The $E_2$ page of the 2-primary Adams spectral sequence in a large range
<p>This dataset contains a description of the cohomology of the 2-primary Steenrod algebra in a large range. Broadly, it contains</p>
<ul>
<li>complete additive data up to stem 256,</li>
<li>complete multiplicative and secondary data up to stem 200 (including the Adams <span class="math-tex">\(d_2\)</span> differentials).</li>
</ul>
<p>This data was generated using `<a href="https://github.com/SpectralSequences/sseq/tree/master/ext">ext-rs</a>` 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.</p>
<p>This repository is comprised of several files.</p>
<ul>
<li>`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:
<ul>
<li>`stem`: The stem <span class="math-tex">\(n\)</span>.</li>
<li>`Adams_filtration`: The homological degree <span class="math-tex">\(s\)</span>.</li>
<li>`dimension`: The rank of the cohomology at the given stem and homological degree.</li>
</ul>
</li>
</ul>
<p>We use `rank.csv` to define a basis for the cohomology. The symbol <span class="math-tex">\(x_{n,s,i}\)</span> refers to the <span class="math-tex">\(i\)</span>th basis element in bidegree <span class="math-tex">\((n,s)\)</span>, where <span class="math-tex">\(n\)</span> is the stem and <span class="math-tex">\(s\)</span> is the homological degree.</p>
<ul>
<li>`d2.csv`: The Adams <span class="math-tex">\(d_2\)</span> 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:
<ul>
<li>`stem`: The stem <span class="math-tex">\(n\)</span>.</li>
<li>`Adams_filtration`: The homological degree <span class="math-tex">\(s\)</span>.</li>
<li>`index`: The index <span class="math-tex">\(i\)</span> in the basis of the bidegree <span class="math-tex">\((n,s)\)</span> given by `rank.csv`.</li>
<li>`target`: The value of <span class="math-tex">\(d_2(x_{n,s,i})\)</span>.</li>
</ul>
</li>
<li>`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:
<ul>
<li>`stem_1`, `Adams_filtration_1`, `index_1`: The stem <span class="math-tex">\(n_1\)</span>, homological degree <span class="math-tex">\(s_1\)</span>, and index <span class="math-tex">\(i_1\)</span> of the first element.</li>
<li>`stem_2`, `Adams_filtration_2`, `index_2`: The stem <span class="math-tex">\(n_2\)</span>, homological degree <span class="math-tex">\(s_2\)</span>, and index <span class="math-tex">\(i_2\)</span> of the second element.</li>
<li>`product`: The vector representing the product <span class="math-tex">\(x_{n_1, s_1, i_1} \cdot x_{n_2, s_2, i_2}\)</span> in bidegree <span class="math-tex">\((n_1 + n_2, s_1 + s_2)\)</span>.</li>
</ul>
</li>
<li>`differentials.txt.zst`: The differentials in our minimal resolution, in a human-readable plain text format. The output was generated using <a href="https://github.com/SpectralSequences/sseq/blob/v0.0.4/ext/examples/differentials.rs">this</a> example code. It is compressed using Zstandard for convenience, and decompresses to a 64 GiB file.</li>
<li>`change_of_basis.csv`: The correspondence between our basis and the Bruner-Rognes (BR) basis defined in <a href="https://arxiv.org/abs/2109.13117">arXiv:2109.13117</a>. Each row expresses one of our basis elements in terms of the BR basis. The columns are as follows:
<ul>
<li>`stem`: The stem <span class="math-tex">\(n\)</span>.</li>
<li>`Adams_filtration`: The homological degree <span class="math-tex">\(s\)</span>.</li>
<li>`index`: The index <span class="math-tex">\(i\)</span> in the basis given by `rank.csv` in bidegree <span class="math-tex">\((n,s)\)</span>.</li>
<li>`BR`: The vector in the BR basis corresponding to our element <span class="math-tex">\(x_{n,s,i}\)</span>.</li>
</ul>
</li>
</ul>
<p>All CSV files are in UTF-8 encoded plain text. Unless otherwise noted, the entries are ordered by stem, then homological degree, then index.</p>
<p>v1.0.1: We fixed an error in the `products.csv` file. Due to a software error, some products involving elements in bidegrees <span class="math-tex">\((0,9)\)</span>, <span class="math-tex">\((9,5)\)</span>, <span class="math-tex">\((11,5)\)</span>, <span class="math-tex">\((11,6)\)</span>, <span class="math-tex">\((20,5)\)</span>, <span class="math-tex">\((25,9)\)</span>, <span class="math-tex">\((30,9)\)</span>, <span class="math-tex">\((31,9)\)</span>, and <span class="math-tex">\((33,5)\)</span> were missing. We emphasize that none of the data that was already present in the original file was incorrect.</p>
<p>Moreover, we have checked that our multiplication table is compatible with Bruner and Rognes' data (<a href="https://doi.org/10.11582/2022.00015">doi:10.11582/2022.00015</a>), and satisfies the relations described by Lin in (<a href="https://doi.org/10.11582/2022.00015">doi:10.5281/zenodo.7786290</a>).</p>
opencc-by-4.0Mar 2023View details →
