78 research outputs found
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- 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- 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-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
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