Cochain level May-Steenrod operations

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander--Whitney's chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, the need to have an effectively computable definition of Steenrod operations has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup-$i$ products on the simplicial and cubical cochains of spaces.Comment: Published in Forum Mathematicum, 202

Cellular Cohomology in Homotopy Type Theory

We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by attaching spheres of progressively higher dimension, and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of cohomology theories (including the ones defined through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant

A computer algebra system for the study of commutativity up-to-coherent homotopies

The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a derived sense. The primary examples of such algebras treated by ComCH are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomology operations at all prime

Persistence Steenrod modules

It has long been envisioned that the strength of the barcode invariant offiltered cellular complexes could be increased using cohomology operations.Leveraging recent advances in the computation of Steenrod squares, we introducea new family of computable invariants on mod 2 persistent cohomology termed$Sq^k$-barcodes. We present a complete algorithmic pipeline for theircomputation and illustrate their real-world applicability using the space ofconformations of the cyclo-octane molecule.<br