AUTOMATED DIFFERENTIAL COMPUTATION IN THE ADAMS SPECTRAL SEQUENCE JOEY BEAUVAIS-FEISTHAUER

2025-05-02 0 0 1.64MB 16 页 10玖币

侵权投诉

AUTOMATED DIFFERENTIAL COMPUTATION IN THE

ADAMS SPECTRAL SEQUENCE

JOEY BEAUVAIS-FEISTHAUER

Abstract. We describe an algorithm for the automated deduction of

many d2diﬀerentials in the Adams spectral sequence. We discuss our

implementation and the results of the computation.

1. Introduction

A central problem in homotopy theory is the computation of stable homo-

topy groups. One of our most eﬃcient tools is the Adams spectral sequence,

which uses information about certain Ext groups to compute stable homo-

topy groups. In our case of interest, its E2page is a diﬀerential graded

algebra isomorphic to the cohomology of the Steenrod algebra, and it con-

verges to the stable homotopy groups of spheres.

The determination of the additive and multiplicative structure of this

diﬀerential graded algebra is a hard problem, but manageable in practice.

Being deﬁned entirely in terms of homological algebra, its calculation can

be automated using computers. For instance, Bruner and Rognes [BR21a]

have computed it up to total degree 184, which is the data that we use for

our results in this paper. They have since then extended their computation

to total degree 200 [BR22], and the author is currently participating in a

similar computation up to stem 256 [BFCC].

While we have a good grasp of the algebraic structure of its E2page, the

main challenge when working with the Adams spectral sequence in general

is the computation of its diﬀerentials. In particular, the ﬁrst obstacle is

computing the diﬀerentials on the E2page. A wide variety of ad hoc tech-

niques have been used to compute some of their values [IWX20]. We have

developed a technique to systematize this procedure.

We introduce the new results that our algorithm found in Section 2. We

then give a brief exposition of the theory behind our algorithm in Sec-

tions 3 and 4, followed by a breakdown of the algorithm in Section 5.

2020 Mathematics Subject Classiﬁcation. Primary 55T15; Secondary 55Q45, 55-04.

Key words and phrases. Adams spectral sequence, stable homotopy groups of spheres,

software for spectral sequence computation.

The author was partially supported by National Science Foundation grant DMS-

1904241.

arXiv:2210.15169v1 [math.AT] 27 Oct 2022

2 JOEY BEAUVAIS-FEISTHAUER

Related work. We should mention that some of these results were already

present in unpublished work of Bruner from around 2005 [Bru05]. However,

we believe that our method is more conceptual and generalizes better to

other spectral sequences. Furthermore, we apply this algorithm to a larger

range, which necessarily gives us stronger results.

Recently, Dexter Chua has developed a tool to compute d2diﬀerentials

algorithmically, directly from homological algebra [Chu21]. In practice we

have found that, using Chua’s algorithm, computing the diﬀerentials on the

E2page is no harder than computing its product structure. This arguably

means that, similarly to the computation of the additive and multiplica-

tive structure of the E2page, the computation of its diﬀerentials can be

considered “solved”.

However, we believe our method of propagation remains relevant. Firstly,

the algorithm allows us to not only determine many d2diﬀerentials but

explore their interrelationship. Noticing this qualitative behavior is what

led us to formulate Conjecture 2.3. Secondly, although we only discuss

the results that our procedure gave while examining the Adams spectral

sequence, and more speciﬁcally its E2page, this same procedure is readily

applicable to any other diﬀerential graded algebra. It has already been used

to study the d3diﬀerentials of the Adams spectral sequence, which have

been found to behave very diﬀerently from the d2diﬀerentials. The code

can also be easily adapted to work with any spectral sequence equipped with

a product structure, such as those of May type.

Conventions.

(1) We index the Adams spectral sequence using bidegrees (n, s) where

nis the stem and sis the homological degree, i.e. the usual Carte-

sian coordinates on an Adams chart. We use capital latin letters

A, B, . . . to refer to individual bidegrees, e.g. A= (20,4). Addition

of bidegrees is componentwise addition. Also, A0denotes the bide-

gree of the output of the diﬀerential on a given bidegree A. In other

words, if A= (n, s), then A0= (n−1, s + 2). As an example, if we

are in the E2page of the Adams spectral sequence, and A= (20,4),

then A0= (20 −1,4 + 2) = (19,6).

Notice that, for any bidegrees Aand B,

A0+B=A+B0= (A+B)0.

(2) We denote the diﬀerential bigraded algebra by E, its homogeneous

component in bidegree Aby EA, and the diﬀerential on EA(as

a linear map) by dA. The symbol DAalways denotes an aﬃne

subspace of Hom(EA, EA0) (see Section 3), and we assume that dA

belongs to DAunless otherwise noted.

AUTOMATED ADAMS DIFFERENTIAL COMPUTATION 3

2. Results

Here are some results that our algorithm has yielded. The data we used

for the computations was generated in 2020 by Bruner and Rognes, using

Bruner’s software [Bru18]. It is a complete description of the cohomology

of the Steenrod algebra through total degree 184, along with its primary

multiplicative structure. As mentioned in the introduction, Bruner and

Rognes have since extended their computation through total degree 200. We

have preliminary results on the application of our algorithm to the larger

computation of Bruner and Rognes. For expository precision, we restrict

our discussion only to the smaller computation of Bruner and Rognes.

It is well-known that d2(h4) = h0h2

3. Adams gives the historically ﬁrst

proof [Ada60], but this diﬀerential can also be computed by other argu-

ments. For example, it can be derived by recognizing h4= Sq0(h3) and

using Bruner’s theorem on the interaction between diﬀerentials and alge-

braic squaring operations [BMMS86]. The more concretely-minded reader

might prefer to read this diﬀerential oﬀ a machine computation, such as the

one in [Chu21].

Similarly, it is known that d2(∆2d2

0) = d0jm. This diﬀerential can again

be obtained by several arguments. For example, it follows by comparison

with the Adams spectral sequence for tmf [BR21b], but it can also be read

oﬀ of a machine computation.

Given only the knowledge of d2(h4) = h0h2

3, our algorithm is able to infer

the value of the diﬀerentials on a large portion of the E2page. Given only

the values of d2(h4) and d2(∆2d2

0), we were able to compute over 95% of the

diﬀerentials up to stem 140. In particular, we found the ﬁrst known proofs

for the values of the diﬀerentials on the following elements:

stem ﬁltration x d2(x)

104 18 ∆4h1h3MP ∆2h2

1+M∆h2

2d2

107 13 ∆D1n0

107 18 M∆2d0M∆h2

2d0e0

108 14 ∆2∆1h1h3MP ∆1h2

108 15 M∆2h40

109 12 x109,12 0

109 14 ∆2A0∆g2g2

109 14 ∆2(A+A0)M∆2h0h4

110 18 ∆4h2

3h3

0x109,17

110 18 M∆2e0M∆h2

2e2

See [IWX20] for an explanation of the notation.

Theorem 2.1.

(1) The product structure of the Adams E2page implies that all elements

represented by a black dot in Figure 1 do not support a d2diﬀerential.

4 JOEY BEAUVAIS-FEISTHAUER

(2) The product structure of the Adams E2page, together with the fact

that d2(h4) = h0h2

3, implies the existence of all cyan diﬀerentials in

Figure 2.

(3) The product structure of the Adams E2page, together with the facts

that d2(h4) = h0h2

3and d2(∆2d2

0) = d0jm, implies:

(a) that all elements represented by a black dot in Figure 2 do not

support a d2diﬀerential.

(b) the existence of all cyan diﬀerentials and all magenta diﬀeren-

tials.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

AUTOMATEDDIFFERENTIALCOMPUTATIONINTHEADAMSSPECTRALSEQUENCEJOEYBEAUVAIS-FEISTHAUERAbstract.Wedescribeanalgorithmfortheautomateddeductionofmanyd2dierentialsintheAdamsspectralsequence.Wediscussourimplementationandtheresultsofthecomputation.1.IntroductionAcentralprobleminhomotopytheoryisthecomputationo...

展开>> 收起<<

AUTOMATED DIFFERENTIAL COMPUTATION IN THE ADAMS SPECTRAL SEQUENCE JOEY BEAUVAIS-FEISTHAUER.pdf

共16页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

AUTOMATED DIFFERENTIAL COMPUTATION IN THE ADAMS SPECTRAL SEQUENCE JOEY BEAUVAIS-FEISTHAUER

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: