#  $RO(C_{2})$-graded Cohomology of $C_{2}$-equivariant Eilenberg-Mac Lane
spaces

UĞUR YİĞİT  Istanbul Medeniyet University, Istanbul, TURKEY
<EMAIL_ADDRESS>Department of Mathematics, Istanbul Medeniyet
University, H1-20, Istanbul, TURKEY 34700

###### Abstract.

In this paper, we calculate $RO(C_{2})$-graded cohomology of
$C_{2}$-equivariant Eilenberg-Mac Lane spaces $K(\underline{Z/2},n+\sigma)$
for $n\geq 0$. These can be used to give the relation between equivariant
lambda algebra and equivariant Adams resolution and equivariant unstable Adams
spectral sequence, which are defined in author‘s dissertation.

###### Key words and phrases:

Equivariant Cohomology, Equivariant Steenrod algebra, Equivariant Eilenberg-
Mac Lane Spaces

###### 2000 Mathematics Subject Classification:

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $C_{2}$-Equivariant Steenrod Algebra
  4. 4 Equivariant Eilenberg-Mac Lane Spaces
  5. 5 Cohomology of Eilenberg-Mac Lane Spaces

##  1\. Introduction

An ordinary cohomology theory $H_{G}^{\star}(-:\underline{M})$ on $G$-spaces
with Mackey functor $\underline{M}$ coefficients and graded by real orthogonal
representations is defined by Lewis, May and Mcclure [8]. In this paper, we
compute the $RO(C_{2})$-graded cohomology of the $C_{2}$-equivariant
Eilenberg-Mac Lane spaces with the constant Mackey functor
$\underline{M}=\underline{Z/2}$ coefficients, which are crucial to give the
relation between the equivariant lambda algebra and the equivariant unstable
Adams resolution and equivariant unstable Adams spectral sequence, which is
given by Mahowald [12] in the classical case. Throughout this paper,
$H^{\star}(-)$ denotes the ordinary $RO(C_{2})$-graded cohomology of a
$C_{2}$-space with the constant Mackey functor coefficients
$\underline{\mathbb{Z}/2}$.

To compute the $RO(C_{2})$-graded cohomology of the $C_{2}$-equivariant
Eilenberg-Mac Lane spaces with the constant Mackey functor
$\underline{M}=\underline{Z/2}$ coefficients, we use Borel theorem 17 for the
path-space fibration

$\Omega K(\underline{Z/2},V)\longrightarrow
P(K(\underline{Z/2},V))\longrightarrow K(\underline{Z/2},V).$

for $V=\sigma+n$, where $n\geq 0$.

If we knew $H^{\star}(K(\underline{Z/2},n\sigma))$ for $n\geq 2$, one could
use the Eilenberg-Moore spectral sequence [4, Chapter 5], the Borel theorem,
and the $RO(G)$-graded Serre spectral sequence of Kronholm [7, Theorem 1.2.]
for the path-space fibration

$\Omega K(\underline{Z/2},V)\longrightarrow
P(K(\underline{Z/2},V))\longrightarrow K(\underline{Z/2},V).$

This paper is organized as follows. In section 2, we provide the basic
equivariant topology tools, and $C_{2}$-equivariant cohomology $M^{C_{2}}_{2}$
of a point, and equivariant connectivity of $G$-spaces. In section 3, we
descripe equivariant Steenrod squares, $C_{2}$-equivariant Steenrod algebra
$\mathcal{A}_{C_{2}}$ and axioms of it. In section 4, we give the definition
of the equivariant Eilenberg-Mac Lane spaces with some properties, and the
fixed point sets of the equivariant Eilenberg-Mac Lane spaces that is very
useful to compute the cohomology of them. In section 5, we compute the
$RO(C_{2})$-graded $C_{2}$-equivariant cohomology of some $C_{2}$-equivariant
Eilenberg-Mac Lane spaces $K_{V}$ for real orthogonal representations
$V=\sigma+n,\ n\geq 0$. Also, we give some conjectures and future directions
for the other cases.
Notation. We provide here notation used in this paper for convenience.

  * $\bullet$

$V=r\sigma+s$, a real orthogonal representation of $C_{2}$, which is a sum of
$r$-copy of the sign representation $\sigma$ and $s$-copy of the trivial
representation $1$.

  * $\bullet$

$\rho=\sigma+1$, the regular representation of $C_{2}$.

  * $\bullet$

$RO(C_{2})$, the real representation ring of $C_{2}$.

  * $\bullet$

$S^{V}$, the equivariant sphere which is the one-point compactification of
$V$.

  * $\bullet$

$\pi_{V}^{C_{2}}(X)$, the $V$-th $C_{2}$-equivariant homotopy group of a
topological $C_{2}$-space $X$.

  * $\bullet$

$\pi_{r\sigma+s}^{S}$, the $C_{2}$-equivariant stable homotopy groups of
spheres.

  * $\bullet$

$\Sigma^{\sigma}(X)$, the $\sigma$-th suspension of $X$.

  * $\bullet$

$\Omega^{\sigma}(X)$, all continuous functions from $S^{\sigma}$ to $X$.

  * $\bullet$

$H_{G}^{\star}(-:\underline{M})$, RO($G$)-graded ordinary equivariant
cohomology with Mackey functor $\underline{M}$ coefficients.

  * $\bullet$

$M^{C_{2}}_{2}$, RO($C_{2}$)-graded $C_{2}$-equivariant cohomology of a point.

  * $\bullet$

$\mathcal{A}_{C_{2}}$, $C_{2}$-equivariant Steenrod algebra.

  * $\bullet$

$K(\underline{M},V)$ or shortly $K_{V}$, the $V$th equivariant Eilenberg-Mac
Lane space with a Mackey functor $\underline{M}$.

  * $\bullet$

$\underline{\pi}_{V}^{G}(X)$, $C_{2}$-equivariant homotopy of a $G$-space $X$
as a Mackey functor.

  * $\bullet$

$Sq_{C_{2}}^{k}$, $C_{2}$-equivariant Steenrod squaring operations for $k\geq
0$.

  * $\bullet$

$RP^{\infty}_{tw}$, the space of lines in the complete universe
$\mathcal{U}=(\mathbb{R}^{\rho})^{\infty}$, which is equivalent to
$K(\underline{Z/2},\sigma)$.

Acknowledgements. I would like to thank Michael A. Hill for valuable
conversations and providing me some suggestions for calculations, and William
Kronholm for producing the action of Steenrod squares on the cohomology ring
of $RP_{tw}^{\infty}$. Lastly and most importanly, I would like to state my
gratitude to my advisor, Douglas C. Ravenel, for his patience, support, and
encouragement throughout my graduate studies, and numerous beneficial
conversations and suggestions. The work in this paper was part of the author’s
dissertation while at the University of Rochester.

##  2\. Preliminaries

In this section we give the main tools that are used in the rest of the
article. Let $X$ be a $G$-space, where $G=C_{2}$ is a cyclic group with
generator $\gamma$ such that $\gamma^{2}=e$. The group $C_{2}$ has two
irreducible real representations, namely the trivial representation denoted by
$1$ (or $\mathbb{R}$) and the sign representation denoted by $\sigma$ (or
$\mathbb{R_{-}}$). The regular representation is isomorphic to
$\rho_{C_{2}}=1+\sigma$ (it is denoted by $\rho$ if there is no confusion).
Thus the representation ring $RO(C_{2})$ is free abelian of rank $2$, so every
representation $V$ can be expressed as $V=r\sigma+s$.

######  Definition 1.

A $G$-universe is a countably infinite-dimensional $G$-representation which
contains the trivial $G$-representation and which contains infinitely many
copies of each of its finite-dimensional subrepresentations. Also, a  complete
G-universe is just a G-universe that contains infinitely many copies of every
irreducible $G$-representation.

######  Definition 2.

A $G$-spectrum $E$ on a $G$-universe $\mathcal{U}$ is a collection $E_{V}$ of
based $G$-spaces together with basepoint-preserving $G$-maps

$\sigma_{V,W}:\Sigma^{W-V}E_{V}\longrightarrow E_{W}$

whenever $V\subset W\subset\mathcal{U}$, where $W-V$ denotes the orthogonal
complement of $V$ in $W$. It is required that $\sigma_{V,V}$ is identity, and
the commutativity of the diagram

${\Sigma^{W-V}\Sigma^{V-U}E_{U}}$${\Sigma^{W-V}E_{V}}$${E_{W}}$$\scriptstyle{\Sigma_{W-V}\sigma_{U,V}}$$\scriptstyle{\sigma_{U,W}}$$\scriptstyle{\sigma_{V,W}}$

for $U\subset V\subset W\subset\mathcal{U}$.

######  Definition 3.

If the adjoint structure maps

$\tilde{\sigma}_{V,W}:E_{V}\longrightarrow\Omega^{W-V}E_{W}$

are weak homotopy equivalences for $V\subset W\subset\mathcal{U}$, then a
$G$-spectrum is called $G-\Omega$-spectrum.

A $G$-spectrum indexed on a complete(trivial) $G$-universe is called
genuine(naive).

For an actual representation $V$ of $G$ and a $G$-space $X$, the $V$-th
homotopy group of $X$ is the Mackey functor $\underline{\pi}_{V}(X)$
determined by

$\underline{\pi}_{V}(X)(G/H)=[S^{V},X]^{H}$

for every $H<G$.

For a virtual representation $V\in RO(G)$ and a $G$-spectrum $E$, the $V$-th
homotopy group of $E$ is the Mackey functor $\underline{\pi}_{V}(E)$
determined by

$\underline{\pi}_{V}(E)(G/H)=colim_{n}\pi_{0}(\Omega^{V+W_{n}}E_{W_{n}})^{H}$

where $\\{W_{n}|n\in\mathbb{N}\\}$ is an increasing sequence of
representations

$\cdots\subset W_{n}\subset W_{n+1}\subset\cdots$

such that any finite dimensional representation $V$ of $G$ admits an
equivariant embedding in some $W_{n}$.

Lewis, May and Mcclure [8] defined an ordinary cohomology theory
$H_{G}^{\star}(-:\underline{M})$ on $G$-spaces with Mackey functor
$\underline{M}$ coefficients and the graded by real orthogonal
representations.

Throughout this paper, the Mackey functor will typically be the constant
Mackey functor $\underline{M}=\underline{Z/2}$, which can be given the
following diagram in Lewis notation.

(2.1)
$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id}$$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id}$$\scriptstyle{0}$

The ordinary equivariant cohomology $M^{C_{2}}_{2}$ of a point with this
coefficient is given in the Figure 1 below. Every $\bullet$ in the figure
represents a copy of $Z/2$.

As you see in the Figure 1 below, there are two elements of interest. The
inclusion map of the fixed point set (the north and south poles)
$a:S^{0}\longrightarrow S^{\sigma}$ defines an element in
$\pi_{-\sigma}^{C_{2}}(S^{-0})$, and we will use the same symbol for its mod
$2$ Hurewicz image. It is called an Euler class. One can show that

$H_{1}^{C_{2}}(S^{\sigma};Z/2)=H^{C_{2}}_{1-\sigma}(S^{-0};Z/2)=Z/2$

and we denote its generator by $u$. Dually, we have $a\in
H_{C_{2}}^{\sigma}(S^{-0};Z/2)$ and $u\in H_{C_{2}}^{\sigma-1}(S^{-0};Z/2)$.
These are the analog of elements $\rho$ and $\tau$ in real motivic homotopy
theory, respectively.

[grid=crossword, entrysize=6mm]-6…6-6…6 1 [color=red]01 ∙ u [color=red]01 ∙
u^2 [color=red]01 ∙ [color=red]01 ∙ [color=red]01 ∙ [color=red]01 ∙

1-5 ∙ [R]a [color=red]01 ∙ [U]ua [color=red]01 ∙ [color=red]01 ∙ [color=red]01
∙ [color=red]01 ∙

1-4 ∙ [R]a^2 [color=red]01 ∙ [color=red]01 ∙ [color=red]01 ∙ [color=red]01 ∙

1-3 ∙ [color=red]01 ∙ [color=red]01 ∙ [color=red]01 ∙

1-2 ∙ [color=red]01 ∙ [color=red]01 ∙

1-1 ∙ [color=red]01 ∙ 10 ∙

-6 -6 [color=blue]11 [color=blue]11 [color=blue]11 [color=blue]11 [color=blue]11 [color=blue]11

-6 -5 [color=blue]11 [color=blue]11 [color=blue]11 [color=blue]11 [color=blue]11

-5 -4 [color=blue]11 [color=blue]11 [color=blue]11 [color=blue]11

-4 -3 [color=blue]11 [color=blue]11 [color=blue]11

-3 -2 [color=blue]11 [color=blue]11

-2 -1 [color=blue]11

-1-8 ∙ [RU]θau

[color=red]0-1 ∙ [R]θau2 [color=red]0-1 ∙ [color=red]0-1 ∙ [color=red]0-1 ∙

-13 ∙ [LU]θa2u [color=red]0-1 ∙ [color=red]0-1 ∙ [color=red]0-1 ∙

-12 ∙ [color=red]0-1 ∙ [color=red]0-1 ∙

-11 ∙ [color=red]0-1 ∙ -10 ∙

44 [color=blue]-1-1 [color=blue]-1-1 [color=blue]-1-1 [color=blue]-1-1

43 [color=blue]-1-1 [color=blue]-1-1 [color=blue]-1-1

32 [color=blue]-1-1 [color=blue]-1-1

21 [color=blue]-1-1

Figure 1. The equivariant cohomology $M_{2}^{C_{2}}$ of a point

The coordinate $(x,y)$ represents degree $(x-y)+\sigma y$, which is convenient
with the motivic bidegree. Red and blue lines represent multiplication by $u$
and $a$, respectively.

Now, we will give the definition of equivariant connectivity of $G$-spaces.

######  Definition 4.

[10]

  1. (i)

A function $\nu^{*}$ from the set of conjugacy classes of subgroups of $G$ to
the integers is called a dimension function. The value of $\nu^{*}$ on the
conjugacy class of $K\subset G$ is denoted by $\nu^{K}$. Let $\nu^{*}$ and
$\mu^{*}$ be two dimension functions. If $\nu^{K}\geq\mu^{K}$ for every
subgroup $K$, then $\nu^{*}\geq\mu^{*}$. Associated to any $G$-representation
$V$ is the dimension function $|V^{*}|$ whose value at $K$ is the real
dimension of the $K$-fixed subspace $V^{K}$ of $V$. The dimension function
with constant integer value $n$ is denoted $n^{*}$ for any integer $n$.

  2. (ii)

Let $\nu^{*}$ be a non-negative dimension function. If for each subgroup $K$
of $G$, the fixed point space $Y^{K}$ is $\nu^{K}$-connected, then a $G$-space
$Y$ is called $G$-$\nu^{*}$-connected. If A $G$-space $Y$ is
$G$-$0^{*}$-connected, then it is called $G$-connected. Also, if it is
$G$-$1^{*}$-connected, it is called simply $G$-connected. A $G$-space $Y$ is
homologically $G$-$\nu^{*}$-connected if, for every subgroup $K$ of $G$ and
every integer $m$ with $0\leq m\leq\nu^{K}$ , the homology group
$H_{m}^{K}(Y)$ is zero.

  3. (iii)

Let $\nu^{*}$ be a non-negative dimension function and let $f:Y\longrightarrow
Z$ be a $G$-map. If, for every subgroup $K$ of $G$,

$(f^{K})_{*}:\pi_{m}(Y^{k})\longrightarrow\pi_{m}(Z^{K})$

is an isomorphism for every integer $m$ with $0\leq m<\nu^{K}$ and an
epimorphism for $m=\nu^{K}$, then $f$ is called $G$-$\nu^{*}$-equivalence. A
$G$-pair $(Y,B)$ is said to be $G$-$\nu^{*}$-connected if the inclusion of $B$
into $Y$ is a $G$-$\nu^{*}$-equivalence. The notions of homology
$G$-$\nu^{*}$-equivalence and of homology $G$-$\nu^{*}$-connectedness for
pairs are defined similarly, but with homotopy groups replaced by homology
groups.

  4. (iv)

Let $V$ be a $G$-representation. For each subgroup $K$ of $G$, let $V(K)$ be
the orthogonal complement of $V^{K}$ ; then $V(K)$ is a $K$-representation. If
$\pi_{V(K)+m}^{K}(Y)$ is zero for each subgroup $K$ of $G$ and each integer
$m$ with $0\leq m\leq|V^{K}|$, the $G$-space $Y$ is called $G$-$V$-connected.
Similarly, if $H_{V(K)+m}^{G}(Y)$ is zero for each subgroup $K$ of $G$ and
each integer $m$ with $0\leq m\leq|V^{K}|$, then the $G$-space $Y$ is called
homologically $G$-$V$-connected.

  5. (v)

Let $V$ be a $G$-representation. A $G$-$0^{*}$-equivalence $f:Y\longrightarrow
Z$ is said to be a $G$-$V$-equivalence if, for every subgroup $K$ of $G$, the
map

$f_{*}:\pi_{V(K)+m}^{K}(Y)\longrightarrow\pi_{V(K)+m}^{K}(Z)$

is an isomorphism for every integer $m$ with $0\leq m<|V^{K}|$ and an
epimorphism for $m=|V^{K}|$. A homology $G$-$V$-equivalence is defined
similarly. A $G$-pair $(Y,B)$ is called $G$-$V$-connected (respectively,
homologically $G$-$V$-connected) if the inclusion of $B$ into $Y$ is a
$G$-$V$-equivalence (respectively, homology $G$-$V$-equivalence).

##  3\. $C_{2}$-Equivariant Steenrod Algebra

The analog of the mod $2$ Steenrod algebra is defined by Voevodsky [19] in the
motivic case, and Po Hu and Igor Kriz [5] in the equivariant case. The two
descriptions are essentially the same.

One has squaring operations $Sq^{k}_{C_{2}}$ for $k\geq 0$, whose degrees

$|Sq^{k}_{C_{2}}|=\left\\{\begin{array}[]{ll}i(1+\sigma)&for\ k=2i\\\
i(1+\sigma)+1&for\ k=2i+1.\\\ \end{array}\right.$

$Sq^{0}_{C_{2}}=1$ as in the classical case. The $C_{2}$-equivariant Steenrod
algebra acts on the coefficient ring $M^{C_{2}}_{2}$ by

(3.1) $Sq^{k}_{C_{2}}(u)=\left\\{\begin{array}[]{ll}u&for\ k=0\\\ a&for\
k=1\\\ 0&else.\\\ \end{array}\right.$

(3.2)
$Sq^{2m+\delta}_{C_{2}}(u^{2l+\epsilon})=\binom{2l+\epsilon}{2m+\delta}u^{2l+\epsilon-m-\delta}a^{2m+\delta}$

The difficulty in deriving the formula 3.2 is the $C_{2}$-equivariant Cartan
formula 3.7, 3.8. Since

$|Sq_{C_{2}}^{2m+\delta}|=m(1+\sigma)+\delta\ \ \ \ \text{for}\
0\leq\delta\leq 1,$

we have

(3.3) $\left\\{\begin{array}[]{ll}\Delta(Sq_{C_{2}}^{2m+1})=\sum_{0\leq i\leq
2m+1}Sq_{C_{2}}^{i}\otimes Sq_{C_{2}}^{2m+1-i}\\\ \\\
\Delta(Sq_{C_{2}}^{2m})=\sum_{0\leq j\leq m}Sq_{C_{2}}^{2i}\otimes
Sq_{C_{2}}^{2m-2j}+u\sum_{1\leq j\leq m}Sq_{C_{2}}^{2j-1}\otimes
Sq_{C_{2}}^{2m-2j+1}.\\\ \end{array}\right.$

The terms divisible by $u$ make things difficult. Here we are using
cohomological degree, so $|u|=\sigma-1$. Note that

$|u^{-m}Sq_{C_{2}}^{2m+\delta}|=m(1-\sigma)+m(1+\sigma)+\delta=2m+\delta$

and define

$\mathcal{S}q^{2m+\delta}:=u^{-m}Sq_{C_{2}}^{2m+\delta}.$

We will see that these operations satisfy the classical Cartan formula. We
have

$\displaystyle\Delta(\mathcal{S}q^{2m+1})$
$\displaystyle=u^{-m}\Delta(Sq_{C_{2}}^{2m+1})$
$\displaystyle=u^{-m}\sum_{0\leq i\leq 2m+1}Sq_{C_{2}}^{i}\otimes
Sq_{C_{2}}^{2m+1-i}$ $\displaystyle=\sum_{0\leq i\leq 2m+1}u^{-\lfloor
i/2\rfloor}Sq_{C_{2}}^{i}\otimes
u^{-\lfloor(2m+1-i)/2\rfloor}Sq_{C_{2}}^{2m+1-i}$ $\displaystyle=\sum_{0\leq
i\leq 2m+1}\mathcal{S}q_{C_{2}}^{i}\otimes\mathcal{S}q_{C_{2}}^{2m+1-i}$

since $\lfloor i/2\rfloor+\lfloor(2m+1-i)/2\rfloor=m$. And also,

$\displaystyle\Delta(\mathcal{S}q^{2m})$
$\displaystyle=u^{-m}\Delta(Sq_{C_{2}}^{2m})$ $\displaystyle=u^{-m}\sum_{0\leq
j\leq m}Sq_{C_{2}}^{2j}\otimes Sq_{C_{2}}^{2m-2j}+u^{1-m}\sum_{1\leq j\leq
m}Sq_{C_{2}}^{2j-1}\otimes Sq_{C_{2}}^{2m-2j+1}$ $\displaystyle=\sum_{0\leq
j\leq m}u^{-j}Sq_{C_{2}}^{2j}\otimes u^{j-m}Sq_{C_{2}}^{2m-2j}+\sum_{1\leq
j\leq m}u^{1-j}Sq_{C_{2}}^{2j-1}\otimes u^{j-m}Sq_{C_{2}}^{2m-2j+1}$
$\displaystyle=\sum_{0\leq j\leq
m}\mathcal{S}q_{C_{2}}^{2j}\otimes\mathcal{S}q_{C_{2}}^{2m-2j}+\sum_{1\leq
j\leq m}\mathcal{S}q_{C_{2}}^{2j-1}\otimes\mathcal{S}q_{C_{2}}^{2m-2j+1}$
$\displaystyle=\sum_{0\leq i\leq
2m}\mathcal{S}q_{C_{2}}^{i}\otimes\mathcal{S}q_{C_{2}}^{2m-i}.$

Now, if we use homological degree, then

$|\mathcal{S}q^{m}|=-m,|a|=-\sigma,\ \text{and}\ |u|=1-\sigma.$

We know that

(3.4) $Sq^{m}_{C_{2}}(u)=\left\\{\begin{array}[]{ll}u&for\ m=0\\\ a&for\
m=1\\\ 0&else.\\\ \end{array}\right.$

Consider the total Steenrod operation

(3.5) $\mathcal{S}q_{t}=\sum_{i\geq 0}t^{i}\mathcal{S}q^{i},$

where $t$ is a dummy variable. Although this sum is infinite, it yields a
finite sum when applied to any monomial in $a$ and $u$. The classical Cartan
formula satisfied by operations $\mathcal{S}q^{i}$ implies that it is a ring
homomorphism, meaning that

$\mathcal{S}q_{t}(xy)=\mathcal{S}q_{t}(x)\mathcal{S}q_{t}(y).$

Then 3.4 implies that

$\displaystyle\mathcal{S}q_{t}(u)=u+ta$ $\displaystyle\mathcal{S}q_{t}(u^{l})$
$\displaystyle=(u+ta)^{l}$ $\displaystyle=\sum_{0\leq m\leq
l}\binom{l}{m}t^{m}u^{l-m}a^{m}$ $\displaystyle=\sum_{0\leq m\leq
l}t^{m}\mathcal{S}q^{m}(u^{l}).$

Hence, $\mathcal{S}q^{m}(u^{l})$ is the coefficient of $t^{m}$ in the first
sum above.

It follows that

$\displaystyle\mathcal{S}q^{2m+\delta}(u^{2l+\epsilon})$
$\displaystyle=\binom{2l+\epsilon}{2m+\delta}u^{2l+\epsilon-2m-\delta}a^{2m+\delta}$
$\displaystyle\mathcal{S}q^{2m+\delta}(u^{2l+\epsilon})$
$\displaystyle=u^{m}\mathcal{S}q^{2m+\delta}(u^{2l+\epsilon})$
$\displaystyle=\binom{2l+\epsilon}{2m+\delta}u^{2l+\epsilon-m-\delta}a^{2m+\delta}.$

As a result, we have the following:

######  Lemma 5.

$Sq^{2m+\delta}_{C_{2}}(u^{2l+\epsilon})=\binom{2l+\epsilon}{2m+\delta}u^{2l+\epsilon-m-\delta}a^{2m+\delta}.$

The natural action of the Steenrod algebra in homology is on the right, not on
the left. Classically, the $\mod p$ cohomology of a space or a spectrum $X$ is
a left module over the Steenrod algebra $\mathcal{A}$, so there is a map

$c_{X}:A\otimes H^{\ast}X\rightarrow H^{\ast}X.$

The Steenrod algebra has a multiplication

$\phi^{\ast}:\mathcal{A}\otimes\mathcal{A}\rightarrow\mathcal{A}$

(the symbol $\phi^{\ast}$ and its dual $\phi_{\ast}$ are taken from Milnor’s
paper [14]) and the following diagram commutes

(3.6) ${\mathcal{A}\otimes\mathcal{A}\otimes H^{\ast}X\ \
}$${\mathcal{A}\otimes H^{\ast}X}$${\mathcal{A}\otimes
H^{\ast}X}$${H^{\ast}X.}$$\scriptstyle{\phi^{\ast}\otimes
H^{\ast}X}$$\scriptstyle{\mathcal{A}\otimes
c_{X}}$$\scriptstyle{c_{X}}$$\scriptstyle{c_{X}}$

Milnor defines a right action of $\mathcal{A}$ on $H_{\ast}X$ by the rule

$\langle xa,y\rangle=\langle x,ay\rangle\in\mathbb{F}_{p}$

for $x\in H_{\ast}X,\ a\in\mathcal{A}$ and $y\in H^{\ast}X$, where the
brackets denotes the evaluation of the cohomology class on the right on the
homology class on the left. Milnor denotes by $\lambda^{\ast}$ the resulting
map

$H_{\ast}X\otimes\mathcal{A}\rightarrow H_{\ast}X.$

The same thing happens in the $C_{2}$-equivariant case. For example, we have

$(u^{2})Sq_{C_{2}}^{3}=(u^{2})Sq_{C_{2}}^{1}Sq_{C_{2}}^{2}=0$

because $(u^{2})Sq_{C_{2}}^{1}=0$. And,

$(u^{2})\chi(Sq_{C_{2}}^{3})=(u^{2})Sq_{C_{2}}^{2}Sq_{C_{2}}^{1}=(ua^{2})Sq_{C_{2}}^{1}=a^{3},$

where $\chi(-)$ means the conjugate Steenrod operations. Hence, 3.2 should
really read as

$(u^{2l+\epsilon})Sq^{2m+\delta}_{C_{2}}=\binom{2l+\epsilon}{2m+\delta}u^{2l+\epsilon-m-\delta}a^{2m+\delta}.$

For example,

$\displaystyle Sq^{l}_{C_{2}}(u^{-1})$
$\displaystyle=\binom{-1}{l}a^{l}u^{-1-l}$
$\displaystyle=\left\\{\begin{array}[]{ll}\binom{-1}{0}u^{-1}=u^{-1}&\text{for}\
l=0\\\ \\\ \binom{-1}{1}au^{-2}=au^{-2}&\text{for}\ l=1\\\ \\\
\binom{-1}{2}a^{2}u^{-3}=a^{2}u^{-3}&\text{for}\ l=2\\\ \\\ 0&\text{for}\
l\geq 3\\\ \end{array}\right.$

Action on the other elements is determined by the Cartan formula (iv) given
below. We now give axioms for the squares $Sq^{k}_{C_{2}}$. For the motivic
case, you can check Voevodsky paper [19]. But, the Adem relation is fixed by
Joël Riou in [17].

  * (i)

$Sq^{0}_{C_{2}}=1$ and $Sq^{1}_{C_{2}}=\beta_{C_{2}}$, Bockstein homomorphism.

  * (ii)

$\beta Sq^{2k}_{C_{2}}=Sq^{2k+1}_{C_{2}}$.

  * (iii)

$\beta Sq^{2k+1}_{C_{2}}=0$.

  * (iv)

(Cartan formula)

(3.7)
$Sq^{2k}_{C_{2}}(xy)=\sum_{r=0}^{k}Sq^{2r}_{C_{2}}(x)Sq^{2k-2r}_{C_{2}}(y)+u\sum_{s=0}^{k-1}Sq^{2s+1}_{C_{2}}(x)Sq^{2k-2s-1}_{C_{2}}(y)$
(3.8)
$Sq^{2k+1}_{C_{2}}(xy)=\sum_{r=0}^{2k+1}Sq^{r}_{C_{2}}(x)Sq^{2k+1-r}_{C_{2}}(y)+a\sum_{s=0}^{k-1}Sq^{2s+1}_{C_{2}}(x)Sq^{2k-2s-1}_{C_{2}}(y)$

  * (v)

(Adem relation) If $0<i<2j$, then when $i+j$ is even

$Sq^{i}_{C_{2}}Sq^{j}_{C_{2}}=\sum_{k=0}^{[i/2]}\binom{b-1-k}{i-2k}u^{\epsilon}Sq^{i+j-k}_{C_{2}}Sq^{k}_{C_{2}}$

where

$\epsilon=\left\\{\begin{array}[]{ll}1&for\ k\ is\ odd\ and\ i\ and\ j\ are\
even\\\ 0&else\\\ \end{array}\right.$

when $i+j$ is odd

$Sq^{i}_{C_{2}}Sq^{j}_{C_{2}}=\sum_{k=0}^{[i/2]}\binom{j-1-k}{i-2k}Sq^{i+j-k}_{C_{2}}Sq^{k}_{C_{2}}+a\sum_{k=odd}\varepsilon\
Sq^{i+j-k}_{C_{2}}Sq^{k}_{C_{2}}$

where

$\varepsilon=\left\\{\begin{array}[]{ll}\binom{j-1-k}{i-2k}&for\ i\ is\ odd\\\
\binom{j-1-k}{i-2k-1}&for\ j\ is\ odd\\\ \end{array}\right.$

  * (vi)

If $x$ has a degree $k\sigma+k$, then $Sq^{2k}_{C_{2}}(x)=x^{2}$.

  * (vii)

(instability) If $x$ has a degree $V$, $V<k\sigma+k$ then
$Sq^{2k}_{C_{2}}(x)=0$, where $V<V^{{}^{\prime}}$ if and only if
$V^{{}^{\prime}}=V+W$ for some actual representations $W$ with positive
degree.

Note that setting $u=1$ and $a=0$ reduces the Cartan formula (iv) to the
classical Cartan formula, and Adem relation (v) to the classical Adem
relation.

######  Examples 6.

We have

$Sq^{1}_{C_{2}}Sq^{n}_{C_{2}}=\left\\{\begin{array}[]{ll}Sq^{n+1}_{C_{2}}&for\
n\ is\ even\\\ 0&for\ n\ is\ odd\\\ \end{array}\right.$
$Sq^{2}_{C_{2}}Sq^{n}_{C_{2}}=\left\\{\begin{array}[]{ll}Sq^{n+2}_{C_{2}}+uSq^{n+1}_{C_{2}}Sq^{1}_{C_{2}}&for\
n\equiv 0\ mod\ 4\\\ Sq^{n+1}_{C_{2}}Sq^{1}_{C_{2}}&for\ n\equiv 1\ mod\ 4\\\
uSq^{n+1}_{C_{2}}Sq^{1}_{C_{2}}&for\ n\equiv 2\ mod\ 4\\\
Sq^{n+2}_{C_{2}}+Sq^{n+1}_{C_{2}}Sq^{1}_{C_{2}}&for\ n\equiv 3\ mod\ 4\\\
\end{array}\right.$

and

$Sq^{3}_{C_{2}}Sq^{n}_{C_{2}}=\left\\{\begin{array}[]{ll}Sq^{n+3}_{C_{2}}+aSq^{n+1}_{C_{2}}Sq^{1}_{C_{2}}&for\
n\equiv 0\ mod\ 4\\\ Sq^{n+2}_{C_{2}}Sq^{1}_{C_{2}}&for\ n\equiv 1\ mod\ 4\\\
aSq^{n+1}_{C_{2}}Sq^{1}_{C_{2}}&for\ n\equiv 2\ mod\ 4\\\
Sq^{n+2}_{C_{2}}Sq^{1}_{C_{2}}&for\ n\equiv 3\ mod\ 4\\\ \end{array}\right.$

Now, let $Sq^{I}_{C_{2}}$ denote $Sq^{i_{1}}_{C_{2}}Sq^{i_{2}}_{C_{2}}\cdots
Sq^{i_{n}}_{C_{2}}$ for a sequence of integers $I=(i_{1},\cdots,i_{n})$. The
sequence $I$ is said to be admissible if $i_{s}\geq 2i_{s+1}$ for all $s\geq
1$, where $i_{s+1}=0$.

The operations $Sq^{I}_{C_{2}}$ with admissible $I$ are called admissible
monomials. We also call $Sq^{0}_{C_{2}}$ admissible, where
$Sq^{0}_{C_{2}}=Sq^{I}_{C_{2}}$ for empty $I$.

######  Lemma 7.

The admissible monomials form a basis for the $C_{2}$-equivariant Steenrod
algebra $\mathcal{A}_{C_{2}}$ as a $H^{\star}(pt)$-module.

###### Proof.

The proof follows from the Adem relations and the Cartan formula as in the
classical case. ∎

For the graded $\mathcal{A}_{C_{2}}$-module structure and Hopf algebra
structure of equivariant Steenrod algebra, one can look [16]. We will now give
unstable module structure of it.

######  Definition 8.

An $\mathcal{A}_{C_{2}}$-module is unstable if it satisfies the preceeding
instability condition (vii).

We define the excess of $Sq^{k}_{C_{2}}$ to be the degree of $Sq^{k}_{C_{2}}$

$e(Sq^{k}_{C_{2}})=\left\\{\begin{array}[]{ll}i\rho&for\ k=2i\\\ i\rho+1&for\
k=2i+1.\\\ \end{array}\right.$

So, $e(Sq^{k}_{C_{2}})=|Sq^{k}_{C_{2}}|$. Then the excess of
$Sq^{I}_{C_{2}}=Sq^{i_{1}}_{C_{2}}Sq^{i_{2}}_{C_{2}}\cdots Sq^{i_{k}}_{C_{2}}$
to be

$e(Sq^{I}_{C_{2}})=\sum_{j}e(Sq^{i_{j}}_{C_{2}})-\rho e(Sq^{i_{j+1}}_{C_{2}})$

where $\rho(r\sigma+s)=(r+s)\rho$.

######  Examples 9.

  * •

The monomial with $e(Sq^{I}_{C_{2}})=0$ is $Sq^{0}_{C_{2}}$.

  * •

The monomials with $e(Sq^{I}_{C_{2}})=1$ are $Sq^{1}_{C_{2}}$,
$Sq^{2}_{C_{2}}Sq^{1}_{C_{2}}$, $Sq^{4}_{C_{2}}Sq^{2}_{C_{2}}Sq^{1}_{C_{2}}$,
$\cdots$

  * •

There is no monomial with $e(Sq^{I}_{C_{2}})=\sigma$.

  * •

The monomials with $e(Sq^{I}_{C_{2}})=2$ are $Sq^{3}_{C_{2}}Sq^{1}_{C_{2}}$,
$Sq^{6}_{C_{2}}Sq^{3}_{C_{2}}Sq^{1}_{C_{2}}$,
$Sq^{12}_{C_{2}}Sq^{6}_{C_{2}}Sq^{3}_{C_{2}}\\-Sq^{1}_{C_{2}}$, $\cdots$

  * •

The monomials with $e(Sq^{I}_{C_{2}})=\rho$ are $Sq^{2}_{C_{2}}$,
$Sq^{4}_{C_{2}}Sq^{2}_{C_{2}}$, $Sq^{8}_{C_{2}}Sq^{4}_{C_{2}}Sq^{2}_{C_{2}}$,
$\cdots$

  * •

There is no monomial with $e(Sq^{I}_{C_{2}})=2\sigma$,

  * •

The monomials with $e(Sq^{I}_{C_{2}})=3$ are
$Sq^{7}_{C_{2}}Sq^{3}_{C_{2}}Sq^{1}_{C_{2}}$,
$Sq^{11}_{C_{2}}Sq^{5}_{C_{2}}Sq^{2}_{C_{2}}Sq^{1}_{C_{2}}$, $\cdots$

  * •

The monomials with $e(Sq^{I}_{C_{2}})=2+\sigma$ are $Sq^{3}_{C_{2}}$,
$Sq^{4}_{C_{2}}Sq^{1}_{C_{2}}$, $Sq^{5}_{C_{2}}Sq^{2}_{C_{2}}$,
$Sq^{6}_{C_{2}}Sq^{3}_{C_{2}}$, $Sq^{6}_{C_{2}}Sq^{2}_{C_{2}}Sq^{1}_{C_{2}}$,
$Sq^{8}_{C_{2}}Sq^{4}_{C_{2}}Sq^{1}_{C_{2}}$, $\cdots$

  * •

There is no monomial with $e(Sq^{I}_{C_{2}})=1+2\sigma$.

######  Remark 10.

There is no monomial with $e(Sq^{I}_{C_{2}})=r\sigma+s$ if $r>s$.

Let $\mathbf{t}_{j,k}=Sq^{j2^{k-1}}_{C_{2}}\cdots Sq^{j}_{C_{2}}$. Then the
set of elements with total excess 1 is

$\left\\{\mathbf{t}_{1,k_{1}}|k_{1}>0\right\\}.$

The set of elements with total excess 2 is

$\left\\{\mathbf{t}_{1+2^{k_{1}},k_{2}+1}\mathbf{t}_{1,k_{1}}|k_{1},k_{2}\geq
0\right\\}.$

The set of elements with total excess 3 is

$\left\\{\mathbf{t}_{1+2^{k_{2}}+2^{k_{1}+k_{2}},k_{3}+1}\mathbf{t}_{1+2^{k_{1}},k_{2}}\mathbf{t}_{1,k_{1}}|k_{1},k_{2},k_{3}\geq
0\right\\}.$

The $C_{2}$-equivariant mod $2$ dual Steenrod algebra (one can check [16], or
[5] for details) is

$\mathcal{A}^{C_{2}}=M_{2}^{C_{2}}[\tau_{i},\xi_{i}]/(\tau_{i}^{2}+a\tau_{i+1}\eta_{R}(u)\xi_{i+1})$

such that

$\displaystyle\eta_{R}(u)=u+a\tau_{0}$ $\displaystyle\eta_{R}(a)=a$
$\displaystyle|\xi_{i}|=(2^{i}-1)\rho$ $\displaystyle|\tau_{i}|=1+|\xi_{i}|$
$\displaystyle\Delta(\xi_{i})=\sum_{j=0}^{i}\xi_{i-j}^{2^{j}}\otimes\xi_{j},\
\text{where}\ \xi_{0}=1$ $\displaystyle\Delta(\tau_{i})=\tau_{i}\otimes
1+\sum_{j=0}^{i}\xi_{i-j}^{2^{j}}\otimes\tau_{j}.$

##  4\. Equivariant Eilenberg-Mac Lane Spaces

For each Mackey functor $\underline{M}$, there is an Eilenberg-Mac Lane
$G$-spectrum $H\underline{M}$ which has the property as Mackey functors

$\underline{\pi}_{n}^{G}(H\underline{M})=\left\\{\begin{array}[]{ll}\underline{M}&n=0\\\
0&n\in\mathbb{Z},n\neq 0\\\ \end{array}\right.$

One can check [13, Chapter XIII, page 162] for the proof of the existence.

Let $\underline{M}$ be a Mackey functor, the $V$th space in the
$\Omega$-spectrum for $H\underline{M}$ is called an equivariant Eilenberg-Mac
Lane space of type $K(\underline{M},V)$, which is a classifying space for the
functor $H^{V}_{G}(-;\underline{M})$. That is, given any real orthogonal
representations $V$ , $W$, there is a $G$-homotopy equivalence
$K(\underline{M};V)\simeq\Omega^{W}K(\underline{M},V+W)$ satisfying various
compatibility properties. Such spaces are constructed in [9], or one can look
[3] for a construction with a different method. Here, I will give the
definition of them for consistency.

######  Definition 11.

[9] Let $V$ be a real orthogonal representation with $|V^{G}|\geq 1$ and
$\underline{M}$ be a Mackey functor. An equivariant Eilenberg-Mac Lane space
$K(\underline{M},V)$ is a based, $(|V^{*}|-1)$-connected $G$-space with the
$G$-homotopy type of a $G$-CW complex such that
$\underline{\pi}_{V}^{G}(K(\underline{M},V))=\underline{M}$, and for
$\underline{\pi}_{V+k}^{G}(K(\underline{M},V))=0$ $k\neq 0$.

######  Remark 12.

One can ask what $\underline{\pi}_{V+n\sigma}^{G}(K(\underline{M},V))$ is for
$n>0$. Our main interest is $K(\underline{Z/2},V)$. Then,

$\underline{\pi}_{V+n\sigma}^{C_{2}}(K(\underline{Z/2},V))(C_{2}/e)=\pi_{V+n\sigma}^{e}(K(\underline{Z/2},V))=0$

and

$\displaystyle\underline{\pi}_{V+n\sigma}^{C_{2}}(K(\underline{Z/2},V))(C_{2}/C_{2})$
$\displaystyle=\pi_{V+n\sigma}^{C_{2}}(K(\underline{Z/2},V))$
$\displaystyle\cong\tilde{H}_{V+n\sigma}^{C_{2}}(S^{V};\underline{Z/2})$
$\displaystyle\cong\tilde{H}_{n\sigma}^{C_{2}}(S^{0,0};\underline{Z/2})$
$\displaystyle\cong H_{n\sigma}^{C_{2}}(\ast;\underline{Z/2})$

So, as a Mackey functor, the homotopy
$\underline{\pi}_{V+n\sigma}^{G}(K(\underline{M},V))$ is one of the

$\textstyle{Z/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id}$
or
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Id}$

depending on the dimension of the representation $V$ and $n$.

As mentioned before, one can check [9] for existence and some properties of
these spaces.

Another approach to construct equivariant Eilenberg-Mac Lane spaces is Dos
Santos [3] approach. As we know in the classical case, the free abelian group
on the $n$-sphere is a model for the Eilenberg-Mac Lane space
$K(\mathbb{Z},n)$, and the free $\mathbb{F}_{2}$-vector space on the
$n$-sphere is a model for the Eilenberg-Mac Lane space $K(\mathbb{F}_{2},n)$.
Dos Santos constructed a topological abelian group $M\otimes X$ in [3,
Definition 2.1.], which is the equivariant generalization of previous sentence
for a Mackey functor $M$, and proved an $RO(G)$-graded version of equivariant
Dold-Thom theorem proved by Lima-Filho for $\mathbb{Z}$-graded case in [11].

Let $M$ be a $\mathbb{Z}[G]$-module, $\underline{M}$ be the Mackey functor
associated to $M$: the value of $\underline{M}$ on $G/H$ is $M^{H}$ and the
value on the projection $G/K\longrightarrow G/H$, for $K<H<G$, is the
inclusion of $M^{H}\hookrightarrow M^{K}$. We define $M\otimes X$ as the
$\mathbb{Z}[G]$-module with a topology as follows([3, Definition 2.1.]): Let
$(X,\ast)$ be a based $G$-set, $M\otimes X$ denote the $\mathbb{Z}[G]$-module
$\bigoplus_{x\in X-\\{\ast\\}}M$. The action of $g\in G$ is given by
$(g.m)_{x}=g.m_{g^{-1}.x}$, where $m_{x}$ denotes the $x$th coordinate of
$m\in\bigoplus_{x\in X-\\{\ast\\}}M$. Given $(X,\ast)$ a based $G$-space,
$M\otimes X$ can be equivalently defined as the quotient

$M\otimes X=\amalg_{n\geq 0}M^{n}\times X^{n}/\backsim,$

where $\backsim$ is the equivalence relation generated by:

  1. (i)

$(r,\phi^{\ast}x)\backsim(\phi_{\ast}r,x)$, for each based map
$\phi:\\{0,\cdots,n\\}\longrightarrow\\{0,\cdots,m\\}$, $n,m\in\mathbb{N}$,
where $\phi^{\ast}x=x\circ\phi$, and
$(\phi_{\ast}r)_{i}=\sum_{k\in\phi^{-1}(i)}r_{k}$.

  2. (ii)

$((r,r^{\prime}),(x,\ast))\backsim(r,x)$, for each $r\in M^{n}$,
$r^{\prime}\in M$, $x\in X$.

We give the discrete topology to $M$ and endow $M\otimes X$ with the quotient
topology corresponding to the relation $\backsim$.

We can define Eilenberg-Mac Lane spaces as $K_{V}=M\otimes S^{V}$. In our
case,

$K_{m+n\sigma}=\mathbb{Z}/2\otimes S^{m+n\sigma}.$

######  Theorem 13.

[3] Let $X$ be a based $G$-CW-complex and let $V$ be a finite dimensional
$G$-representation, then $M\otimes X$ is an equivariant infinite loop space
and there is a natural equivalence

$\pi_{V}^{G}(M\otimes X)\cong\tilde{H}_{V}^{G}(X;\underline{M})$

As a corollary to this theorem we have that $M\otimes S^{V}$ is a
$K(\underline{M},V)$ space (as Definition 11). Thus we have a simple model for
the equivariant Eilenberg-Mac Lane spectrum $H\underline{M}$.

######  Examples 14.

  * (i)

$K(\underline{Z/2},1)$ is $RP^{\infty}$, with trivial action.

  * (ii)

Recall that $RP^{\infty}_{tw}=\mathbb{P}(\mathcal{U})$ is the space of lines
in the complete universe (Definition 1)

$\mathcal{U}=(\mathbb{R}^{\rho})^{\infty}$

[13]. The cohomology of $RP^{\infty}_{tw}$ is calculated by Kronholm in
[WKronholm2010]. The space $RP^{\infty}_{tw}$ is equivalent to
$K(\underline{Z/2},\sigma)$, since it is equivalent to $\mathbb{Z}/2\otimes
S^{\sigma}$.

######  Theorem 15.

[6] $H^{\star}(RP^{\infty}_{tw})\cong H^{\star}(pt)[c,d]/(c^{2}=ac+ud)$, where
deg($c$)=$\sigma$, and deg($d$)=$\rho$.

Now, we will give a structure of fixed points of equivariant Eilenberg-Mac
Lane spaces, which is useful to calculate the cohomology of them.

######  Theorem 16.

[2, Corollary 10]

  * (i)

$(K(\underline{Z/2},r\sigma+s))^{e}\simeq K(Z/2,r+s)$.

  * (ii)

$(K(\underline{Z/2},r\sigma+s))^{C_{2}}\simeq K(Z/2,s)\times\cdots\times
K(Z/2,r+s)$.

##  5\. Cohomology of Eilenberg-Mac Lane Spaces

In classical case the cohomology of Eilenberg- Mac Lane spaces $K_{n}$ with
$\mathbb{Z}/2$-coefficients, which is given by Serre in [18] is a polynomial
ring

$H^{\ast}(K_{n};\mathbb{Z}/2)=P(Sq^{I}(\iota_{n})|e(I)<n)$

where $I$ are admissible sequences, $\iota_{n}$ is the fundamental class, and
$e(Sq^{I})=\sum_{j}(i_{j}-2i_{j+1})$. We thought that we can give similar
description for $RO(C_{2})-$graded $C_{2}$-equivariant cohomology of
$C_{2}$-equivariant Eilenberg-Mac Lane spaces, but these are more complicated
than we expect.

Let $\mathbf{s}_{V,l}$ is the operation that sends $x$ to $x^{2^{l}}$ for
$x\in H^{V}$. It is possible to express $\mathbf{s}_{V,l}$ as a linear
combination of Steenrod operations.

$\mathbf{s}_{V,0}=1$

If $x\in H^{a+b\sigma}$, and $b=r_{1}+\lfloor\frac{a+b}{2}\rfloor$, then
$(u^{-r_{1}}x)^{2}=Sq_{C_{2}}^{a+b}(u^{-r_{1}}x)$, so

$x^{2}=u^{2r_{1}}Sq_{C_{2}}^{a+b}(u^{-r_{1}}x)$

By using $C_{2}$-equivariant Cartan formula and the formula 3.2

$Sq^{2m+\delta}_{C_{2}}(u^{2l+\epsilon})=\binom{2l+\epsilon}{2m+\delta}u^{2l+\epsilon-m-\delta}a^{2m+\delta}$

one has general formula for $Sq_{C_{2}}^{a+b}(u^{-r_{1}}x)$. By iterating this
method one can find a formula for every $x^{2^{l}}$, so $\mathbf{s}_{V,l}$
exist. For example, if $x\in H^{3+\sigma}$, then

$\displaystyle(ux)^{2}=Sq^{4}_{C_{2}}(ux)$
$\displaystyle=\sum_{r=0}^{2}Sq^{2r}_{C_{2}}(u)Sq^{4-2r}_{C_{2}}(x)+\sum_{s=0}^{1}Sq^{2s+1}_{C_{2}}(u)Sq^{3-2s}_{C_{2}}(x)$
$\displaystyle=uSq^{4}_{C_{2}}(x)+uaSq^{3}_{C_{2}}(x)$

Thus

$x^{2}=u^{-1}Sq^{4}_{C_{2}}(x)+u^{-1}aSq^{3}_{C_{2}}(x).$

The set of elements $x_{i}$ whose finite distinct products form a basis for a
graded ring $A$ is called a simple system of generators. For example, a
polynomial algebra $k[x]$ has a simple system of generators
$\left\\{x^{2^{i}}|\ i\geq 0\right\\}$.

######  Theorem 17.

(Borel) Let $F\rightarrow E\rightarrow B$ be a $C_{2}$-fibration with $E$
contractible. Suppose that $H^{\star}(F)$ has a simple system $\\{x_{i}\\}$ of
transgressive generators. Then $H^{\star}(B)$ is a polynomial ring in the
$\left\\{\Sigma(x_{i})\right\\}$.

$E_{2}$-page of $RO(G)$-graded Serre spectral sequence of Kronholm [7] depends
only on the total degree of representations, not the dimension of twisted
part. The proof of the theorem is completely same as the classical case. See,
for example, [15, Page 88, Theorem 1].

A simple system of generators for $H^{\star}(K_{\sigma})\cong
H^{\star}(pt)[c,d]/(c^{2}=ac+ud)$ is

$\left\\{c,d^{2^{l}}|l\geq
0\right\\}=\left\\{c,\mathbf{s}_{1+\sigma,l}(d)|l\geq 0\right\\}$

By applying the Borel Theorem to the path space fibration

$K_{\sigma}\rightarrow P(K_{\rho})\rightarrow K_{\rho}$

we have

$H^{\star}(K_{\rho})=P(x_{\rho},\mathbf{s}_{\rho,l}(x_{1+\rho})|l\geq 0).$

A simple system of generator for $H^{\star}(K_{\rho})$ is

$\left\\{x_{\rho}^{2^{j}},(\mathbf{s}_{\rho,l}(x_{1+\rho}))^{2^{j}}|j,l\geq
0\right\\}=\left\\{\mathbf{s}_{\rho,j}(x_{\rho}),\mathbf{s}_{2^{l}\rho+1,j}\mathbf{s}_{\rho,l}(x_{1+\rho})|j,l\geq
0\right\\}$

where $|\mathbf{s}_{\rho,l}(x_{1+\rho})|=2^{l}\rho+1$. Then,

$H^{\star}(K_{1+\rho})=P(\mathbf{s}_{\rho,j}(x_{1+\rho}),\mathbf{s}_{2^{l}\rho+1,j}\mathbf{s}_{\rho,l}(x_{2+\rho})|j,l\geq
0).$

A simple system of generators for $H^{\star}(K_{1+\rho})$ is

$\displaystyle\left\\{(\mathbf{s}_{\rho,j}x_{1+\rho})^{2^{k}},(\mathbf{s}_{2^{j}\rho+1,j}\mathbf{s}_{\rho,l}x_{2+\rho})^{2^{k}}|j,l,k\geq
0\right\\}$
$\displaystyle=\left\\{\mathbf{s}_{2^{j}\rho+1,k}\mathbf{s}_{\rho,j}x_{2+\sigma},\mathbf{s}_{2^{j}(2^{l}\rho+1)+1,k}\mathbf{s}_{2^{l}\rho+1,j}\mathbf{s}_{\rho,l}x_{2+\rho}|j,l,k\geq
0\right\\}$

where
$|\mathbf{s}_{2^{l}\rho+1,j}\mathbf{s}_{\rho,l}x_{2+\rho}|=2^{j}(2^{l}\rho+1)+1$.
Then,

$H^{\star}(K_{2+\rho})=P(\mathbf{s}_{2^{j}\rho+1,k}\mathbf{s}_{\rho,j}x_{2+\rho},\mathbf{s}_{2^{j}(2^{l}\rho+1)+1,k}\mathbf{s}_{2^{l}\rho+1,j}\mathbf{s}_{\rho,l}x_{3+\rho}|j,l,k\geq
0).$

Thus, by iterating this process, one can find the $RO(C_{2})$-graded
cohomology of $C_{2}$-equivariant Eilenberg-Mac Lane spaces $K_{n+\sigma}$

$H^{\star}(K_{n+\sigma})$

for $n\geq 0$.

######  Conjecture 18.

We know that

$\displaystyle H^{\ast}(K_{1})=P(x_{1})$ $\displaystyle
H^{\ast}(K_{\sigma})=P(x_{\sigma},x_{1+\sigma})/(x_{\sigma}^{2}+ax_{\sigma}+ux_{1+\sigma}).$

If we knew $H^{\star}(K_{1+k(\sigma-1)})$ for all $k\geq 0$, then we could use
the Borel theorem to find $H^{\star}(K_{1+m+k(\sigma-1)})$ for $m\geq 0$.

We conjectured that $H^{\star}(K_{1+k(\sigma-1)})$ is a polynomial algebra on
$k+1$ generators with $k$ relations, saying that the square of each of the
first k generators is a linear combination of the other generators. The
dimensions of the first $k$ generators of the $H^{\star}(K_{1+k(\sigma-1)})$
are obtained by adding $\sigma-1$ to those of the generators of the
$H^{\star}(K_{1+(k-1)(\sigma-1)})$, and the dimension of the last generator is
$k\sigma+2^{k}-k$.

######  Example 19.

Let $k=3$. Then

$H^{\ast}(K_{3\sigma-2})=P(x_{3\sigma-2},x_{3\sigma-1},x_{3\sigma+1},x_{3\sigma+5})/(x_{3\sigma-2}^{2}+\cdots,x_{3\sigma-1}^{2}+\cdots,x_{3\sigma+1}^{2}+\cdots)$

where the other terms in the relations are linear. The resulting simple system
of generators is

$\\{x_{3\sigma-2},x_{3\sigma-1},x_{3\sigma+1}\\}\cup\\{x_{3\sigma+5}^{2^{i}}|i\geq
0\\}.$

Now, we give another useful lemma for computations, which is the cohomology
analogous of the Lemma 2.7. in [1].

There is a forgetful map

$\Phi^{e}:H^{V}_{C_{2}}(X;\underline{Z/2})\longrightarrow H^{|V|}(X^{e};Z/2)$

from the equivariant cohomology to the non-equivariant cohomology with
$Z/2$-coefficients. And also, we have a fixed point map

$\Phi^{C_{2}}:H^{V}_{C_{2}}(X;\underline{Z/2})\longrightarrow
H^{|V^{C_{2}}|}(X^{\Phi C_{2}};Z/2)$

where $X^{\Phi C_{2}}$ is a geometric fixed point of a $G$-space $X$. Now, I
will state the lemma, whose proof is the analog of Lemma 2.7. in [1].

######  Lemma 20.

Let $X$ be a genuine $C_{2}$-spectrum, and suppose that $\\{b_{i}\\}$ is a set
of elements of $H^{\star}(X)$ such that

  * (i)

$\\{\Phi^{e}(b_{i})\\}$ is a basis of $H^{\ast}(X^{e})$, and

  * (ii)

$\\{\Phi^{C_{2}}(b_{i})\\}$ is a basis of $H^{\ast}(X^{\Phi C_{2}})$

Then $H^{\star}(X)$ is free over $H^{\star}(pt)$ with the basis $\\{b_{i}\\}$.

One project is to finish calculations of the $RO(C_{2})-$graded
$C_{2}$-equivariant cohomology of $C_{2}$-equivariant Eilenberg-Mac Lane
spaces by using Caruso theorem 16 and lemma 20, and then Eilenberg-Moore
spectral sequences of Michael A. Hill [4, Chapter 5].

######  Conjecture 21.

$H^{\star}(K_{r\sigma+s})$ is a polynomial algebra on certain
$C_{2}$-equivariant Steenrod operations $Sq_{C_{2}}^{I}(\iota_{r\sigma+s})$
divided by certain powers of $u$, where $e(I)<r\sigma+s$, and
$\iota_{r\sigma+s}$ is the fundamental class, and $V<V^{{}^{\prime}}$ if and
only if $V^{{}^{\prime}}=V+W$ for some actual representations $W$ with
positive degree.

######  Example 22.

$H^{\star}(K(\underline{Z/2},1+\sigma))$ is the polynomial algebra generated
by elements $Sq^{I}(\iota_{1+\sigma)}$, where $I$ is admissible and
$e(I)<1+\sigma$. So, it is a polynomial algebra

$P(Sq^{0}(\iota_{1+\sigma}),Sq^{2}Sq^{1}(\iota_{1+\sigma}),Sq^{4}Sq^{2}Sq^{1}(\iota_{1+\sigma}),\cdots).$

Then, it is shortly

$P(x_{\rho},x_{1+2\rho},x_{1+4\rho},x_{1+8\rho},\cdots).$

By now, we have calculated the cohomology of $K(\underline{Z/2},n+\sigma)$ for
$n\geq 0$. To calculate other cases, if knew $H^{\ast}(K_{n\sigma})$ for
$n\geq 2$, we could use the Eilenberg-Moore spectral sequence [4, Chapter 5]
anf the Borel theorem for the path-space fibration

$\Omega K(\underline{Z/2},V)\longrightarrow
P(K(\underline{Z/2},V))\longrightarrow K(\underline{Z/2},V).$

For example, for the path-space fibration

$K(\underline{Z/2},\sigma)\longrightarrow
P(K(\underline{Z/2},1+\sigma))\longrightarrow K(\underline{Z/2},1+\sigma).$

$E_{\infty}$-term of the Eilenberg-Moore spectral sequence is

$E_{\infty}=E(x_{\sigma},x_{\rho},x_{2\rho},x_{4\rho},\cdots)$

with the relations

$x_{\sigma}^{2}=ax_{\sigma}+ux_{1+\sigma}$ $x_{1+\sigma}^{2}=x_{2+2\sigma}$
$x_{2+2\sigma}^{2}=x_{4+4\sigma}$ $\vdots$ $x^{2}_{2^{i}\rho}=x_{2^{i+1}\rho}$
$\vdots$

for $i\geq 0$. As a result,
$H^{\star}(K(\underline{Z/2},\sigma),\underline{Z/2})$ is a quadratic
extension of a polynomial algebra, as it has already known.

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