Adams-Novikov charts and data at the prime 3

This repository contains miscellaneous documentation, charts, and data related to Guozhen Wang's MinimalResolution program, as modified by me to run at p = 3. The program computes the algebraic Novikov spectral sequence converging to the E₂ page of the Adams-Novikov spectral sequence for the sphere, including multiplicative information.

Key for reading the chart anss_E2_158.pdf: This depicts the E₂ page of the Adams-Novikov spectral sequence for the sphere at p=3. Blue dots denote β₁-divisible classes, brown lines denote α₁ multiplication, dashed gray lines denote <α₁, α₁, -> brackets, and concentric circles indicate 3-divisibility.

Higher differentials are complete through the 108 stem, and are inferred from the homotopy groups computed in the green book (Table A3.4). When a differential has a source or target of dimension > 1, in general we do not claim to specify which linear combination is involved. (Similarly, we do not attempt to indicate whether a differential hits a generator, or 2 times the generator.)

The differentials beyond the 108 stem are just the easy differentials propagated via β₁-multiplications. Propagating other differentials, as well as determining new families of differentials, is work in progress.

Data files: The files in data/ came from running the code here as follows:

./mr_st 185 90
./BPtab 185
./mr_BP 185 40

This computes the ANSS E₂ page through internal degree 185.

Interpreting the data files: Information about multiplication by p and α₁ can be read fairly straightforwardly from the files 185_BPAANSS_a0.txt and 185_BPAANSS_h0.txt, respectively. Names like v0^2[2-5] denote specific generators in a minimal resolution; they are illustrated in the chart in this repository.

It is less obvious how to interpret the information about beta multiplication; the purpose of the rest of this readme is to document how to do that, and the process is formalized in the python code in beta.py. If you just want to look up specific multiplications, read the next section for a brief explanation of how to run that code.

If you have issues running the code, or have other questions/ comments related to this, I'd love to hear from you-- my email address is my github username at northwestern.edu.

How to use beta.py to view beta (and other) multiplications

I'll assume you've cloned this repository and started a python interpreter in the top level of this repository.

>>> from beta import *
>>> data = ANSSData() # parses files in data/ (may be slow)
>>> data.in_deg(3,1) # [1-0] is the only generator in stem 3, ANSS filtration 1
['[1-0]']
>>> data.beta1("[1-0]") # multiply [1-0] by beta1
2[3-0]
>>> data.alpha1("[1-0]")
0
>>> data.alpha1(data.beta2("[2-2]")) # multiplications can be composed
2[5-4]

Available multiplications are three, alpha1, beta1, beta2, beta33, beta4, beta5, and beta63.

To parse files from your own run of Guozhen's program, use ANSSData(file_prefix) where file_prefix is a string such that e.g. file_prefix + "_BPBocSS_table.txt" is a valid (relative) path to your output files.

Notes on specific files

For the rest of this readme I will be talking about the files in the data/ directory.

• 185_BPAANSS_table.txt describes the algebraic Novikov spectral sequence (algNSS) converging to the E_2 page of the Adams-Novikov spectral sequence (ANSS). Lines containing a single element name indicate that the element is a permanent cycle. Lines containing y <- x | dr indicate an algebraic Novikov differential d_r(x) = y. The degree pair deg=(a,b) means stem = a and Adams filtration = b. Warning: the grading convention in this file has been changed from Guozhen's original! At odd primes, the E_2 page of the algebraic Novikov spectral sequence coincides with the E_2 page of the Adams spectral sequence, so by taking all elements in this table, including those involved in differentials, one obtains a list of elements in the Adams E_2 page.

• 185_BPBocSS_table.txt describes the Bockstein spectral sequence from the ANSS E_2 page for the mod 3 Moore space, to the ANSS E_2 page for the sphere. These are different generators from the algNSS table. The single degree listed is Adams-Novikov internal degree (i.e., stem + ANSS filtration).

• 185_BPB2A_table.txt provides a translation between these two sets of generators: a line x -> y indicates that x is the Bockstein name for the image of the algebraic Novikov element y along the natural map q: E_2(S) --> E_2(S/p). In other words, the inverse of this mapping specifies q.

Using the data files to determine beta multiplications

The files 185_BPBocSS_theta*.txt describe multiplication on E_2(S/3) by an element a_i/j whose image under the boundary map delta: E_2(S/3) --> E_2(S) is β_i/j. This repository contains information about six different multiplication types:

• theta2 --> β₁
• theta3 --> β₂
• theta4 --> β_3/3
• theta5 --> β₄
• theta6 --> β₅
• theta7 --> β_6/3

It should be easy to change the mult_theta function in BP_init.cpp and the thetas function in BP.cpp to generate other beta's.

Thus the full procedure for determining β_i/j * x for x in E_2(S) is:

1. Apply q: E_2(S) --> E_2(S/3) to x by applying the inverse of the mapping described in 185_BPB2A_table.txt.
2. Multiply by a_i/j using the appropriate 185_BPBocSS_theta*.txt file.
3. Apply the boundary map delta: E_2(S/3) --> E_2(S) as described in the "Top cell classes and the boundary map" section below.

Top cell classes and the boundary map

The boundary map δ: E_2(S/p) --> E_2(S) arises from the long exact sequence in ANSS E_2 pages associated to the cofiber sequence S --> S --> S/p. Differentials in the Bockstein spectral sequence are related to a filtered version of this map. More precisely, the Bockstein spectral sequence comes from the filtration E_2(S) <-- 3 E_2(S) <-- 9 E_2(S) <--.... A Bockstein differential d_r(x) = 3^r y means that 3 δ(x) = 3^r y mod 3^r+1.

The line y <- x | dr in the file 185_BPBocSS_table.txt indicates d_r(x) = y, and hence δ(x) = y/3. A permanent cycle in the Bockstein spectral sequence (indicated in the file 185_BPBocSS_table.txt by a line containing a single element and a degree) is sent to zero under δ, and thus is in the image of the natural map q: E_2(S) --> E_2(S/p) (these are the "bottom cell classes"). Bockstein d0's can be ignored as extraneous information about the computation; if we are told that d_0(x) = y, we may treat y = 0, as that is true in the Bockstein E_1 page.

In order to specify δ in a more useful way, we need to convert y/3 (specified in terms of Bockstein generators) back to algNSS notation. The file 185_BPBocSS_a0.txt gives the multiplication-by-3 table for Bockstein generators. Thus to calculate the algNSS name for y/3, we invert that mapping and then apply the name translation in 185_BPB2A_table.txt. (In more detail: the B2A conversion is only defined for non-3-divisible names, so we must use 185_BPBocSS_a0.txt to divide by the highest possible power of 3, and then multiply the converted algNSS element by all but one of those powers of 3.) Beware that multiplication by 3 might not correspond to adding a power of v0 to the generator name; see the next section for more on what the names really mean.

More details on generator names

There are two different sets of generator names in these files. The names in the 185_BPAANSS_*.txt file refers to generators in the algNSS, which is built by constructing a BP_*BP-comodule resolution of BP_* and filtering by powers of the augmentation ideal I = (v_0, v_1, ...). A basis of permanent cycles is chosen, and the name of each generator (probably a long linear combination of cobar terms) is abbreviated in the tables by the name of its "leading term", where the terms in the linear combination are ranked according to an ordering we will now describe. Write the homological-degree n part of the cobar complex as a direct sum of copies of BP_*; for v in BP_*, the element v[n-i] in the cobar resolution denotes v in the i-th copy of BP_*. Then the elements v_0^{a_0} v_1^{a_1} ... [n-i] are ordered using the lexicographic ordering on the tuple of exponents (a_0, a_1, ...). For example, the monomial v1^4[1-0] in the output files might stand for a linear combination 2 v1^4[1-0] + v2[1-0], where v1^4[1-0] and v2[1-0] are elements in homological degree 1 in the cobar complex.

The names in the 185_BPBocSS_*.txt files represent a different set of generators, which arise when calculating the Bockstein spectral sequence E_2(S/p) => E_2(S). A different resolution is used, but the conventions about leading terms and lexicographic ordering are the same as above. For example, the fact that [8-6] is a Bockstein permanent cycle means that there is a term in the cobar complex x = [8-6] + (higher lexicographic terms such as v2^3[8-0]) such that x + p*y for some y is a cycle in the cobar complex.

These conventions lead to some quirks.

• In 185_BPBocss_a0.txt, we see lines like

  [5-16] -> v0^1[5-16]+v0^1v2^7[5-1]+v0^1v2^7[5-1]+ <boundaries>

  which leads to the relation v0^1[5-16] / 3 = [5-16] + v2^7[5-1] (since 3 * v2^7[5-1] = v0^1v2^7[5-1] + <boundaries>), not [5-16] as one might have expected by looking at the names. This is because the cycle with leading term v0^1[5-16] is not v0 times the cycle with leading term [5-16]. Instead, the cycle represented by [5-16] is (up to boundaries) [5-16] + 2v2^7[5-1] whereas the cycle represented by v0^1[5-16] is (up to boundaries) just v0^1[5-16].

• There are also lines like

  v1^1[1-0] -> v0^1v1^1[1-0]+v0^1v1^1[1-0]+...;

  i.e., 3 * v1^1[1-0] = 2 v0^1v1^1[1-0] + ... instead of v0^1v1^1[1-0] + .... This is not a contradiction, since the coefficient of v1^1[1-0] in the cycle it represents does not have to be the same as the coefficient of v0^1v1^1[1-0] in the cycle that element represents.