}(H\underline{\mathbb{Z}})=\mathbb{Z}[u_{\lambda},a_{\lambda}]/(pa_{\lambda})\oplus\left(\bigoplus_{i>0}p\mathbb{Z}\langle
u_{\lambda}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/p\langle\Sigma^{-1}u_{\lambda}^{-j}a_{\lambda}^{-k}\rangle\right).$

Here the generators $u_{2\sigma},a_{\sigma},u_{\lambda},a_{\lambda}$ are
defined in Definition 5.1.

We can obtain generating elements in
$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$ by tracing the pre-image of
generators in $\pi_{\bigstar}^{C_{2}}(H\underline{\mathbb{Z}})$ and
$\pi_{\bigstar}^{C_{p}}(H\underline{\mathbb{Z}})$.

First we consider the case when $p$ is inverted. According to the first
equation in Proposition 5.5, we have

(1)

$\pi_{0}^{C_{2}}(H\underline{\mathbb{Z}})\cong\pi_{c(1+\sigma-\gamma)}^{D_{2p}}(H\underline{\mathbb{Z}}),\text{
}\forall c\in\mathbb{Z}.$

We use $u_{\gamma-\sigma}$ to denote the generator of
$\pi_{1+\sigma-\gamma}^{D_{2p}}(H\underline{\mathbb{Z}})$. Then
$u_{\gamma-\sigma}$ is invertible.

(2)

$\pi_{k+m\sigma}^{C_{2}}(H\underline{\mathbb{Z}})\cong\pi_{k+m\sigma+c(1+\sigma-\gamma)}^{D_{2p}}(H\underline{\mathbb{Z}}),\text{
}\forall c\in\mathbb{Z}.$

Thus the pre-images of
$u_{2\sigma},a_{\sigma}\in\pi_{\bigstar}^{C_{2}}(H\underline{\mathbb{Z}})$ are
$u_{2\sigma}u_{\gamma-\sigma}^{c},a_{\sigma}u_{\gamma-\sigma}^{c}\in\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$,
for any $c\in\mathbb{Z}$.

Therefore, $\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})$ can be obtained
by adding the invertible $u_{\gamma-\sigma}$ to
$\pi_{\bigstar}^{C_{2}}(H\underline{\mathbb{Z}})$:

$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})[\frac{1}{p}]=\mathbb{Z}[\frac{1}{p}][u_{2\sigma},a_{\sigma},u_{\gamma-\sigma}^{\pm}]/(2a_{\sigma})$
$\oplus\left(\bigoplus_{i>0}2\mathbb{Z}[\frac{1}{p}][u_{\gamma-\sigma}^{\pm}]\langle
u_{2\sigma}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/2[u_{\gamma-\sigma}^{\pm}]\langle\Sigma^{-1}u_{2\sigma}^{-j}a_{\sigma}^{-k}\rangle\right).$

The case when $2$ is inverted is similar. According to the second equation in
Proposition 5.5, we have:

(1)

$\pi_{0}^{C_{p}}(H\underline{\mathbb{Z}})\cong\pi_{2c(1-\sigma)}^{D_{2p}}(H\underline{\mathbb{Z}}),\text{
}\forall c\in\mathbb{Z}.$

Thus $u_{2\sigma}$, as the generator of
$\pi_{2-2\sigma}^{D_{2p}}(H\underline{\mathbb{Z}})$, becomes invertible.

(2)

$\pi_{2-\lambda}^{C_{p}}(H\underline{\mathbb{Z}})\cong\pi_{1+\sigma-\gamma+2c(1-\sigma)}^{D_{2p}}(H\underline{\mathbb{Z}}),\text{
}\forall c\in\mathbb{Z}.$

Thus the pre-image of
$u_{\lambda}\in\pi_{\bigstar}^{C_{p}}(H\underline{\mathbb{Z}})$ is
$u_{\gamma-\sigma}u_{2\sigma}^{c}$, for any $c\in\mathbb{Z}$.

(3)

$\pi_{-\lambda}^{C_{p}}(H\underline{\mathbb{Z}})\cong\pi_{-\gamma+2c(1-\sigma)}^{D_{2p}}(H\underline{\mathbb{Z}}),\text{
}\forall c\in\mathbb{Z}.$

Thus the pre-image of
$a_{\lambda}\in\pi_{\bigstar}^{C_{p}}(H\underline{\mathbb{Z}})$ is
$a_{\gamma}u_{2\sigma}^{c}$, for any $c\in\mathbb{Z}$.

Therefore, we have

$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})[\frac{1}{2}]=\mathbb{Z}[\frac{1}{2}][u_{\gamma-\sigma},a_{\gamma},u_{2\sigma}^{\pm}]/(pa_{\gamma})$
$\oplus\left(\bigoplus_{i>0}p\mathbb{Z}[\frac{1}{2}][u_{2\sigma}^{\pm}]\langle
u_{\gamma-\sigma}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/p[u_{2\sigma}^{\pm}]\langle\Sigma^{-1}u_{\gamma-\sigma}^{-j}a_{\gamma}^{-k}\rangle\right).$

Finally, we just need to glue
$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})[1/p]$ and
$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})[1/2]$ together.

######  Theorem 5.7.

Recall that we obtain $a_{\sigma}$, $a_{\gamma}$ and $u_{2\sigma}$ from
Definition 5.1 and choose $u_{\gamma-\sigma}$ to be the generator of

$\pi_{1+\sigma-\gamma}^{D_{2p}}(H\underline{\mathbb{Z}})\cong\mathbb{Z}.$

Then we have

$\pi_{\bigstar}^{D_{2p}}(H\underline{\mathbb{Z}})=\mathbb{Z}[u_{\gamma-\sigma},u_{2\sigma},a_{\sigma},a_{\gamma}]/(2a_{\sigma},pa_{\gamma})$
$\oplus\left(\bigoplus_{i>0}2\mathbb{Z}[u_{\gamma-\sigma}]\langle
u_{2\sigma}^{-i}\rangle\right)\oplus\left(\bigoplus_{i>0}p\mathbb{Z}[u_{2\sigma}]\langle
u_{\gamma-\sigma}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}2p\mathbb{Z}\langle
u_{2\sigma}^{-j}u_{\gamma-\sigma}^{-k}\rangle\right)$
$\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/p[u_{\gamma-\sigma}]\langle
a_{\gamma}^{j}u_{2\sigma}^{-k}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/2[u_{2\sigma}]\langle
a_{\sigma}^{j}u_{\gamma-\sigma}^{-k}\rangle\right)$
$\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/2[u_{\gamma-\sigma}^{\pm}]\langle\Sigma^{-1}u_{2\sigma}^{-j}a_{\sigma}^{-k}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/p[u_{2\sigma}^{\pm}]\langle\Sigma^{-1}u_{\gamma-\sigma}^{-j}a_{\gamma}^{-k}\rangle\right).$

##  6 Computation of $\pi_{\bigstar}^{A_{5}}(H\underline{\mathbb{Z}})$

In this section, we will use the splitting method to compute
$\pi_{\bigstar}^{G}(H\underline{\mathbb{Z}})$ for $G=A_{5}$. First we list all
subgroups of $A_{5}$.

Subgroups of $A_{5}$ up to conjugacy:

Sylow $2$-subgroup $K_{4}=\langle(12)(34),(13)(24)\rangle$, which is
isomorphic to $C_{2}\times C_{2}$;

Sylow $3$-subgroup $C_{3}=\langle(123)\rangle$;

Sylow $5$-subgroup $C_{5}=\langle(12345)\rangle$;

Normalizers of Sylow subgroups:

$A_{4}=N_{G}K_{4}=\langle(12)(34),(123)\rangle$;

$D_{6}=N_{G}C_{3}=\langle(123),(12)(45)\rangle$;

$D_{10}=N_{G}C_{5}=\langle(12345),(15)(24)\rangle$;

Other subgroups: $C_{2}=\langle(12)(34)\rangle$ and the trivial subgroup.

We also list all irreducible real representations of $A_{5}$ and its
subgroups. Since there are lots of groups, we express the group and dimension
of a representation in the subscript. For example, $V_{H,n}$ is an
$n$-dimension $H$-representation. The trivial representations of all groups
will be written as $1$.

Irreducible representations of $A_{5}$:

$V_{A_{5},3}^{+}$: This is obtained by viewing $A_{5}$ as the group of
rotations of the regular icosahedron in the 3 dimensional space.

$V_{A_{5},3}^{-}$: This is the same as $V_{A_{5},3}^{+}$ except applying the
conjugation by $(12)$ (inside $S_{5}$) first.

For the two representations $V_{A_{5},3}^{+}$ and $V_{A_{5},3}^{-}$, we will
see that all our computations which work for one of them also work for the
other. This further implies that the two representation spheres are
$HA_{G}$-equivalent. We use $V_{A_{5},3}$ to denote both representations.

$V_{A_{5},4}$: Consider the space of functions from $\\{1,2,3,4,5\\}$ to
$\mathbb{R}$ whose images have sum $0$. The action of $A_{5}$ is the natural
permutation.

$V_{A_{5},5}$: Consider the space of functions from antipodal pairs of
vertices of the regular icosahedron, such that the sum of all images of the
function is $0$. The action of $A_{5}$ is induced by the rotations of the
icosahedron.

Irreducible representations of $A_{4}$:

$V_{A_{4},2}$: All elements in $A_{4}$ with order $3$ become rotations of
either $2\pi/3$ or $\pi/3$. All elements with order $2$ act trivially.

$V_{A_{4},3}$: Consider the space of functions from $\\{1,2,3,4\\}$ to
$\mathbb{R}$ whose images have sum $0$. The action of $A_{4}$ is the natural
permutation.

Irreducible representations of $D_{2p}$ ($p=3$ or $5$):

$V_{D_{2p},1}$: The sign representation of $D_{2p}/C_{p}$, with $C_{p}$ acting
trivially.

$V_{D_{2p},2}$: Dihedral representations, where all elements with odd orders
act as rotations, and all elements with order 2 act as reflections.

Irreducible representations of $K_{4}$:

Three different sign representations
$V_{K_{4},1,1},V_{K_{4},1,2},V_{K_{4},1,3}$ which correspond to the three
proper subgroups of $K_{4}$.

In addition, we use $V_{K_{4},3}$ to denote
$V_{K_{4},1,1}+V_{K_{4},1,2}+V_{K_{4},1,3}$ (which is not irreducible).

Irreducible representations of $C_{p}$: ($p=3$ or $5$)

$V_{C_{p},2}$: All elements act as rotations.

Irreducible representations of $C_{2}$:

$V_{C_{2},1}$: Sign representation.

The representations of these groups are related by restrictions:

The restrictions of $V_{A_{5},3}$ in $A_{4},D_{6},D_{10}$ are $V_{A_{4},3}$,
$V_{D_{6},1}+V_{D_{6},2}$ and $V_{D_{10},1}+V_{D_{10},2}$.

The restrictions of $V_{A_{5},4}$ in $A_{4},D_{6},D_{10}$ are $1+V_{A_{4},3}$,
$1+V_{D_{6},1}+V_{D_{6},2}$ and $2V_{D_{10},2}$.

The restrictions of $V_{A_{5},5}$ in $A_{4},D_{6},D_{10}$ are
$V_{A_{4},2}+V_{A_{4},3}$, $1+2V_{D_{6},2}$ and $1+2V_{D_{10},2}$.

The restrictions of $V_{A_{4},2},V_{A_{4},3}$ in $K_{4}$ are $2$ and
$V_{K_{4},3}$.

The restrictions of $V_{D_{2p},1},V_{D_{2p},2}$ in $C_{p}$ are $1$ and
$V_{C_{p},2}$ for $p=3,5$.

Applying Theorem 4.18, eliminating the geometric fixed point spectra with
Proposition 4.9, and simplifying the diagrams by the fact that
$\pi_{*}(H\underline{\mathbb{Z}})=\mathbb{Z}$ is fixed by $G$, we get

######  Proposition 6.1.

For $p=2,3,5$, let $P$ be a Sylow $p$-subgroup of $G=A_{5}$. Assume that all
prime factors of $|G|$ except $p$ are inverted. Then

$\pi_{V}^{G}(H\underline{\mathbb{Z}})\cong\pi_{V|_{P}}^{P}(H\underline{\mathbb{Z}})^{W_{G}P}.$

Now we can use the same strategy as in section 5.2 to obtain explicit
expressions of localized homotopy:

######  Theorem 6.2.

When $2,5$ are inverted, we have

$\pi_{\bigstar}^{A_{5}}(H\underline{\mathbb{Z}})=\mathbb{Z}[u_{V_{4}-V_{3}}^{\pm},u_{2V_{3}-V_{5}}^{\pm}]\otimes$
$\left[\mathbb{Z}[u_{V_{3}},a_{V_{5}-1}]/(3a_{V_{5}-1})\oplus\left(\bigoplus_{i>0}3\mathbb{Z}\langle
u_{V_{3}}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/3\langle\Sigma^{-1}u_{V_{3}}^{-j}a_{V_{5}-1}^{-k}\rangle\right)\right].$

Here $V_{i}$ denotes $V_{A_{5},i}$, $i=3,4,5$. For any virtual representation
$V$, we have $|a_{V}|=-V$, $|u_{V}|=|V|-V$.

######  Theorem 6.3.

When $2,3$ are inverted, we have

$\pi_{\bigstar}^{A_{5}}(H\underline{\mathbb{Z}})=\mathbb{Z}[u_{V_{5}-V_{4}}^{\pm},u_{2V_{3}-V_{4}}^{\pm}]\otimes$
$\left[\mathbb{Z}[u_{V_{3}},a_{V_{4}}]/(5a_{V_{4}})\oplus\left(\bigoplus_{i>0}5\mathbb{Z}\langle
u_{V_{3}}^{-i}\rangle\right)\oplus\left(\bigoplus_{j,k>0}\mathbb{Z}/5\langle\Sigma^{-1}u_{V_{3}}^{-j}a_{V_{4}}^{-k}\rangle\right)\right].$

For $p=2$, we have a similar result:

######  Theorem 6.4.

Let $n_{1},n_{3},n_{4},n_{5}$ be arbitrary integers. Assume that $3,5$ are
inverted. Then

$\pi_{n_{1}+n_{3}V_{A_{5},3}+n_{4}V_{A_{5},4}+n_{5}V_{A_{5},5}}^{A_{5}}(H\underline{\mathbb{Z}})\cong\pi_{(n_{1}+n_{4}+2n_{5})+(n_{3}+n_{4}+n_{5})V_{K_{4},3}}^{K_{4}}(H\underline{\mathbb{Z}})^{A_{4}/K_{4}}.$

The computation of $\pi_{*+*V_{K_{4},3}}^{K_{4}}(H\underline{\mathbb{Z}})$ is
given in Appendix B. Unlike Theorems 6.2 and 6.3, it is quite hard to give an
explicit expression for this ring. We will provide a less explicit, but still
computable expression in Theorems B.7 and B.15.

The unlocalized $\pi_{\bigstar}^{A_{5}}(H\underline{\mathbb{Z}})$ can be
recovered from Theorems 6.2, 6.3, 6.4, B.7, and B.15.

##  7 Mackey functor valued homology

In this section, we will discuss the computation of the Mackey functor valued
homotopy $\underline{\pi}_{\bigstar}^{G}(H\underline{\mathbb{Z}})$. All
arguments and theorems in this section still work when
$\underline{\mathbb{Z}}$ is replaced by other constant Mackey functors.

#### Notation:

We use $\Pi$ to denote the Mackey functor
$\underline{\pi}_{V}^{G}(H\underline{\mathbb{Z}})$ when the virtual
$G$-representation $V$ is fixed. Then we have

$\Pi(G/H)=[G/H_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{G}\cong[S^{V},H\underline{\mathbb{Z}}]^{H}=\pi_{V|_{H}}^{H}(H\underline{\mathbb{Z}}),$
$\Pi|_{H}=\underline{\pi}_{V|_{H}}^{H}(H\underline{\mathbb{Z}}).$

When all prime factors of $|G|$ except $p$ are inverted, since
$H\underline{\mathbb{Z}}=H\underline{\mathbb{Z}}\wedge E\mathscr{F}_{+}$, we
can use Theorem 4.19 and Proposition 4.9 to compute
$[S^{V},H\underline{\mathbb{Z}}]^{H}$ as a subgroup of
$[S^{V},H\underline{\mathbb{Z}}]^{P}$ where $P$ is a Sylow $p$-subgroup of
$H$. In order to get the Mackey functor structure on $\Pi$, we still need to
compute the restrction and transfer maps.

Consider any $H_{1}\subset H_{2}\subset G$. Choose Sylow $p$-subgroups
$P_{1},P_{2}$ of $H_{1},H_{2}$ such that $P_{1}\subset P_{2}$. We will compute
$tr_{H_{1}}^{H_{2}}$ and $res_{H_{1}}^{H_{2}}$ with the Mackey functor
structure of $\Pi|_{P_{2}}$ and conjugation maps.

The restriction map is simple:

######  Proposition 7.1.

When computing $\Pi(G/H_{1})$ and $\Pi(G/H_{2})$ as subgroups of
$\Pi(G/P_{1})$ and $\Pi(G/P_{2})$, $res_{H_{1}}^{H_{2}}$ is determined by
$res_{P_{1}}^{P_{2}}$.

#### Proof:

The inclusion $\Pi(G/H_{i})\hookrightarrow\Pi(G/P_{i})$ is given by
$res_{P_{i}}^{H_{i}}$, $i=1,2$. The proposition follows from the fact that

$res_{P_{1}}^{P_{2}}\circ res_{P_{2}}^{H_{2}}=res_{P_{1}}^{H_{1}}\circ
res_{H_{1}}^{H_{2}}.$

$\Box$

The transfer map is more complicated:

Write the underlying $P_{2}$-set of $H_{2}/H_{1}$ as the disjoint union of
$P_{2}$-orbits:

$H_{2}/H_{1}=\bigsqcup_{i=1}^{t}P_{2}/Q_{i}.$

Choose $g_{1},g_{2},...,g_{t}\in H_{2}$ such that each $Q_{i}$ corresponds to
$g_{i}H_{1}$ inside $H_{2}/H_{1}$. Then we have $g_{i}^{-1}Q_{i}g_{i}\subset
H_{1}$, $i=1,2,...,t$.

######  Proposition 7.2.

When computing $\Pi(G/H_{1})$ and $\Pi(G/H_{2})$ as subgroups of
$\Pi(G/P_{1})$ and $\Pi(G/P_{2})$, $tr_{H_{1}}^{H_{2}}$ is determined by

$\bigoplus_{i}tr_{Q_{i}}^{P_{2}}\circ c_{g_{i}}\circ
res_{g_{i}^{-1}Q_{i}g_{i}}^{P_{1}}.$

#### Proof:

Notice that

$tr_{H_{1}}^{H_{2}}:[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{H_{2}}\rightarrow[S^{V},H\underline{\mathbb{Z}}]^{H_{2}}$

is induced by the $H_{2}$-stable map

$S^{0}=\Sigma^{-W}S^{W}\rightarrow\Sigma^{-W}(H_{2}/H_{1})_{+}\wedge
S^{W}=(H_{2}/H_{1})_{+}.$

Here $W$ is an $H_{2}$-representation with an embedding
$H_{2}/H_{1}\hookrightarrow W$. Taking a tubular neighborhood of $H_{2}/H_{1}$
inside $S^{W}$ and collapsing the complement to a point, we get the map
$S^{W}\rightarrow(H_{2}/H_{1})_{+}\wedge S^{W}$ which defines the middle map
above.

Now consider the commutative diagram

$\textstyle{[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{H_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{tr_{H_{1}}^{H_{2}}}$$\textstyle{[S^{V},H\underline{\mathbb{Z}}]^{H_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{P_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[S^{V},H\underline{\mathbb{Z}}]^{P_{2}}}$

where the vertical maps come from taking the underlying $P_{2}$-maps, hence
are inclusions by Theorem 4.3. The bottom horizontal map is induced by the
underlying $P_{2}$-stable map of $S^{0}\rightarrow(H_{2}/H_{1})_{+}$.

For each $i=1,2,...,t$, compose the $P_{2}$-stable map
$S^{0}\rightarrow(H_{2}/H_{1})_{+}$ with the projection
$H_{2}/H_{1}\rightarrow P_{2}/Q_{i}$. The composition can be expressed by

$S^{0}=\Sigma^{-W}S^{W}\rightarrow\Sigma^{-W}(P_{2}/Q_{i})_{+}\wedge
S^{W}=(P_{2}/Q_{i})_{+},$

which exactly induces $tr_{Q_{i}}^{P_{2}}$.

Thus the bottom map

$[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{P_{2}}\rightarrow[S^{V},H\underline{\mathbb{Z}}]^{P_{2}}$

is given by

$[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{P_{2}}\cong\bigoplus_{i=1}^{t}[(P_{2}/Q_{i})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{P_{2}}\xrightarrow{\oplus
tr_{Q_{i}}^{P_{2}}}[S^{V},H\underline{\mathbb{Z}}]^{P_{2}}.$

Moreover, the map

$[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{H_{2}}\rightarrow[(P_{2}/Q_{i})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{P_{2}}$

is obtained by taking projection and the underlying $Q_{i}$-map. Thus we can
write it as

$[(H_{2}/H_{1})_{+}\wedge
S^{V},H\underline{\mathbb{Z}}]^{H_{2}}\cong[S^{V},H\underline{\mathbb{Z}}]^{g_{i}H_{1}g_{i}^{-1}}\xrightarrow{res_{g_{i}P_{1}g_{i}^{-1}}^{g_{i}H_{1}g_{i}^{-1}}}[S^{V},H\underline{\mathbb{Z}}]^{g_{i}P_{1}g_{i}^{-1}}\xrightarrow{res_{Q_{i}}^{g_{i}P_{1}g_{i}^{-1}}}[S^{V},H\underline{\mathbb{Z}}]^{Q_{i}}.$

Commute the conjugation by $g_{i}$ with the restriction map. The composition
above becomes

$c_{g_{i}}\circ res_{g_{i}^{-1}Q_{i}g_{i}}^{P_{1}}\circ res_{P_{1}}^{H_{1}}.$

In conclusion, the left vertical map of the commutative diagram above can be
factored as

$(c_{g_{i}}\circ res_{g_{i}^{-1}Q_{i}g_{i}}^{P_{1}})_{i}\circ
res_{P_{1}}^{H_{1}},$

while the bottom horizontal map is given by $\oplus_{i}tr_{Q_{i}}^{P_{2}}$.
Notice that $res_{P_{1}}^{H_{1}}$ is the map we used to compute $\Pi(G/H_{1})$
by Theorem 4.19. Thus $tr_{H_{1}}^{H_{2}}$ is determined by

$\bigoplus_{i}tr_{Q_{i}}^{P_{2}}\circ c_{g_{i}}\circ
res_{g_{i}^{-1}Q_{i}g_{i}}^{P_{1}}.$

$\Box$

######  Remark 7.3.

In practice, usually we do not need to compute both restriction and transfer
maps. It’s possible that one of them is induced by the other together with the
Mackey functor structure, especially when the Mackey functor is cohomological.
Recall that the definition and properties of a cohomological Mackey functor
are given in section 4.4.

Theoretically, whenever our splitting method in section 4 applies, we can
always compute the homotopy of $H\underline{\mathbb{Z}}$ as an $RO(G)$-graded
Green functor instead of an $RO(G)$-graded ring since Theorem 4.19 also
preserves the multiplicative structure. But it would be too complicated to
finish the complete computation by hand.

For the rest of this section, we will give two examples to show how
Proposition 7.1 and Proposition 7.2 work in practice.

###  7.1 One example

We will focus on $G=A_{5}$ since it is the most complicated group for which
$\underline{\pi}_{\bigstar}^{G}(H\underline{\mathbb{Z}})$ is computable by our
splitting methods. The computation for other smaller groups (like $D_{2p}$)
will be similar and more simple.

When $H$ is a subgroup of $A_{5}$, a similar argument as in Proposition 6.1
gives us

######  Proposition 7.4.

Assume that all prime factors of $|A_{5}|$ except $p$ are inverted. For any
$H\subset A_{5}$, let $P$ be a Sylow $p$-subgroup of $H$. Then we have

$res_{P}^{H}:\Pi(A_{5}/H)\xrightarrow{\sim}\Pi(A_{5}/P)^{W_{H}P}.$

We choose a specific virtual representation
$V=V_{A_{5},3}+V_{A_{5},4}-V_{A_{5},5}-2$ and compute
$\underline{\pi}_{V}^{G}(H\underline{\mathbb{Z}})$. We will provide details in
the application of Proposition 7.1 and how to use the Mackey functor structure
to simplify our computation.

We have $V|_{C_{3}}=0$, $V|_{K_{4}}=V_{K_{4},3}-3$,
$V|_{C_{5}}=V_{C_{5},2}-2$. The restrictions of $V$ as Mackey functors over
$C_{3},K_{4},C_{5}$ are given below:

$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{3}$
$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{2}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$
$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{5}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$

The Weyl group actions are all trivial in this case.

Using Proposition 7.4 and Proposition 7.1, we can obtain the values of all
levels in $\Pi$ and the restriction maps when two of the three prime factors
$2,3,5$ are inverted.

The first diagram below is the standard expression of an $A_{5}$-Mackey
functor. The other three are the expressions of $\Pi$ (with only restriction
maps) when 6, 10, or 15 is inverted respectively.

$\textstyle{M(A_{5}/A_{5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/D_{6})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/A_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/D_{10})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/K_{4})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/C_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/C_{5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/C_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(A_{5}/e)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\text{ }\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$551$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$11$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$11$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$15$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$1$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$1$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$5$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$1$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\text{ }\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$111$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$11$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$11$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$11$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$1$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$1$\textstyle{\text{
}\text{ }\mathbb{Z}\text{ }\text{