New formulas for cup-ii products and fast computation of Steenrod squares

Operations on the cohomology of spaces are important tools enhancing thedescriptive power of this computable invariant. For cohomology with mod 2coefficients, Steenrod squares are the most significant of these operations.Their effective computation relies on formulas defining a cup-$i$ construction,a structure on (co)chains which is important in its own right, havingconnections to lattice field theory, convex geometry and higher category theoryamong others. In this article we present new formulas defining a cup-$i$construction, and use them to introduce a fast algorithm for the computation ofSteenrod squares on the cohomology of finite simplicial complexes. Inforthcoming work we use these formulas to axiomatically characterize thecup-$i$ construction they define, showing additionally that all other formulasin the literature define the same cup-$i$ construction up to isomorphism.<br

A finitely presented E∞{E}_{\infty}-prop II: cellular context

We construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_\infty$-operad. We use one of these to construct a cellular $E_\infty$-bialgebra structure on the interval and derive from it a natural cellular $E_\infty$-coalgebra structure on the geometric realization of a simplicial set which, passing to cellular chains, recovers up to signs the Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse and McClure-Smith. We use another prop, a quotient of the first, to relate our constructions to earlier work of Kaufmann and prove a conjecture of his. This is the second of two papers in a series, the first investigates analogue constructions in the category of chain complexes

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

A cochain level proof of Adem relations in the mod 2 Steenrod algebra

In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod's original cochain definition of the Square operations

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

Inefficient Chronic Activation of Parietal Cells in Ae2(a,b)(−/−) Mice

In parietal cells, basolateral Ae2 Cl(−)/HCO(3)(−) exchanger (Slc4a2) appears to compensate for luminal H(+) pumping while providing Cl(−) for apical secretion. In mouse and rat, mRNA variants Ae2a, Ae2b1, Ae2b2, and Ae2c2 are all found in most tissues (although the latter at very low levels), whereas Ae2c1 is restricted to the stomach. We studied the acid secretory function of gastric mucosa in mice with targeted disruption of Ae2a, Ae2b1, and Ae2b2 (but not Ae2c) isoforms. In the oxyntic mucosa of Ae2(a,b)(−/−) mice, total Ae2 protein was nearly undetectable, indicating low gastric expression of the Ae2c isoforms. In Ae2(a,b)(−/−) mice basal acid secretion was normal, whereas carbachol/histamine-stimulated acid secretion was impaired by 70%. These animals showed increased serum gastrin levels and hyperplasia of G cells. Immunohistochemistry and electron microscopy revealed baseline activation of parietal cells with fusion of intracellular H(+)/K(+)-ATPase-containing vesicles with the apical membrane and degenerative changes (but not substantial apoptosis) in a subpopulation of these cells. Increased expression of proliferating cell nuclear antigen in the oxyntic glands suggested enhanced Ae2(a,b)(−/−) parietal cell turnover. These data reveal a critical role of Ae2a-Ae2b1-Ae2b2 isoforms in stimulated gastric acid secretion whereas residual Ae2c isoforms could account to a limited extent for basal acid secretion