adebray / steenrod_squarer

This repo consists of a program for calculating Steenrod squares in mod 2 cohomology: you specify the Steenrod squares on generators and it handles multiplication and the Cartan formula for you.

Warning: this code works properly when the cohomology ring is free, and when there are simple relations, but for more complicated rings there can be errors. There is definitely a bug for quaternion groups.

Usage

./steenrod_squarer.py config/file.json

Here config/file.json is a configuration file: you specify the generators of the cohomology ring and their Steenrod squares (along with the Steenrod squares of the Thom class, if you are working with a Thom spectrum). You can also specify relations in the ring. There are many examples in the config/ directory of the repo.

The program steenrod_squarer.py contains several useful utilities. Someday I'll organize these into a library.

• str_to_poly: internally, elements of a cohomology ring are represented as members of a class called polynomial. The function str_to_poly takes a string representing a monomial, such as ab, and converts it into a member of the polynomial class. For powers, just repeat a variable, such as bb for b^2. If you want non-monomials such as ab + b^2, you can use the + operator: str_to_poly('ab') + str_to_poly('bb').

• identify_a1_summands: takes as input a list of polynomials and determines which ones generate free A(1)-module summands.

• compute_span_of takes as input a list of polynomials and repeatedly applies Sq^1 and Sq^2 until eventually reaching 0, and outputs a list of everything that can be reached in this way.

Please let me know if you find any bugs, or if anything seems confusing or incorrect, etc.