Po Hu, Igor Kriz, Petr Somberg, and Foling Zou

###### Abstract.

In the present paper, we construct a $\mathbb{Z}/p$-equivariant analog of the $\mathbb{Z}/2$-equivariantspectrum $BP\mathbb{R}$ previously constructed by Hu and Kriz. We prove that this spectrum has some ofthe properties conjectured by Hill, Hopkins, and Ravenel. Our main construction method is an $\mathbb{Z}/p$-equivariantanalog of the Brown-Peterson tower of $BP$, based on a previous description of the $\mathbb{Z}/p$-equivariant Steenrodalgebra with constant coefficients by the authors. We also describe several variants of our construction andcomparisons with other known equivariant spectra.

Hu acknowledges support by NSF grant 2301520. Kriz acknowledges the support of Simons Foundation award 958219. Somberg acknowledges supportfrom the grant GACR 19-28628X

With contributions by Guchuan Li

## 1. Introduction

Hill, Hopkins, and Ravenel [7] conjectured that there exists a $\mathbb{Z}/p$-equivariant structure on$BP^{\wedge(p-1)}$ for $p>2$ such that the geometric fixed point spectrum is $H\mathbb{Z}/p$. This equivariantspectrum is expected to have a number of properties. The purposeof this paper is to give a construction of such a spectrum and prove some of theproperties. Our construction is based on a mild extension of the computation of the $\mathbb{Z}/p$-equivariant Steenrod algebrawith respect to the constant Mackey functor $\underline{\mathbb{Z}/p}$ [9, 16]. Following Sankar-Wilson[16], we denote by $T$ the second desuspension of the $\mathbb{Z}/p$-equivariant degree $p$map

$S^{\beta}\rightarrow S^{2}$ |

where $\beta$ is the basic irreducible representation of $\mathbb{Z}/p$. Denoting by $\widetilde{\mathcal{L}}_{p-1}$the $\mathbb{Z}/p$-Mackey functor whose non-equivariant part is the reduced regular representation$\mathcal{L}_{p-1}$ of$\mathbb{Z}/p$ and the $\mathbb{Z}/p$-fixed part is $0$, there is a map of $\mathbb{Z}/p$-equivariant spectra

(1) | $H\underline{\mathbb{Z}}\rightarrow\Sigma^{2p^{n-1}+(p^{n}-p^{n-1}-1)\beta}H%\underline{\mathbb{Z}}\wedge T$ |

(which comes from the fact that after smashing with $H\underline{\mathbb{Z}}$, the right-hand side of (1)becomes a summand of the left-hand side as an $H\underline{\mathbb{Z}}$-module.Now we also have a βconnecting mapβ

$H\underline{\mathbb{Z}}\wedge T\rightarrow\Sigma^{\beta-1}H\widetilde{\mathcal%{L}}_{p-1}$ |

(see the cofibration sequence (21) below). Composing with (1), we get a map

(2) | $Q_{n}^{\prime}:H\underline{\mathbb{Z}}\rightarrow\Sigma^{2p^{n-1}-1+(p^{n}-p^{%n-1})\beta}H\widetilde{\mathcal{L}}_{p-1}$ |

(see formula (22) below).This map is a $\mathbb{Z}/p$-equivariant analog of the integral $Q_{n}$-elements which form the first k-invariantof the Brown-Peterson construction of $BP$ [3].

The approach of our construction is to construct a $\mathbb{Z}/p$-equivariant spectrum $BP\mathbb{R}$ by mimicking, in a minimalway, the Brown-Peterson construction [3] in the category of $\mathbb{Z}/p$-equivariant spectra. It is important to note that the appearance of the non-constant Mackeyfunctor occurring on the right-hand side of (2) is a reflection of the fact that in $(BP\mathbb{R}_{\{e\}})_{*}$,$v_{n}$ lies in a copy of the representation $\mathcal{L}_{p-1}$. We denote the generator of this representationby $r_{n}$. In fact, in the equivariant analogue of the $BP$ tower, a key feature is the interplay betweenthe Mackey functors $\underline{\mathbb{Z}}$, $\widetilde{\mathcal{L}}_{p-1}$ and theMackey functor $\underline{\mathcal{L}}_{p}$, which is the principal projectiveMackey functor on a fixed element (i.e. has the integral regular representation as the non-equivariant part, and$\mathbb{Z}$ as the fixed points).

The non-equivariantspectrum underlying our βminimalβ $BP\mathbb{R}$ following the Brown-Petersonconstruction is somewhat smaller than the $\bigwedge_{p-1}BP$ conjectured byHill, Hopkins, and Ravenel [7]. In fact, at each $v_{n}$, we are forced to put into the non-equivariantcoefficients the representations

(3) | $(\mathbb{Z}\oplus\mathcal{L}_{p-1}r_{n}\oplus r_{n}^{2}\mathcal{L}_{p}[r_{n}])%\otimes\mathbb{Z}[Nr_{n}]$ |

where $\mathcal{L}_{p}$ is the integral regular $\mathbb{Z}/p$-representation and $N$ denotes the multiplicative norm(see formula (9) below).

The non-equivariant spectrum conjectured by Hill, Hopkins, and Ravenel [7] (which we denoteby $BP\mathbb{R}^{HHR}$) should have, instead of (3),

(4) | $Sym(\mathcal{L}_{p-1}\cdot r_{n}).$ |

This, in fact, coincides with (3) for $p=3$. For $p\geq 5$, (4) is bigger, but one canconstruct a candidate for the spectrum $BP\mathbb{R}^{HHR}$ by forming a wedge of $BP\mathbb{R}$ with a wedgeof even suspensions of copies of the multiplicative norm of $BP$ from $\{e\}$ to $\mathbb{Z}/p$.(See Section 5 below.)

We should remark that we can currently only make the equivariant analogue of the Brown-Petersonconstruction in the category of Borel-complete spectra. In this category, the obstructions to continuing theconstruction are either non-equivariant (and vanish for the same reason as in [3], i.e. evenness), orare non-torsion with respect to multiplication by the class represented by the inclusion

$S^{0}\rightarrow S^{\beta},$ |

which can therefore be treated on the level of geometric fixed points. On geometric fixed points, however,the equivariant Brown-Peterson tower splits into a wedge equivalences on summands, which is whythose obstructions also vanish.

The fact that we work in the Borel-complete category, however, should not be a substantial restriction,since the spectrum $BP\mathbb{R}$ should be Borel-complete anyway (similarly as for $p=2$). In fact, wehave a $\mathbb{Z}/p$-equivariant analog of the Borel cohom*ology spectral sequence [8], whichwe present in Section 3. This allows us to calculate the$RO(\mathbb{Z}/p)$-graded coefficients $BP\mathbb{R}_{\star}$ completely (Theorem 2 below).Not having a βgenuineβ version of equivariant Brown-Petersonconstruction amounts to not having, at the moment, an analogue of the slice spectral sequence.

In Section 5, we also discuss analogues of the $BP\mathbb{R}$ construction on Johnson-Wilson-type spectra.Using the obstruction theory of Robinson [15] and Baker [2],we construct a $\mathbb{Z}/p$-action related to $BP\mathbb{R}$ on a completion of the smash product of $(p-1)$ copies of $E(n)$(or its variants, such as $E_{n}$). We conjecture that there is an βorientationβ map from $BP\mathbb{R}^{HHR}$to these spectra (however, we do not yet have a ring structure on the source).

It is also known that $\mathbb{Z}/p$ acts on $E_{n}$ when $(p-1)\mid n$ via the subgroup ofthe Morava stabilizer group. The case$n=p-1$ has been especially studied(see [13, 17]). It is reasonable to conjecture that $BP\mathbb{R}^{HHR}$ maps into this equivariant$E_{n}$ in a way which is identity on the $\mathcal{L}_{p-1}$ containing $v_{n}$ inthe non-equivariant coefficients, for which we give evidence on the level of formal group laws.

The present paper incorporates numerous observations, suggestions, and references communicated to us by Guchuan Li.

## 2. Preliminary computations

We will use theintegral trivial, regular resp.reduced regular representation $\mathcal{L}_{1}$, $\mathcal{L}_{p}$, $\mathcal{L}_{p-1}$of $\mathbb{Z}/p$. Note that all are isomorphic to their duals integrally.In case of $\mathcal{L}_{p-1}$, we have an isomorphism

$\mathbb{Z}^{p}/(1,1,\dots,1)\rightarrow\{(a_{1},\dots,a_{p})\in\mathbb{Z}^{p}%\mid\sum a_{i}=0\}$ |

by sending

$(1,0,\dots,0)\mapsto(1,-1,0,\dots,0).$ |

We will work with integral Mackey functors here. We denote by $\underline{\mathbb{Z}}=\underline{\mathcal{L}}_{1}$ the constantMackey functor, and by $\widetilde{\mathcal{L}}_{p-1}$ the Mackey functor equal to the integral reduced regular representationon the free orbit and $0$ on the fixed orbit. We also have the co-constant Mackey functor $\overline{\mathcal{L}}_{1}$ where again both the non-equivariant and fixed parts are $\mathbb{Z}$ andthe corestriction is $1$ while the restriction is $p$. We have short exact sequences

(5) | $0\rightarrow\underline{\mathcal{L}}_{1}\rightarrow\underline{\mathcal{L}}_{p}%\rightarrow\widetilde{\mathcal{L}}_{p-1}\rightarrow 0$ |

(6) | $0\rightarrow\widetilde{\mathcal{L}}_{p-1}\rightarrow\underline{\mathcal{L}}_{p%}\rightarrow\overline{\mathcal{L}}_{1}\rightarrow 0.$ |

We also note that for a non-trivial irreducible representation $\beta$ of $\mathbb{Z}/p$ ($p>2$), we have

(7) | $H\overline{\mathcal{L}}_{1}=\Sigma^{2-\beta}H\underline{\mathcal{L}}_{1}.$ |

We also have

(8) | $H\widetilde{\mathcal{L}}_{p-1}\wedge_{H\underline{\mathbb{Z}}}H\widetilde{%\mathcal{L}}_{p-1}=H\overline{\mathcal{L}}_{1}\vee\bigvee_{p-2}H\underline{%\mathcal{L}}_{p}.$ |

We will work $p$-locally, so whenever we say β$\mathbb{Z}$,β we mean $\mathbb{Z}_{(p)}$. Then all equivariant spectraare $(\beta-\beta^{\prime})$-periodic for non-trivial irreducible $\mathbb{Z}/p$-representations $\beta,\beta^{\prime}$. Thisis due to the fact that there exists a $\mathbb{Z}/p$-equivariant map of spaces

$S^{\beta}\rightarrow S^{\beta^{\prime}}$ |

of some degree $k\in\mathbb{Z}/p^{\times}$, which is a $p$-local equivalence (cf. [16]).Thus, instead of $RO(\mathbb{Z}/p)$-grading,we can consider $R$-grading where $R=\mathbb{Z}\{1,\beta\}$ for a chosen non-trivial irreducible $\mathbb{Z}/p$-representation$\beta$. This is also true for ordinary $\mathbb{Z}/p$-equivariant hom*ology with coefficients over a $\underline{\mathbb{Z}}$-Mackeymodule for a different reason (see [9]).

###### Proposition 1.

1. The $R$-graded coefficients of $H\underline{\mathcal{L}}_{1}$ are $\mathbb{Z}$ in degrees $2k-k\beta$, $k\geq 0$and $\mathbb{Z}/p$ in degrees $2k-\ell\beta$ with $0\leq k<\ell$ (this is called the good wedge)and $\mathbb{Z}$ in degrees $2k-k\beta$, $k<0$ and $\mathbb{Z}/p$ in degrees $-1-2k+\ell\beta$ where $2<2k+1<2\ell$(this is called the derived wedge).

2. The $R$-graded coefficients of $H\widetilde{\mathcal{L}}_{p-1}$ are $\mathbb{Z}/p$ in degrees $2k+1-\ell\beta$where $0<2k+1<2\ell$ (this is the good wedge) and $\mathbb{Z}/p$ in degrees $-2k+\ell\beta$ where$0<k\leq\ell$ (this is the derived wedge).

(In the remaining $R$-dergrees, the coefficients are $0$.)

$\square$

The pattern of the $R$-graded coefficients of $H\underline{\mathcal{L}}_{1}=H\underline{\mathbb{Z}}$ is shown in Figure 1 below (where a solid dotmeans a copy of $\mathbb{Z}/p$ and an empty square means a copy of $\mathbb{Z}$.

In our tower illustration, we shall use the shortcut for this pattern shown in Figure 2.

Figure 3 shows the pattern of the $R$-graded coefficients of $\Sigma^{\beta-1}H\widetilde{\mathcal{L}}_{p-1}$.The shift is so that the corner of the βgood wedgeβ is at the $(0,0)$-point.

Figure 4 shows the shortcut we use for this pattern.

Figure 5 shows the $R$-graded coefficients of $H\underline{\mathbb{Z}/p}$.

Our shortcut for this pattern is shown in Figure 6.

Figure 7 shows the $R$-graded coefficients of$\Sigma^{\beta-1}H\widetilde{L}_{p-1}=\Sigma^{\beta-1}H(\widetilde{L}_{p-1}/p)$.

Our shorthand for this pattern is shown in Figure 8.

## 3. The Borel cohom*ology spectral sequence

The Borel cohom*ology $F(E\mathbb{Z}/p_{+},X)$ for a $\mathbb{Z}/p$-equivariant spectrum $X$ will be denotedby $X^{c}$. Coefficients of $H\underline{\mathcal{L}}_{1}^{c}$, $H\widetilde{\mathcal{L}}_{p-1}^{c}$ are obtained byinverting $\sigma^{-2}$ in the good wedge in Proposition 1 (which acts by isomorphismwherever dimensionally possible).

hom*otopy classes in a summand of the non-equivariant hom*otopy groups of a $\mathbb{Z}/p$-equivariantspectrum which is isomorphic to $\mathcal{L}_{p}$ is called negligible. Such classes cannot receive or supportdifferentials in the Borel cohom*ology spectral sequence because of the theory of Mackey functors.

We will completely calculate the Borel cohom*ology spectral sequence of $BP\mathbb{R}$ at an odd prime $p$modulo negligible $BP$-summands in $BP\mathbb{R}_{\{e\}}$. Modulo negligible $BP$-summands, $(BP\mathbb{R}_{\{e\}})_{*}$can be written as

(9) | $B=\bigotimes_{n>0}(\mathcal{L}_{1}\oplus\mathcal{L}_{p-1}\cdot r_{n}\oplus r_{%n}^{2}\mathcal{L}_{p}[r_{n}])[\Phi(v_{n})]$ |

where $\Phi(v_{n})$ has the same dimension as $v_{n}^{p}$ and$r_{n}$ is a generator of a $\mathcal{L}_{p-1}$-subrepresentation which contains $v_{n}$.

Recall that in [8], Hu and Kriz started with the $E^{1}$-term of the Borel cohom*ology spectral sequence and included $d^{1}$in the differential pattern. This has the advantage of allowing to treat $v_{0}$ on the same level as the other $v_{n}$βs, which makes thepattern more natural.

To describe the appropriate analog for $p$ odd, we preview the slightly more complicated assortment of elements anddifferentials (14), (15), (16) we will encounter. From the point of view of this pattern, the most naturalway to start is to put

(10) | $p=\Phi(v_{0}),$ |

and include the second, but not the first, differential of (14) for $n=0$. Thus, the elements $v_{0}$, $\sigma^{\frac{2}{p}-2}$ offractional degrees

$|v_{0}|=\frac{2}{p}-1-\frac{\beta}{p},\;|\sigma^{\frac{2}{p}-2}|=\beta(\frac{1%}{p}-1)-\frac{2}{p}+2$ |

are not present in our spectral sequence, but the exterior generator

$v_{0}\sigma^{\frac{2}{p}-2}$ |

of degree

$|v_{0}\sigma^{\frac{2}{p}-2}|=1-\beta$ |

is included, making the $E^{1}$-term

$B[\sigma^{2},\sigma^{-2}][b]\otimes\Lambda(v_{0}\sigma^{\frac{2}{p}-2}).$ |

The $d^{1}$ differential has

$d^{1}(v_{0}\sigma^{\frac{2}{p}-2})=pb,$ |

but also takes group cohom*ology on (9). This gives $E^{2}$-term

(11) | $E^{2}=B^{\prime}[\sigma^{2},\sigma^{-2}][b]/pb$ |

where if we put

(12) | $v_{I}=\prod v_{n}^{i_{n}},\;I=(i_{1},i_{2},\dots),\;i_{n}\in\{0,1\},$ |

(note that $v_{n}^{2}$ is negligible), and

(13) | $|I|=\sum_{n}i_{n},$ |

then

$B^{\prime}=\bigl{(}\mathbb{Z}\{v_{I}:I\text{ even}\}\oplus\mathbb{Z}/p\{v_{I}:%I\text{ odd}\}\bigr{)}[\Phi(v_{n})].$ |

The geometric fixed points of $BP\mathbb{R}$ are $H\mathbb{Z}/p$, so we need a resolution with respect to the dualof the polynomial algebra $\mathbb{Z}[\sigma^{-2}]$, which is a divided power polynomial algebra. This implies two setsof differentials

(14) | $\sigma^{-2p^{n-1}}\mapsto v_{n}b^{?},\;v_{n}\sigma^{-2(p-1)p^{n-1}}\mapsto\Phi%(v_{n})b^{?}.$ |

To figure out the exact power of $b$ in (14), we need to need to figure out the exact equivariant degreeof $v_{n}$. The $\mathbb{Z}$-graded component must be $2p^{n-1}-1$ in order for the differential (14) to decreasedegree by $1$. Given the fact that $v_{n}$ sits in a $\widetilde{\mathcal{L}}_{p-1}$, and given Proposition 1,its equivariant degree should be of the form $k+\ell\beta$ where $k+2\ell=|v_{n}|-1=2p^{n}-3$. Thus, we concludethat

(15) | $|v_{n}|=2p^{n-1}-1+(p^{n}-p^{n-1}-1)\beta.$ |

Similarly, given the fact that $\Phi(v_{n})$ sits in a copy of $\underline{\mathcal{L}}_{1}$, we have

(16) | $|\Phi(v_{n})|=2p^{n}-2+(p^{n}-1)(p-1)\beta.$ |

From this, we conclude that the differential pattern is

(17) | $\sigma^{-2p^{n-1}}\mapsto v_{n}b^{p^{n}-1},\;v_{n}\sigma^{-2(p-1)p^{n-1}}%\mapsto\Phi(v_{n})b^{(p^{n}-1)(p-1)+1}.$ |

(The differentials on $\sigma^{-2kp^{n-1}}$ for $2\leq k\leq p-1$ are determined by the Leibniz rule.)This is called the even differential pattern.

The reason this does not tell the whole story is that we have (8), so$v_{I}$ sits in a $\sigma^{2\ell}$-shifted copy of $\widetilde{\mathcal{L}}_{p-1}$ resp. $\underline{\mathcal{L}}_{1}$depending on whether $|I|$ is even resp. odd. In fact, working out the shifts precisely, one gets that

(18) | $|v_{I}|=\sum_{i_{n}=1}(2p^{n-1}-1)+\left(\sum_{i_{n}=1}(p^{n}-p^{n-1})-\lceil%\frac{|I|}{2}\rceil\right)\beta.$ |

When multiplying (17) by $v_{I}$ with $i_{n}=0$ and $|I|$ odd, the $\sigma^{-2}$-power will bein a copy of $\widetilde{\mathcal{L}}_{p-1}$, (so non-equivariantly, its dimension goes down by $1$),while the $v_{n}$ will be in a copy of $\underline{\mathcal{L}}_{1}$, so non-equivariantly, its dimension goes up by $1$.Since the non-equivariant dimension of $b$ is $-2$, we then obtain the odd differential pattern

(19) | $\sigma^{-2p^{n-1}}\mapsto v_{n}b^{p^{n}},\;v_{n}\sigma^{-2(p-1)p^{n-1}}\mapsto%\Phi(v_{n})b^{(p^{n}-1)(p-1)}.$ |

Of course, in the spectral sequence, the differential of every monomial has multiple summands, and we mustselect the summand which generates lowest $b$-torsion. Since, luckily,

$p^{n}<(p^{n}-1)(p-1)$ |

and

$(p^{n}-1)(p-1)+1<p^{n+1}-1,$ |

the lowest $b$-torsion is always generated by $v_{n}$ or $\Phi(v_{n})$ with the lowest $n$. For this reason, theTate spectral sequence (obtained by inverting $b$) converges to $\mathbb{Z}/p[b,b^{-1}]$ and similarly as in thecase of $p=2$, writing monomials

$v_{I}\Phi(v_{J})$ |

where $I$ is as above and $J=(j_{1},j_{2},\dots)$, $j_{n}\in\mathbb{N}_{0}$,

$\Phi(v_{J})=\prod_{n}\Phi(v_{n})^{j_{n}},$ |

letting $r=r(I)$ resp. $s=s(J)$ be the lowest $n$ for which $i_{n}\neq 0$ resp. $j_{n}\neq 0$,(set to $\infty$ when not applicable),we obtain the following

###### Theorem 2.

Modulo negligible $BP_{*}[\sigma^{2},\sigma^{-2}]$-summands, the $R$-graded coefficients of $BP\mathbb{R}$ area sum of

$\mathbb{Z}_{(p)}[b],$ |

βeven-pattern summandsβ where $|I|$ is odd and $r\leq s$:

$\bigoplus_{\ell\nequiv-1\mod p}\mathbb{Z}/p\cdot v_{I}\Phi(v_{J})\cdot\sigma^{%2p^{r-1}\ell}[b]/(b^{p^{r}-1})$ |

or $|I|$ is even, $I\neq 0$ and $r>s$:

$\bigoplus_{\ell\in\mathbb{Z}}\mathbb{Z}_{(p)}\cdot v_{I}\Phi(v_{J})\cdot\sigma%^{2p^{s}\ell}[b]/(b^{(p^{s}-1)(p-1)+1},pb)$ |

and βodd-pattern summandsβ where $|I|$ is even and $I\neq 0$ and $r\leq s$:

$\bigoplus_{\ell\nequiv-1\mod p}\mathbb{Z}_{(p)}\cdot v_{I}\Phi(v_{J})\cdot%\Sigma^{2p^{r-1}\ell}[b]/(b^{p^{r}},pb)$ |

or $|I|$ is odd and $r>s$ :

$\bigoplus_{\ell\in\mathbb{Z}}\mathbb{Z}/p\cdot v_{I}\Phi(v_{J})\cdot\sigma^{2p%^{s}\ell}[b]/(b^{(p^{s}-1)(p-1)}).$ |

$\square$

## 4. Construction

In this section, we will construct $BP\mathbb{R}$ at an odd prime $p$ by realizing spectrally a $\mathbb{Z}/p$-equivariantanalog of the Brown-Peterson resolution. Throughout this section, we will work in Borel cohom*ology, andomit this from the notation. With this convention, the integral hom*ology of $BP\mathbb{R}$, modulo negligiblecopies of

$H\mathbb{Z}_{*}BP[\sigma^{2},\sigma^{-2}],$ |

is

$H\underline{\mathbb{Z}}[\underline{\theta}_{1},\underline{\theta}_{2},\dots]%\wedge_{H\underline{\mathbb{Z}}}\Lambda_{H\widetilde{\mathcal{L}}_{p-1}}[%\widehat{\xi}_{1},\widehat{\xi}_{2},\dots].$ |

(The second term denotes the exterior algebra.)The k-invariants are defined as cohom*ological operations (in Borel cohom*ology), and our job is to prove that theycan be realized spectrally (i.e. on the realization of the partial resolution at each step).

While introducing some negligible terms is necessary, the corresponding non-equivariant hom*ology will remaineven-degree and non-torsion, and there is therefore no obstruction to spectral realization of the negligiblek-invariants, by Brown-Petersonβs βeven-odd argumentβ [3]. On the other hand, the non-negligiblek-invariants are non-torsion either rationally or non-torsion with respect to $b$ and hence the obstructionto spectral realization is $0$ because it vanishes rationally, resp. on geometric (i.e., in our case, Tate) fixed points.

The Borel cohom*ology version of the $\mathbb{Z}/p$-equivariant Brown-Peterson resolution is essentially describedas a product over $H\underline{\mathbb{Z}}$ of the following sequences:

Starting with $H\underline{\mathbb{Z}}$, we have

(20) | $\begin{array}[]{l}H\underline{\mathbb{Z}}\wedge H\underline{\mathbb{Z}}=\\\bigwedge_{n\geq 1}(H\underline{\mathbb{Z}}\vee(H\underline{\mathbb{Z}}\wedge T%\{\widehat{\xi}_{n}\underline{\xi}_{n}^{k}\mid k=0,\dots,p-2\})\vee H%\underline{\mathbb{Z}/p}\{\widehat{\xi}_{n}\underline{\xi}_{n}^{p-1}\})[%\underline{\theta}_{n}].\end{array}$ |

The $H\underline{\mathbb{Z}/p}$ arises, in the notation of [9], as two copies of$H\underline{\mathbb{Z}}$ stuck together by $p$-multiplication, in other words from the fact that

$\beta(\underline{\mu}_{n})=\underline{\theta}_{n}.$ |

Now one has a cofibration sequence

(21) | $\Sigma^{\beta-2}H\widetilde{\mathcal{L}}_{p-1}\rightarrow H\underline{\mathbb{%Z}}\wedge T\rightarrow\Sigma^{\beta-1}H\widetilde{\mathcal{L}}_{p-1}$ |

(split on $R$-graded coefficients). Then, there is an operation

(22) | $Q_{n}^{\prime}:H\underline{\mathbb{Z}}\rightarrow\Sigma^{\beta}H\widetilde{%\mathcal{L}}_{p-1}\{v_{n}\}$ |

which applies (21) to the $\underline{\xi}_{n}$-wedge summand of (20). (Herethe notation $v_{n}$ refers to the $R$-graded indexing of the previous section.)

Comment:The operations $Q_{n}^{\prime}$ form the first stage of our equivariant version of the $BP\mathbb{R}$ tower.One may wonder how these elements are identified. To this end, it is beneficial to consider whathappens on the level of geometrical fixed points of hom*ology. We have

(23) | $\begin{array}[]{l}\Phi^{\mathbb{Z}/p}(H\underline{\mathbb{Z}}\wedge BP\mathbb{%R})_{\star}=\\[4.30554pt]\Phi^{\mathbb{Z}/p}(H\underline{\mathbb{Z}})\wedge\Phi^{\mathbb{Z}/p}(BP%\mathbb{R})_{*}[b,b^{-1}]=\\[4.30554pt]A_{*}[\sigma^{-2}][b,b^{-1}].\end{array}$ |

On the other hand, we have

(24) | $\begin{array}[]{l}\Phi^{\mathbb{Z}/p}(H\underline{\mathbb{Z}}\wedge H%\underline{\mathbb{Z}})_{\star}=\\[8.61108pt]\Phi^{\mathbb{Z}/p}(H\underline{\mathbb{Z}})\wedge\Phi^{\mathbb{Z}/p}(H%\underline{\mathbb{Z}})_{*}[b,b^{-1}]=\\[8.61108pt]A_{*}[\sigma^{-2},\rho^{-2}][b,b^{-1}].\end{array}$ |

Thus, to the eyes of $\mod p$ hom*ology, to get from $H\underline{\mathbb{Z}}$ to $BP\mathbb{R}$ on geometricalfixed points, we must kill the $\rho^{-2}$-powers in (24), while preserving the othergenerators.

To describe how this is done, let us recall briefly the $p=2$ case.As remarked in [8], for $p=2$, a BP-tower construction of $BP\mathbb{R}$ on the levelof geometric fixed points is modelled on the cobar complex of $\mathbb{Z}/2[\rho^{-2}]$, which is the samething as the cohom*ological Koszul complex of the divided polynomial power algebra $\mathbb{Z}/2[\rho^{-2}]^{\vee}$,which is an exterior algebra. Thus, the generators are in total degrees $2^{i}-1$, $i=1,2,\dots$, which is correct,since the dimension of $v_{i}$ is $(2^{i}-1)(1+\alpha)$, and the $\alpha$ may be disregarded on the levelof geometrical fixed points. In particular, for $p=2$, on geometrical fixed points, the $BP\mathbb{R}$ tower is split (i.e.the maps are equivalences of cancelling wedge summands), and will work for any choice of $Q^{\prime}_{n}$ whichsends $\rho^{-2^{n}}\mapsto 1$ (up to multiplying by a power of $a$). We also note that modulo possibleerror terms in the derived wedge, this condition, together with its dimension, essentially determines $Q^{\prime}_{n}$,since on the good wedge, inverting $a$ is injective.

For $p>2$, the construction is precisely analogous (the only difference being that for $p=2$, error terms in the derivedwedge do not matter, since we have an a priori model of $BP\mathbb{R}$ coming from $M\mathbb{R}$ via spectral algebra, while for$p>2$, no a priori geometric model is known, so we circumvent the difficulty by working in Borel cohom*ology).

On the level of geometric fixed points, however, again, the $BP\mathbb{R}$-tower is modelled on the cohom*ologicalKoszul complex of the divided polynomial power algebra

(25) | $\mathbb{Z}/p[\rho^{-2}]^{\vee}.$ |

Now (25) is a tensor product of truncated polynomial algebras of the form

(26) | $\mathbb{Z}/p[x]/x^{p}$ |

where $x$ is the dual of

(27) | $\rho^{-2p^{n-1}},\;n=1,2,\dots$ |

This means that our primary operation corresponds to an exterior generator of cohom*ological degree $1$ and topological degree$2p^{n-1}$. (The total degree $2p^{n-1}-1$ corresponds to the geometric fixed point version of $v_{n}$.)Then there is a secondary generator in cohom*ological degree $2$ and topologicaldegree $2p^{n}$, which gives total degree $2p^{n}-2$, correspondingto the geometric fixed point version of $\Phi(v_{n})$.

Recalling the short exact sequence of $\underline{\mathbb{Z}}$-modules

$0\rightarrow\underline{\mathbb{Z}}\rightarrow\underline{\mathcal{L}}_{p}%\rightarrow\widetilde{\mathcal{L}}_{p-1}\rightarrow 0,$ |

when taking geometric fixed points (i.e. inverting $b$), $H\widetilde{L}_{p-1}$ becomes identifiedwith $\Sigma H\underline{\mathbb{Z}}$, so we see that ignoring the $\beta$-part, our primary operationis at least of the right degree $2p^{n-1}$.

To assert that we actually have the right element, we need to show how $Q^{\prime}_{n}$ cancels the element(25). To this end, we recall the computation in [9] in Borel cohom*ology:

$\begin{array}[]{ll}\underline{\xi}_{1}&=(\sigma^{-2}-\rho^{-2})b^{1-p}\\\underline{\xi}_{n+1}&=(\sigma^{2p^{n-1}-2p^{n}}\underline{\xi}_{n}-\rho^{-2}%\xi_{n})b^{p^{n}-p^{n+1}}.\\\end{array}$ |

The first equation eliminates the term $\rho^{-2}$ modulo the allowed error terms in (23),while the first term of the second equation can be similarly used by induction to eliminate $\rho^{-2p^{n-1}}$.

We will see below how the transpotence also corresponds to $\Phi(v_{n})$. However, since it is definedby canceling the hom*ology element corresponding tothe βKudo elementβ $x^{p-1}y$ where $x$ is as in (26) and $y$ is its Koszul dual, we have nostatement on the level of the dual Steenrod algebra in that case. Similarly as in the $p=2$ case, again,on the level of geometric fixed points, our $BP\mathbb{R}$-tower splits, i.e. is described as a wedge of equivalences ofwedge summands.

We also note that the Borel cohom*ology pieces we add to our tower match (9). Therefore, by the calculation ofSection 3, adding Borel cohom*ology pieces, on the Tate level, we recover the $b$-non-torsion part of the coefficients,which matches the above proposed geometric fixed points. This shows that both constructions correspond correctly.

Now to continue our construction, smashing the cofibration sequence

$S^{\beta-2}\rightarrow S^{0}\rightarrow T$ |

with $H\widetilde{L}_{p-1}$, we have a cofibration sequence

(28) | $H\widetilde{\mathcal{L}}_{p-1}\rightarrow H\widetilde{\mathcal{L}}_{p-1}\wedgeT%\rightarrow\Sigma^{\beta-1}H\widetilde{\mathcal{L}}_{p-1}.$ |

The $\underline{\xi}_{n}$-multiple of $Q_{n}^{\prime}$ kills on the $R$-graded coefficients of themiddle term of the $|\underline{\xi}_{n}|$-suspension of (28)everything except the $|\underline{\xi}_{n}|$-suspension of a copy of $H\underline{\mathcal{L}}_{p}$coming from the first term of (28).Thus, we obtain a k-invariant

(29) | $\overline{Q}_{n}:H\widetilde{\mathcal{L}}_{p-1}\{v_{n}\}\rightarrow H%\underline{\mathcal{L}}_{p}\{v_{n}^{2}\}.$ |

On the other hand, the fiber $F$ of (28) contains a copy of a fiber of

(30) | $H\underline{\mathbb{Z}}\cdot\underline{\mu}_{n}\rightarrow H\widetilde{%\mathcal{L}}_{p-1}\wedge T\cdot\widehat{\xi}_{n}\underline{\xi}_{n}^{p-2}$ |

which has a factor of $H\underline{\mathbb{Z}}$. We define a k-invariant

(31) | $Q_{n}^{\prime\prime}:F\rightarrow H\underline{\mathbb{Z}}\cdot\Phi(v_{n})$ |

isomorphically on that factor.

In more detail, we have a cofibration sequence

(32) | $H\underline{\mathbb{Z}}\rightarrow HT\rightarrow\Sigma^{\beta-1}H\underline{%\mathbb{Z}}$ |

and thus also

(33) | $H\widetilde{\mathcal{L}}_{p-1}\rightarrow HT\wedge_{H\underline{\mathbb{Z}}}H%\widetilde{\mathcal{L}}_{p-1}\rightarrow\Sigma^{\beta-1}H\widetilde{\mathcal{L%}}_{p-1}.$ |

On the other hand, we have a cofibration sequence

(34) | $H\widetilde{\mathcal{L}}_{p-1}\rightarrow H\underline{\mathcal{L}}_{p}%\rightarrow\Sigma^{2-\beta}H\underline{\mathbb{Z}},$ |

or

(35) | $H\underline{\mathbb{Z}}\rightarrow\Sigma^{\beta-1}H\widetilde{\mathcal{L}}_{p-%1}\rightarrow\Sigma^{-1}H\underline{\mathcal{L}}_{p}.$ |

Taking the derived pullback of (33) via the first map (35), we obtain a cofibrationsequence

(36) | $H\widetilde{\mathcal{L}}_{p-1}\rightarrow HX\rightarrow H\underline{Z}.$ |

However, (36) must split since there is no essential $H\underline{\mathbb{Z}}$-module map

$H\underline{\mathbb{Z}}\rightarrow\Sigma H\widetilde{\mathcal{L}}_{p-1}$ |

(by dimensional reasons).

This is the even pattern in the Borel cohom*ology $\mathbb{Z}/p$-equivariantBrown-Peterson resolution. An illustration is given in Figure 9 of the Appendix.

The odd pattern is obtained essentially by smashing this over $H\underline{\mathbb{Z}}$ with$H\widetilde{\mathcal{L}}_{p-1}$, but the negligible term appears at a slightly different place.

The first k-invariant comes from composing the βconnecting mapβ

$\widetilde{\mathcal{L}}_{p-1}\rightarrow\underline{\mathcal{L}}_{p}$ |

with the second map (28), applied to the$\widehat{\xi}_{n}$-summand of (20) smashed over $H\underline{\mathbb{Z}}$ with$H\widetilde{\mathcal{L}}_{p-1}$. This gives a negligible k-invariant

$\overline{\overline{Q}}:H\widetilde{\mathcal{L}}_{p-1}\rightarrow H\underline{%\mathcal{L}}_{p}\{v_{n}\}$ |

and its fiber $F^{\prime}$ has a quotient of $H\underline{\mathbb{Z}}\cdot\underline{\xi}_{n}$. We obtaina k-invariant

(37) | $Q_{n}^{\prime}:F^{\prime}\rightarrow H\underline{\mathbb{Z}}\cdot v_{n}$ |

by applying an isomorphism on this quotient. The argument follows (32)-(36).

On the other hand, we can apply the second map (21) to the $\underline{\xi}_{n}^{p-2}\cdot\widehat{\xi}_{n}$copy in (20) to obtain the k-invariant

$Q_{n}^{\prime\prime}:F^{\prime\prime}\rightarrow H\widetilde{\mathcal{L}}_{p-1%}\cdot\Phi(v_{n})$ |

where $F^{\prime\prime}$ is the fiber of (37).

An illustration of the odd pattern is given in Figure 10 of the Appendix.

## 5. Comparison with Johnson-Wilson spectra

In this section, we shall investigate Johnson-Wilson spectra analogs of $BP\mathbb{R}$. Denote by$I_{n}$ the ideal $(p,v_{1},\dots,v_{n-1})\subset BP_{*}$.

###### Proposition 3.

The spectrum $(E(n)\wedge\dots\wedge E(n))^{\wedge}_{I_{n}}$ is an $A_{\infty}$-ring spectrum andthe space of its $A_{\infty}$-selfmaps is hom*otopically discrete. (The same statement holds forother βflavorsβ of $E(n)$, notably $E_{n}$.)

###### Proof.

We follow the method of Robinson [15] and Baker [2].We first recall that the ideal generated by $I_{n}$ by inclusions

$BP_{*}\rightarrow(BP\wedge\dots\wedge BP)_{*}$ |

via one of the factors does not depend on the factor. (See [14], Theorem A2.2.6.)

Next, let

$E(n,k)=\bigwedge_{k}E(n),$ |

$\Sigma(n,k)_{*}=(\bigwedge_{k}E(n))_{*}/I_{n}.$ |

One writes $\Sigma(n)_{*}=\Sigma(n,1)_{*}$.The method of Robinson [15], Baker [2] formula (3.10) reduces this task to provinga vanishing result for a certain $\mathcal{E}xt$-group

(38) | $\mathcal{E}xt^{s>0}_{\Sigma(n,2k)}(\Sigma(n,k),\Sigma(n,k))=0$ |

where $s$ denotes the cohom*ological degree.

To define the $\mathcal{E}xt$-groups concerned, we denote by $E$ the smash of $k$ copies of $E(n)$,the ring structore on $E_{*}E$ is given by the composition

(39) | $(E\wedge E)_{*}\otimes(E\wedge E)_{*}\rightarrow(E\wedge E\wedge E\wedge E)_{*%}\rightarrow(E\wedge E)_{*}$ |

where the first map is give by sending the first two coordinate to the first and last coordinate in the target,and the last two coordinates into the middle two coordinates (without changing order internally),and the second map is given by multiplication in the first two and the last two coordinates. The tensorproduct is over $\mathbb{Z}$.The left $E_{*}E$-module structure on $E_{*}$ is defined analogously, replacing thelast two coordinates $E\wedge E$ in the source and middle two coordinates in the target of (39)by $E$. It is customary to refer to this construction as βHochschild cohom*ologyβ([15]), even though this is not correct in full generality.

To prove (38), we recall from [15] that

(40) | $\displaystyle\Sigma(n)_{*}=\left(\bigotimes_{i\geq 1}\right)_{K(n)_{*}}\left(K%(n)_{*}\prod\frac{K(n)_{*}[t_{i}]}{(v_{n}t_{i}^{p^{n}-1}-v_{n}^{p^{i}})}\right)$ |

In our present setting, we have duplicate copies of the coordinates $t_{i}$, but the essential point is that the $E_{*}E/I_{n}=\Sigma(n,2k)$forms a direct limit of Γ©tale extensions of $K(n)_{*}$, and thus the given $\mathcal{E}xt^{>0}=0$.

$\square$

We claim that for all $n$, there is a $\mathbb{Z}/p$-action on $(\bigwedge_{p-1}E(n))^{\wedge}_{I_{n}}$ compatiblewith the $\mathbb{Z}/p$-action on the non-equivariant spectrum underlying the $BP\mathbb{R}$ we constructed above. To this end,we need to make a couple of remarks. First of all, the conjecture of Hill, Hopkins and Ravenel concerned theexistence of a $\mathbb{Z}/p$-equivariant spectrum, which we denote by $BP\mathbb{R}^{HHR}$, which would satisfy

(41) | $BP\mathbb{R}^{HHR}_{\{e\}}=\bigwedge_{p-1}BP.$ |

Our construction gives the pattern (9) above.We see [1] that this gives the right answer at the prime $p=3$, but is smaller than the coefficients of(41) for $p\geq 5$. One notes however that$BP\mathbb{R}^{HHR}$ can be obtained from $BP\mathbb{R}$ by taking a wedge with even suspensionsof copies of the additive norm $\mathbb{Z}/p_{+}\wedge BP$.

Now we claim that our constructiondetermines a $\mathbb{Z}/p$-action on $(\bigwedge_{p-1}BP)_{*}$. which sends the universal formal group lawto strictly isomorphic formal group laws (recall that a strict isomorphism between isomorphic formal group laws ona torsion-free ring is uniquely determined). In a gradedring $R\supseteq BP_{*}$, maps $BP_{*}\rightarrow R$ which carrythe universal formal group law to an isomorphic formal group law are characterized by elements $\overline{t}_{i}\in R$ ofthe same degree as $t_{i}$, which, when substituted for $t_{i}$ into $\eta_{R}(v_{n})$, give the image of $v_{n}$.

In fact, we do not need the whole construction of $BP\mathbb{R}$, its first k-invariant (22)determines the information we need. Non-equivariantly,this gives an operation

(42) | $H\mathbb{Z}\rightarrow\bigvee_{p-1}\Sigma^{2p^{n}-1}H\mathbb{Z}.$ |

We know that this operation is just the integral $Q_{n}$ landing in an $H\mathbb{Z}$ wedge summand of the right hand side.It is, further, invariant under $\mathbb{Z}/p$-action. This identifies the wedge summand which is supported by $v_{n}$.In hom*otopy groups, on $\mathcal{L}_{p-1}$, it is an element which reduces modulo $p$ to an elementof $L_{p-1}$ which is annihilated by $1-\gamma$ where $\gamma$ is the generator of $\mathbb{Z}/p$.

Since, however, we also know from [3] that $v_{n}$ is only determined modulo $I_{n}$, we see that theaction on $v_{n}$ given by the $\mathbb{Z}/p$-action on (42) necessarily sends the universal FGL toisomorphic FGLβs.

Along with (3), this then implies

###### Theorem 4.

There exists a strict action of $\mathbb{Z}/p$ (in the βnaiveβ sense) on

$(\bigwedge_{p-1}E(n))^{\wedge}_{I_{n}}$ |

(and its $I_{n}$-complete variants, replacing, for example, $E(n)$ by $E_{n}$)by morphisms of $A_{\infty}$-ring spectra, which on coefficientscoincides with the action on compositions of $(p-2)$ strict isomorphisms of FGLβs given by thefirst k-invariant (42).

$\square$

It is worth noting that while these considerations producemany equivalent actions on chains of $(p-2)$ strict isomorphismsof FGLβs, for $p>2$, there does not appear to be one canonical construction such as the $-i_{F}(x)$-seriesassociated with $BP\mathbb{R}$ for $p=2$.

It makes sense to conjecture that there exists an $A_{\infty}$-structure on $BP\mathbb{R}^{HHR}$ andan $A_{\infty}$-map from $BP\mathbb{R}^{HHR}$ into the $\mathbb{Z}/p$-equivariant Borel-complete spectrum defined byTheorem 4. At the moment, however, several ingredients are missing toward proving this,the first of which is the proof of the existence of an $A_{\infty}$-ring structure (or even a non-rigid ring structure)on our construction of $BP\mathbb{R}^{HHR}$.

It is important to note that there also exists a $\mathbb{Z}/p$-action on $E_{p-1}$ induced from thethe $\mathbb{Z}/p$-subgroup of the Morava stabilizer group. (For simplicity, let us assume $p\geq 5$.)The $\mathbb{Z}/p$-action on coefficients is discussed in Nave [13], Symonds [17].Essentially, from the representation-theoretical point of view,the non-negligible part is an exterior algebra on $\mathcal{L}_{p-1}$, tensored witha polynomial algebra on $Nv_{n}$. (This is in no contradiction with the hom*otopy-theoretical point of view,where we are dealing with the completion of a Laurent series ring.)

The generating $\mathcal{L}_{p-1}$ contains allof

(43) | $u,u_{n-1}u,\dots,u_{1}u$ |

where $u_{i}$ are the Lubin-Tate generators (i.e. are related to$v_{i}$ by change of normalization) and $u^{-1}$ is a root of $v_{n}$. In fact, it is proved in [13]that $u$ generates this $\mathcal{L}_{p-1}$-representation and the elements (43),in the order listed, are related by applying $1-\gamma$.

It was conjectured in [7] that there should be a $\mathbb{Z}/p$ equivariant map from

(44) | $BP\mathbb{R}^{HHR}\rightarrow(E_{p-1})_{\mathbb{Z}/p}$ |

which would restict to an isomorphism of the $\mathcal{L}_{p-1}$-representationsbetween the $\mathcal{L}_{p-1}$ containing $v_{1}$ and the $u^{-p^{p-1}+1}$-multipleof (43). Such a morphism of representations certainly exists, but making this assignmentimplies that the non-equivariant elements $v_{i}$ in our construction are sent to $0$ for $i>1$.This may seem absurd, since the element $u^{-p^{p-1}+1}$ is of type $v_{p-1}$, andis supposed to be inverted. The answer, however, is of course that there is no real contradiction,since when dealing with a chain of $p-2$ strict isomorphisms of the universal formal group law, therecould be many elements of type $v_{p-1}$, incluing some which contain a summand which is a unitmultiple of $t_{p-1}$. Thus, the element $u^{-p^{p-1}+1}$ could be an image of a generator ofa separate copy of $BP$ in $BP\mathbb{R}_{\{e\}}$. In fact, on the level of non-equivariant coefficients with$\mathbb{Z}/p$-action, such maps are easily shown to exist. The existence of the comparison map (44)on the level of $\mathbb{Z}/p$-equivariant spectra is at present still open.

## 6. Odd prime real orientations and $E_{\infty}$-constructions

One knows (see [11]) that $MU$ is an $E_{\infty}$-ring spectrum. On the other hand, we have thecomplex orientation

(45) | $\Sigma^{-2}\mathbb{C}P^{\infty}\rightarrow MU.$ |

Thus, (after appropriate discussion of cofibrancy), the universal property allows us to extend (45) toan $E_{\infty}$-map

(46) | $C_{\bullet}\Sigma^{-2}\mathbb{C}P^{\infty}\rightarrow MU$ |

where $C_{\bullet}E$ denotes the unital free $E_{\infty}$-ring spectrum on a spectrum $E$ with unit (meaninga morphism $S\rightarrow E$). We remark in fact that (46) is a retraction (i.e. that in the derived category ofspectra, the target splits off as a direct summand, as an ordinary ring spectrum). This isdue to the fact that $C_{\bullet}\Sigma^{-2}\mathbb{C}P^{\infty}$ is, by definition, complex-oriented, so the complexorientation gives a section of (46) in the category of ordinary commutative ring spectra.

A similar story is true, essentially without change, for Real-oriented $\mathbb{Z}/2$-equivariant spectra: Denoting by$\mathbb{S}^{1}$ the unit sphere in $\mathbb{C}$ with the $\mathbb{Z}/2$-action of complex conjugation, we have a Realorientation

(47) | $B\mathbb{S}^{1}\rightarrow M\mathbb{R},$ |

which gives rise to a $\mathbb{Z}/2$-equivariant $E_{\infty}$-map

(48) | $C_{\bullet}\Sigma^{-2}B\mathbb{S}^{1}\rightarrow M\mathbb{R}$ |

which, again, has a section in the category of ordinary $\mathbb{Z}/2$-equivariant commutative ring spectra (and hence,in particular, splits in the category of $\mathbb{Z}/2$-equivariant spectra).

This raises the question what happens for a prime $p>2$. Hahn, Senger and Wilson [6]defined a version of Real orientations of $\mathbb{Z}/p$-equivariant spectra for odd primes $p$ as follows. One denotes

$\mathbb{T}=B_{\mathbb{Z}/p}\mathcal{L}_{p-1}.$ |

The $\mathbb{Z}/p$-equivariant space $\mathbb{T}$ can, indeed, be identified with the subset of$(S^{1})^{\mathbb{Z}/p}$ consisting of all tuples $(z_{1},\dots,z_{p})$ where

$\prod_{j=1}^{p}z_{j}=1.$ |

Note that in particular,

(49) | $\mathbb{T}^{\mathbb{Z}/p}=\mathbb{Z}/p.$ |

Selecting once and for all any (non-equivariant path from $0$ to $1$ in (49) specifies a$\mathbb{Z}/p$-equivariant map

(50) | $\widetilde{\mathbb{Z}/p}\rightarrow\mathbb{T},$ |

which, in turn, by adjunction, gives a $\mathbb{Z}/p$-equivariant map

(51) | $\iota:\Sigma\widetilde{\mathbb{Z}/p}\rightarrow B\mathbb{T}.$ |

The right-hand side is, again, equivalent whether whether we take $B_{\mathbb{Z}/p}$ or the βnaiveβ Schubert cellconstruction. In fact, we have the following

###### Proposition 5.

Let $\mathbb{T}=\{(z_{1},\cdots,z_{p})|z_{i}\in S^{1},\prod z_{i}=1\}$ be the $\mathbb{Z}/p$-equivariantgroup so that $\mathbb{Z}/p$ acts by permuting the coordinates. Then the equivariantclassifying space $B_{\mathbb{Z}/p}\mathbb{T}$ is equivalent to the fiber of

(52) | $B((S^{1})^{\mathbb{Z}/p})\to BS^{1}.$ |

In (52), $B$ denotes the bar construction,$(S^{1})^{\mathbb{Z}/p}$ is the $\{(z_{1},\cdots,z_{p})|z_{i}\in S^{1}\}$ and the map is induced by multiplying the coordinates.

Comment: The bar construction is equivariant and also preserves the βmultiplicative normβ on space level,so the source of (52) is a product of $p$ copies of $\mathbb{C}P^{\infty}$ with action by permutation of coordinates,while the target of (52) is a fixed copy of $\mathbb{C}P^{\infty}$.

###### Proof.

Notice that on fixed points, (52) is $p:BS^{1}\to BS^{1}$, (meaning the bar construction applied to the $p$βth power map on $S^{1}$),so it suffices to show that $(B_{\mathbb{Z}/p}\mathbb{T})^{\mathbb{Z}/p}\simeq B\mathbb{Z}/p$, as the underlyinglevel is easy. By [10, Theorem 10],

(53) | $(B_{\mathbb{Z}/p}\mathbb{T})^{\mathbb{Z}/p}\simeq\coprod_{[\Lambda]}BW_{%\mathbb{T}\rtimes\mathbb{Z}/p}(\Lambda),$ |

where the index is over subgroups $\Lambda\subset\mathbb{T}\rtimes\mathbb{Z}/p$ isomorphic to $\mathbb{Z}/p$ andintersecting $\mathbb{T}$ trivially, up to $\mathbb{T}\text{-conjugation}$.Write $\mathbb{Z}/p=\langle\gamma\rangle$ and take $x=((z_{1},\cdots,z_{p}),\gamma)\in\mathbb{T}\rtimes\mathbb{Z}/p$. Then

$x^{p}=((\sqcap z_{i},\cdots,\sqcap z_{i}),\gamma^{p})=e\text{ and }x^{-1}=((z_%{2}^{-1},\cdots,z_{p}^{-1},z_{1}^{-1}),\gamma^{-1}).$ |

This means any element $x$ of $\mathbb{T}\rtimes\mathbb{Z}/p$ not contained in $\mathbb{T}$ generates a $\Lambda$.However, these are all conjugate. Let $y=((y_{1},\cdots,y_{p}),\gamma)$ to bedetermined. Setting

$yxy^{-1}=((y_{1}x_{p}y_{p}^{-1},y_{2}x_{1}y_{1}^{-1},\cdots,y_{p}x_{p-1}y_{p-1%}^{-1}),\gamma)=((1,1,\cdots,1),\gamma),$ |

we obtain

$\displaystyle y_{p}/y_{1}=x_{p}$ | ||

$\displaystyle y_{1}/y_{2}=x_{1}$ | ||

$\displaystyle\cdots$ | ||

$\displaystyle y_{p-1}/y_{p}=x_{p-1}$ |

It is not hard to see that this allows a solution with $\prod y_{i}=1$.Now, using $\Lambda=\langle((1,1,\cdots,1),\gamma)\rangle$ without loss of generality,one obtains $(B_{\mathbb{Z}/p}\mathbb{T})^{\mathbb{Z}/p}\simeq B\mathbb{Z}/p$ from (53). $\square$

Note that in the proof, the only thing we used about $S^{1}$ is that it is anabelian compact Lie group. However, the same argument often extends to more general groups.For example, we can show that the fiber of

$B(\mathbb{Z}^{\mathbb{Z}/p})\to B\mathbb{Z}$ |

is$B_{\mathbb{Z}/p}\mathcal{L}_{p-1},$recalling $\mathcal{L}_{p-1}=\{(a_{1},\cdots,a_{p})|z_{i}\in\mathbb{Z},\sum a_{i}=0\}$. In other words, $\mathbb{T}\simeq B_{\mathbb{Z}/p}\mathcal{L}_{p-1}$.Although the group $\mathbb{Z}$ is non-compact, the formalism works in this case, since the family consits of finite subgroups.Instead of [10, Theorem 10], we use the fact that $H^{1}(\mathbb{Z}/p,\mathcal{L}_{p-1})=\mathbb{Z}/p$, bythe long exact se quence in cohom*ology induced by the short exact sequence

$0\rightarrow\mathbb{Z}\rightarrow\mathcal{L}_{p}\rightarrow\mathcal{L}_{p-1}%\rightarrow 0.$ |

Hahn Senger, and Wilson [6] define a $\mathbb{Z}/p$-equivariant commutative ring spectrum $E$ to beReal-oriented if the following diagram can be completed:

(54) |

where the horizontal map is the smash of $\Sigma\widetilde{\mathbb{Z}/p}$ with the unit. If this happens, one canshow that the Schubert-cell spectral sequence converging to $E^{*}B\mathbb{T}$ (see Proposition 5)collapses.

In effect, we claim that for a Real-oriented $\mathbb{Z}/p$-equivariant spectrum $E$, modulo βnegligible parts,βi.e. copies of the additive norm from $\{e\}$ to $\mathbb{Z}/p$ on $E_{\{e\}}$, we have

(55) | $E\wedge B\mathbb{T}\sim\bigwedge_{E}(E\vee E\wedge\Sigma\widetilde{\mathbb{Z}/%p})\wedge_{E}E[Nx]$ |

(we mean this up to hom*otopy, not in any coherent sense). In fact, the $E\wedge\Sigma\widetilde{\mathbb{Z}/p}$occurs by the definition of $\mathbb{Z}/p$-equivariant Real orientation, while the class $Nx$ is simply the multiplicative norm(on the level of spaces) of the complex orientation of $E_{\{e\}}$ (whose existence also follows fromthe definition of Real orientation by forgetting the $\mathbb{Z}/p$-action), restricted by the inclusion map

$B\mathbb{T}\rightarrow B((S^{1})^{\mathbb{Z}/p}.$ |

By the observation, the ring $E^{*}[Nx]$ splis off as a canonical summand of $E^{*}B\mathbb{T}$. Multiplication on $B\mathbb{T}$then givesrise to a $p$-valued formal group law in the sense of Buchstaber [4, 5], see Definition 1.2 of[5]. To recapitulate, Buchstaber defines a $p$-valued formal group law as a polynomial of theform

(56) | $\Theta(x,y)=Z^{p}-\theta_{1}(x,y)Z^{p-1}+\dots-\theta_{p}(x,y)$ |

where $\theta_{i}(x,y)$ are power series which satisfies the conditions

$\displaystyle\Theta(x,0)=Z^{n}+\sum_{i=1}^{n}(-1)^{i}{n\choose i}x^{i}Z^{n-i},$ |

(57) | $\Theta(\Theta(x,y),z)=\Theta(x,\Theta(y,z)),$ |

$\Theta(x,y)=\Theta(y,x).$ |

The associativity condition (57) requires some explanation.

If $F(x,y)$ is an ordinary formal group law, then a $p$-valued formal group law can be obtained as

$\Theta(x,y)=\prod_{j=1}^{n}(Z-F_{j}(x,y))$ |

where

$F_{j}(x,y)=\exp_{F}((\log_{F}x)^{1/p}+\zeta_{p}^{j}(\log_{F}y)^{1/p})^{p}.$ |

In the associativity condition, we can similarly write terms in three variables

(58) | $F_{jk}(x,y,z)=\exp_{F}((\log_{F}x)^{1/p}+\zeta_{p}^{j}(\log_{F}y)^{1/p}+\zeta_%{p}^{k}(\log_{F}z)^{1/p})^{p}$ |

and the multivalued formal sum in three variables should be

(59) | $\Theta(x,y,z)=\prod_{j,k=1}^{n}(Z-F_{jk}(x,y,z)).$ |

This can be processed in two different ways into expressions involving only the series $\theta_{j}$:

We can write (58) as

$\begin{array}[]{l}F_{jk}(x,y,z)=\\\exp_{F}((\log_{F}x)^{1/p}+\zeta_{p}^{j}(((\log_{F}y)^{1/p}+\zeta_{p}^{k-j}(%\log_{F}z)^{1/p})^{p})^{1/p})^{p}\end{array}$ |

and thus express (59) in terms of

$\theta_{i}(x,F_{j}(y,z)).$ |

The resulting expression, however, is symmetrical in the $F_{j}(y,z)$, which allows them to be reduced to$\theta_{s}(y,z)$. Permuting the variables $x,y,z$ cyclically and doing the same thing gives anotherexpression. The equality of both expressions is what one means by (57).

For $p=2$, Real-oriented spectra produce an ordinary formal group law due to the fact that the orientation classis a polynomial generator. As we saw in (55) (and as is, in some sense, thetheme of this paper), however, for $p>2$ the orientation classis only an βexterior generatorβ from the point of view of representation theory, which is why we have torestrict to the polynomial ring on the (space-level) multiplicative norm of the non-equivariant orientationclass, which leads only to a $p$-valued formal group law. Note that the collapse (55) is necessaryto assure that the ring $E^{*}[Nx]$ indeed survives as a canonical summand of the cell spectralsequence given by Proposition 5.

For our present purposes, we recall from [11] that the$\mathbb{Z}/p$-equivariant Spanier-Whitehead dual of $\widetilde{\mathbb{Z}/p}$ is given by

(60) | $D\widetilde{\mathbb{Z}/p}=\Sigma^{-\beta}\widetilde{\mathbb{Z}/p}.$ |

Thus,

$\Sigma^{-1-\beta}\widetilde{\mathbb{Z}/p}\wedge B\mathbb{T}$ |

becomes a unital spectrum via $\iota$ and the spectrum

(61) | $C_{\bullet}\Sigma^{-1-\beta}\widetilde{\mathbb{Z}/p}\wedge B\mathbb{T}$ |

becomes Real-oriented in the sense of [6]. It seems therefore reasonable to conjecture thatafter localizing at $p$, $BP\mathbb{R}$ will split off (61).

One also notes that since $(E_{p-1})_{\mathbb{Z}/p}$ is both $E_{\infty}$ and Real-oriented (as proved in[6]), we do know that there is a $\mathbb{Z}/p$-equivariant $E_{\infty}$-map

$C_{\bullet}\Sigma^{-1-\beta}\widetilde{\mathbb{Z}/p}\wedge B\mathbb{T}%\rightarrow E_{p-1}.$ |

## 7. Appendix: Graphical illustration of the construction of $BP\mathbb{R}$ at $p>2$

The purpose of this Appendix is to provide a graphical illustration of the construction given in Section 4. The even (resp. odd)pattern of the tower is depicted in Figure 9 (resp. Figure 10). Let us briefly recall the classical Brown-Peterson $BP$-tower [3]. Their tower isdescribed by taking the hom*ology $H\mathbb{Z}/p_{*}H\mathbb{Z}$ and killing those elements which are not supposed to be in $H\mathbb{Z}/p_{*}BP$. These are, of course,all the copies of $H\mathbb{Z}/p_{*}=\mathbb{Z}/p$ indexed by a generator $\xi_{R}\tau_{E}$ in Milnorβs notation [12] where $E=(e_{1},e_{2},\dots)\neq 0$, $e_{n}\in\{0,1\}$. In the first stage of the tower, we attach copies of $H\mathbb{Z}$ along the Milnor primitives $Q_{n}$, (the corresponding generators being thus labelledby $v_{n}$), which sends $\tau_{n}\mapsto 1$, thereby killingall these elements, creating, however, βerror termsβ due to the relations $\tau_{n}^{2}=0$. These are corrected in subsequent stages ofthe tower, thus making the generators $v_{n}$ polynomial.

This construction has anexact analogue in constructing $BP\mathbb{R}$ for $p=2$ [8]; while the relation for $\tau_{n}^{2}$ is more complicated, it does not affect themultiplication rule of the $v_{n}$βs (this also occurs in the classical $BP$ tower for $p=2$).

In the case of the $BP\mathbb{R}$ tower for $p>2$