# Steenrod Cohomology

This repository contains a description of the cohomology of the 2-primary Steenrod algebra in a
large range. Broadly, it contains

- complete additive data up to stem 256,
- complete multiplicative and secondary data up to stem 200 (including the Adams $d_2$
  differentials).

This data was generated using [`ext-rs`][sseq_repo] running on the Wayne State Grid. The Grid is a
high-performance computing cluster operated by Wayne State University Computing & Information
Technology. See the accompanying document for a more detailed breakdown of the timeline and the
resource usage.

This repository is comprised of several files.

- `rank.csv`: The rank of the cohomology in each bidegree. This data is complete up to and including
  stem 256 and homological degree 132. The bidegrees with rank zero are omitted. The columns are as
  follows:
  - `stem`: The stem $n$.
  - `Adams_filtration`: The homological degree $s$.
  - `dimension`: The rank of the cohomology at the given stem and homological degree.
  
  We use `rank.csv` to define a basis for the cohomology. The symbol $x_{n,s,i}$ refers to the $i$th
  basis element in bidegree $(n,s)$, where $n$ is the stem and $s$ is the homological degree.

- `d2.csv`: The Adams $d_2$ differentials. This data is complete up to and including stem 200 and
  homological degree 128. If the target bidegree is zero-dimensional, the differentials are omitted.
  The columns are as follows:
  - `stem`: The stem $n$.
  - `Adams_filtration`: The homological degree $s$.
  - `index`: The index $i$ in the basis of the bidegree $(n,s)$ given by `rank.csv`.
  - `target`: The value of $d_2(x_{n,s,i})$.

- `products.csv`: The multiplication table of the product structure with respect to our basis. This
  data is complete for all pairs of elements whose product is in stem at most 200 and homological
  degree at most 128. The pairs of elements that multiply to zero are omitted. The columns are as
  follows:
  - `stem_1`, `Adams_filtration_1`, `index_1`: The stem $n_1$, homological degree $s_1$, and index
    $i_1$ of the first element.
  - `stem_2`, `Adams_filtration_2`, `index_2`: The stem $n_2$, homological degree $s_2$, and index
    $i_2$ of the second element.
  - `product`: The vector representing the product $x_{n_1, s_1, i_1} \cdot x_{n_2, s_2, i_2}$ in
    bidegree $(n_1 + n_2, s_1 + s_2)$.

- `differentials.txt.zst`: The differentials in our minimal resolution, in a human-readable plain
  text format. The output was generated using [this][differentials_rs] example code. It is
  compressed using Zstandard for convenience, and decompresses to a 64 GiB file.
  
- `change_of_basis.csv`: The correspondence between our basis and the Bruner-Rognes (BR) basis
  defined in [arXiv:2109.13117][BR]. Each row expresses one of our basis elements in terms of the BR
  basis. The columns are as follows:
  - `stem`: The stem $n$.
  - `Adams_filtration`: The homological degree $s$.
  - `index`: The index $i$ in the basis given by `rank.csv` in bidegree $(n,s)$.
  - `BR`: The vector in the BR basis corresponding to our element $x_{n,s,i}$.

All CSV files are in UTF-8 encoded plain text. Unless otherwise noted, the entries are ordered by
stem, then homological degree, then index.

v1.0.1: We fixed an error in the `products.csv` file. Due to a software error, some products
involving elements in bidegrees $(0,9)$, $(9,5)$, $(11,5)$, $(11,6)$, $(20,5)$, $(25,9)$, $(30,9)$,
$(31,9)$, and $(33,5)$ were missing. We emphasize that none of the data that was already present in
the original file was incorrect.

Moreover, we have checked that our multiplication table is compatible with Bruner and Rognes' data
([doi:10.11582/2022.00015](https://doi.org/10.11582/2022.00015)), and satisfies the relations
described by Lin in ([doi:10.5281/zenodo.7786290](https://doi.org/10.5281/zenodo.7786290)).

[BR]: https://arxiv.org/abs/2109.13117
[differentials_rs]:
    https://github.com/SpectralSequences/sseq/blob/v0.0.4/ext/examples/differentials.rs
[sseq_repo]: https://github.com/SpectralSequences/sseq/tree/master/ext