# The slice spectral sequence for the analog of real -theory

###### Abstract.

We describe the slice spectral sequence of a 32-periodic -spectrum related to the norm

of the real cobordism spectrum . We will give it as a SS of Mackey functors converging to the graded Mackey functor , complete with differentials and exotic extensions in the Mackey functor structure.

The slice spectral sequence for the 8-periodic real -theory spectrum was first analyzed by Dugger. The analog of is 256-periodic and detects the Kervaire invariant classes . A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that does not exist for .

###### Key words and phrases:

equivariant stable homotopy theory, Kervaire invariant Mackey functor, slice spectral sequence###### 2010 Mathematics Subject Classification:

55Q10 (primary), and 55Q91, 55P42, 55R45, 55T99 (secondary)## 1. Introduction

In [HHR] we derived the main theorem about the Kervaire invariant elements from some properties of a -\eqvr spectrum we called constructed as follows. We started with the -spectrum , meaning the usual complex cobordism spectrum equipped with a action defined in terms of complex conjugation.

Then we defined a functor , the norm of [HHR, §2.2.3] which we abbreviate here by , from the category of -spectra to that of -spectra. Roughly speaking, given a -spectrum , is underlain by the fourfold smash power where a generator of acts by cyclically permuting the four factors, each of which is invariant under the given action of the subgroup . In a similar way one can define a functor from -spectra to -spectra for any finite groups .

A -spectrum such as , which is a commutative ring spectrum, has \eqvr homotopy groups indexed by , the orthogonal \rep ring for the group . One element of the latter is , the regular \rep. In [HHR, §9] we defined a certain element and then formed the associated mapping telescope, which we denoted by . The symbol was chosen to suggest a connection with the octonions, but there really is none apart from the fact that the octonions are 8-dimensional like .

is also a -\eqvr commutative ring spectrum. We then proved that it is \eqvrly equivalent to ; we call this result the Periodicity Theorem. Then our spectrum is , the fixed point spectrum of .

It is possible to do this with replaced by for any . The dimension of the periodicity is then . For example it is 32 for the group and for . We chose the group because it is the smallest that suits our purposes, namely it is the smallest one yielding a fixed point spectrum that detects the Kervaire invariant elements .

We know almost nothing about , only that it is periodic with periodic 256, that (the Gap Theorem of [HHR, §8]), and that when exists its image in is nontrivial (the Detection Theorem of [HHR, §11]).

We also know, although we did not say so in [HHR], that more explicit computations would be much easier if we cut down to size in the following way. Its underlying homotopy, meaning that of the spectrum , is known classically to be a polynomial algbera over the integers with four generators (cyclically permuted up to sign by the group action) in every positive even dimension. This can be proved with methods described by Adams in [Ad:SHGH]. For the cyclic group one has generators in each positive even degree. Specific generators for and are defined in [HHR, §5.4.2].

There is a way to kill all the generators above dimension 2k that was described in [HHR, §2.4]. Roughly speaking, let be a wedge of suspensions of the sphere spectrum, one for each monomial in the generators one wants to kill. One can define a multiplication and group action on corresponding to the ones in . Then one has a map whose restriction to each summand represents the corresponding monomial, and a map (where the target is the sphere spectrum, not the space ) sending each positive dimensional summand to a point. This leads to two maps

whose coequalizer we denote by . Its homotopy is the quotient of obtained by killing the polynomial generators above diimension . The construction is \eqvr, meaning that underlies a -spectrum.

In [HHR, §7] we showed that for the spectrum we get is the integer \SESM spectrum ; we called this result the Reduction Theorem. In the non\eqvr case this is obvious. We are in effect attaching cells to to kill all of its homotopy groups in positive dimensions, which amounts to constructing the 0th Postnikov section. In the \eqvr case the proof is more delicate.

Now consider the case , meaning that we are killing the polynomial generators above dimension 2. Classically we know that doing this to (without the -action) produces the connective complex -theory spectrum, some times denoted by , or (2-locally) . Inverting the Bott element via a mapping telescope gives us itself, which is of course 2-periodic. In the -\eqvr case one gets the “real -theory” spectrum first studied by Atiyah in [Atiyah:KR]. It turns out to be 8-periodic and its fixed point spectrum is , which is also referred to in other contexts as real -theory.

The spectrum we get by killing the generators above dimension 2 in the-spectrum will be denoted analogously by . We can invert the image of by forming a mapping telescope, which we will denote by . More generally we denote by the spectrum obtained from by killing all generators above dimension 2. In particular . Then we denote the mapping telescope (after defining a suitable ) by and its fixed point set by .

For , also has a Periodicity, Gap and Detection Theorem, so it could be used to prove the Kervaire invariant theorem.

Thus is a substitute for with much smaller and therefore more tractable homotopy groups. A detailed study of them might shed some light on the fate of in the 126-stem, the one hypothetical Kervaire invariant element whose status is still open. If we could show that , that would mean that does not exist.

The computation of the \eqvr homotopy at this time is daunting. The purpose of this paper is to do a similar computation for the group as a warmup exercise. In the process of describing it we will develope some techniques that are likley to be needed in the case. We start with , kill its polynomial generators (of which there are two in every positive even dimension) above dimension 2 as described previously, and then invert a certain element in . We denote the resulting spectrum by , see LABEL:def-kH below. This spectrum is known to be 32-periodic. In an earlier draft of this paper it was denoted by .

The computational tool for finding these homotopy groups is the slice spectral sequence introduced in [HHR, §4]. Indeed we do not know of any other way to do it. For it was first analyzed by Dugger [Dugger] and his work is described below in §LABEL:sec-Dugger. In this paper we will study the slice spectral sequence of Mackey functors associated with . We will rely extensively on the results, methods and terminology of [HHR].

We warn the reader that the computation for is more intricate than the one for . For example, the slice spectral sequence for , which is shown in Figure LABEL:fig-KR, involves five different Mackey functors for the group . We abbreviate them with certain symbols indicated in Table LABEL:tab-C2Mackey. The one for , partly shown in Figure LABEL:fig-KH, involves over twenty Mackey functors for the group , with symbols indicated in Table LABEL:tab-C4Mackey.

Part of this SS is also illustrated in an unpublished poster produced in late 2008 and shown in Figure 1. It shows the SS converging to the homotopy of the fixed point spectrum . The corresponding SS of Mackey functors converges to the graded Mackey functor .

In both illustrations some patterns of s and families of elements in low filtration are excluded to avoid clutter. In the poster, representative examples of these are shown in the second and fourth quadrants, the SS itself being concentrated in the first and third quadrants. In this paper those patterns are spelled out in §LABEL:sec-C2diffs and §LABEL:sec-C4diffs.

We now outline the rest of the paper. Briefly, the next five sections introduce various tools we need. Our objects of study, the spectra and , are formally introduced in §LABEL:sec-kH. Dugger’s computation for is recalled in §LABEL:sec-Dugger. The final six sections describe the computation for and .

In more detail, §2 collects some notions from \eqvr stable homotopy theory with an emphasis on Mackey functors. Definition LABEL:def-graded introduces new notation that we will ocasionally need.

§LABEL:sec-HZZ concerns the \eqvr analog of the homology of a point namely, the -graded homotoy of the integer \SESM spectrum . In particular Lemma LABEL:lem-aeu describes some relations among certain elements in it including the “gold relation” between and .

§LABEL:sec-gendiffs describes some general properties of spectral sequences of Mackey functors. These include Theorem LABEL:thm-exotic about the relation between differential and exotic extensions in the Mackey functor structure and Theorem LABEL:thm-normdiff on the norm of a differential.

§LABEL:sec-C4 lists some concise symbols for various specific Mackey functors for the groups and that we will need. Such functors can be spelled out explicitly by means of Lewis diagrams (LABEL:eq-Lewis), which we usually abbreviate by symbols shown in Tables LABEL:tab-C2Mackey and LABEL:tab-C4Mackey.

In §LABEL:sec-chain we study some chain complexes of Mackey functors that arise as cellular chain complexes for -CW complexes of the form .

In §LABEL:sec-kH we formally define (in LABEL:def-kH) the -spectra of interest in this paper, and .

In §LABEL:sec-Dugger we describe the slice SS for an easier case, the -spectrum for real -theory, . This is due to Dugger [Dugger] and serves as a warmup exercise for us. It turns out that everything in the SS is formally determined by the structure of its -term and Bott periodicity.

In §LABEL:sec-more we introduce various elements in the homotopy groups of and . They are collected in Table LABEL:tab-pi*, which spans several pages. In §LABEL:sec-slices we determine the -term of the slice SS for and .

In §LABEL:sec-diffs we use the Slice Differentials Theorem of [HHR] to determine some differentials in our SS.

In §LABEL:sec-C2diffs we examine the -spectrum as a -spectrum. This leads to a calculation only slightly more complicated than Dugger’s. It gives a way to remove a lot of clutter from the calculation.

In §LABEL:sec-C4diffs we determine the -term of our SS. It is far smaller than and the results of §LABEL:sec-C2diffs enable us to ignore most of it. What is left is small enough to be shown legibly in the SS charts of Figures LABEL:fig-E4 and LABEL:fig-KH. They illustrate integrally graded (as opposed to -graded) spectral sequences of Mackey functors, which are discussed in §LABEL:sec-C4. In order to read these charts one needs to refer to Table LABEL:tab-C4Mackey which defines the “heiroglyphic” symbols we use for the specific Mackey functors that we neede.

We finish the calculation in §LABEL:sec-higher by dealing with the remaining differentials and exotic Mackey functor extensions. It turns out that they are all formal consequences of differentials of the previous section along with the results of §LABEL:sec-gendiffs.

The result is a complete description of the integrally graded portion of . It is best seen in the SS charts of Figures LABEL:fig-E4 and LABEL:fig-KH. Unfortunately we do not have a clean description, much less an effective way to display the full -graded homotopy groups.

For , the two irreducible orthogonal \reps are the trivial one of degree 1, denoted by the symbol 1, and the sign \rep denoted by . Thus is additvley a free abelain group of rank 2, and the SS of interest is trigraded. In the -graded homotopy of , a certain element of degree (the degree of the regular \rep ) is invertible. This means that each component of is canonically isomorphic to a Mackey functor indexed by an ordinary integer. See Theorem LABEL:thm-ROG-graded for a more precise statement. Thus the full (trigraded)-graded slice SS is determined by bigraded one shown in Figure LABEL:fig-KR.

For , the \rep ring is additively a free abelian group of rank 3, so it leads to a quadrigraded SS. The three irreducible \reps are the trivial and sign \reps 1 and (each having degree one) and a degree two \rep given by a rotation of the plane of order 4. The regular \rep is isomorphic to . As in the case of , there is an invertible element (see Table LABEL:tab-pi*) in of degree . This means we can reduce the quadigraded slice SS to a trigraded one, but finding a full description of it is a problem for the future.

## 2. Recollections about \eqvr stable homotopy theory

We first discuss some structure on the \eqvr homotopy groups of a-spectrum . We will assume throughout that is a finite cyclic -group. This means that its subgroups are well ordered by inclusion and each is uniquely determined by its order. The results of this section hold for any prime , but the rest of the paper concerns only the case . We will define several maps indexed by pairs of subgroups of . We will often replace these indices by the orders of the subgroups, sometimes denoting by .

The homotopy groups can be defined in terms of finite -sets Let

be the set of homotopy classes of \eqvr maps from , the suspension spectrum of the union of with a disjoint base point, to the spectrum . We will often omit from the notation when it is clear from the context. For an orthogonal \rep of , we define

As an -graded contravariant abelian group valued functor of , this converts disjoint unions to direct sums. This means it is determined by its values on the sets for subgroups .

Since is abelian, is normal and is a -module.

Given subgroups , one has pinch and fold maps between the -spectra and . This leads to a diagram {numequation}