resolution

Variations on a theme: the cohomology of the Steenrod algebra, providing the E_2 page of the Adams spectral sequence.

This program computes Ext_A(M,F_p), where

• A is the Steenrod algebra or its subalgebra A(n) or the even subalgebra P,
• M is an A-module
• p is a prime.

Output takes the form of a scrollable chart, with products by the first three Hopf elements drawn (even at odd primes where they are not actually spherical homotopy classes). There are also various output formats.

How to use

Running resolution requires only the file resolution.jar, available at

http://github.com/hoodmane/resolution/resolution.jar

To run the program, call java -jar resolution.jar inputFile.json, or if this is too long for you and you have a unix-style system (or windows and Cygwin bash), run ./resolution inputFile.json.

For trigraded computations in the 2D viewer, there are controls to limit the visible range for the third grading; these are also jointly controlled by the PgUp and PgDn keys on your keyboard, in case you want to quickly step through cross-sections.

If there are features you'd like to see in this program, please get in touch!

Input format

Input is a json file. The simplest possible file that will run is {prime: 2} which will resolve the sphere at 2. The "modules" directory contains a bunch of working json files for standard Steenrod modules which should get you started.

Fields:

• prime: An integer, the prime that you are working at. This field is mandatory, if it is missing resolution will quit with an error.

• generators: A map String → Int of the form generatorName ↦ generator degree. For example, "generators" : {"x0" : 0, "x2" : 2}. If this field is omitted, it defaults to a single generator in degree zero (so computing the E_2 page for the sphere).

• relations: A list of relations. If this field is omitted, then the Steenrod action is assumed to vanish (so computing the E_2 page for a wedge of spheres). Note that resolution makes no attempt to check whether Each relation is a string. The format the string must have is a bit complicated. First I'll give an example: "relations" : ["Sq2(x0) = x2"] indicates that a Sq2 connects x0 to x2. It should have the form <Operator>(<var>) = <linear combination>. Here <var>is one of the generator names specified in the generators field. <Operator> is of one of the forms Sq<n>, b, P<n> or bP<n>. All of these work at both 2 and odd primes. At 2, Pn is the same as Sq2*n and bPn is the same as Sq2*n+1. At odd primes, Sqn refers to the unique generator that raises degree by n if one exists and is illegal otherwise (so to be legal, n should be congruent to 0 or 1 mod 2*p - 2). The right hand side should be a valid linear combination of appropriate degree variables. This is expressed in terms of functions +/- (which take two vectors or two scalars), * (takes a vector and a scalar), binom(m,n) and n! which take scalars, and the function sum(expr,{i,imin,imax,istep}) which expands to the sum of expr over the iterator. There is also the function table which produces multiple relations using an iterator.

• T_max: A positive integer. The maximum value of t to compute, where the ASS has the form Ext^{s,t}(HF_p_* M) ==> pi_{t-s}(M). Defaults to 50.

• xscale : A double. Adjusts the xscale of the viewer. Defaults to 1. Note that the y/x aspect ratio currently can't be adjusted in the GUI.

• yscale : A double. Adjusts the yscale of the viewer. Defaults to |v1| = 2*p-2. Note that the y/x aspect ratio currently can't be adjusted in the GUI.

• scale : Adjust the starting scale of the viewer. Defaults to 1. This can be adjusted by scrolling in the GUI.

• tex_output : A string filename. Makes a file with the given name with spectralsequences tex code to display the computed spectral sequence drawing. For large drawings, the code may run slowly if at all. Compile with lualatex for best results.

• json_output : A string filename. Writes the calculated spectral sequence into a json file. This file can be viewed later using resolution without recomputing the resolution page. Note that this is not stored in a way that facilitates computing more stems -- so far if you want that, everything needs to be recomputed.

• windowed : A boolean. If this is false, then no gui will open. In that case, in order for the program to do anything, at least one of the tex_output or json_output fields need to be present. Useful for running computations on remote computers.

Compiling

The easy way to compile is to import the folder as a netbeans repository. There is also a make script to help compile by hand, but it's a bit more finnicky.

Acknowledgements

Amelia Perry:

Most of the features in this program were developed during a period of collaboration with Michael Andrews, and so he has guided its direction to a large extent. Thanks also to Mark Behrens for feature suggestions.