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

Assessing molecular outflows and turbulence in the protostellar cluster Serpens South

Molecular outflows driven by protostellar cluster members likely impact their surroundings and contribute to turbulence, affecting subsequent star formation. The very young Serpens South cluster consists of a particularly high density and fraction of protostars, yielding a relevant case study for protostellar outflows and their impact on the cluster environment. We combined CO $J=1-0$ observations of this region using the Combined Array for Research in Millimeter-wave Astronomy (CARMA) and the Institut de Radioastronomie Millim\'{e}trique (IRAM) 30 m single dish telescope. The combined map allows us to probe CO outflows within the central, most active region at size scales of 0.01 pc to 0.8 pc. We account for effects of line opacity and excitation temperature variations by incorporating $^{12}$CO and $^{13}$CO data for the $J=1-0$ and $J=3-2$ transitions (using Atacama Pathfinder Experiment and Caltech Submillimeter Observatory observations for the higher CO transitions), and we calculate mass, momentum, and energy of the molecular outflows in this region. The outflow mass loss rate, force, and luminosity, compared with diagnostics of turbulence and gravity, suggest that outflows drive a sufficient amount of energy to sustain turbulence, but not enough energy to substantially counter the gravitational potential energy and disrupt the clump. Further, we compare Serpens South with the slightly more evolved cluster NGC 1333, and we propose an empirical scenario for outflow-cluster interaction at different evolutionary stages.Comment: 26 pages, 15 figures, accepted for publication in the Astrophysical Journa

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