A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra

Brumfiel, G. W.; Medina-Mardones, Anibal M.; Morgan, John

doi:10.48550/arxiv.2006.09354

Cited by 4 publications

(4 citation statements)

References 18 publications

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“…Proofs at the chain level of these formulae have been given for the even prime in [MM20b] and [BMMM20]. Similar proofs for odd primes will appear elsewhere.…”

Section: Andmentioning

confidence: 82%

Cochain level May-Steenrod operations

Kaufmann,

Medina-Mardones

2020

Preprint

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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. Later, following a viewpoint developed by Adem, Steenrod defined his homonymous operations for all primes using the homology of symmetric groups. In recent years, thanks to the development of new applications of cohomology -most notably in Applied Topology and Quantum Field Theory-having a definition of Steenrod operations that can be effectively computed in specific examples has become a key issue. This article provides such definition, at all primes, for spaces expressed via simplicial or cubical cell structures. Two further concrete contributions of this work are the first explicit description of an E∞-structure on cubical chains, and an open-source implementation of all constructions used.

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“…Proofs at the chain level of these formulae have been given for the even prime in [MM20b] and [BMMM20]. Similar proofs for odd primes will appear elsewhere.…”

Section: Andmentioning

confidence: 82%

Cochain level May-Steenrod operations

Kaufmann,

Medina-Mardones

2020

Preprint

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“…To tap into the secondary structure associated with these relations, one needs to provide effective proofs for them, that is to say, construct explicit cochains enforcing them when passing to cohomology. Such effective proofs were recently given respectively in [30] and [31], and we expect the additional structure they unlock will also play an important role in computational topology.…”

Section: Secondary Operationsmentioning

confidence: 89%

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

Medina-Mardones

2021

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Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective computation relies on a cup-i construction, a structure on (co)chains which is important in its own right, having connections to, among others, lattice field theory, convex geometry and higher category theory. In this article we present new formulas defining a cup-i construction, and use them to introduce a fast algorithm for the computation of Steenrod squares on the cohomology of simplicial complexes. Contents1. Introduction 2.1. Chain complexes 2.2. Group actions 2.3. Simplicial topology 3. Cup-i products and Steenrod squares 4. New formulas for cup-i products 5. New algorithm for Steenrod squares 6. Performance comparison 7. Proof of Lemma 9 8. Secondary operations References

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“…The Cartan and Adem Relations are of particular importance, and constructing cochains enforcing them is open problem for p > 2. The case p = 2 was treated by Brumfiel, Morgan and the author [31,32] and will soon be implemented in ComCH.…”

Section: Discussionmentioning

confidence: 99%

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

Medina-Mardones¹

2021

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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.

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A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra

Cited by 4 publications

References 18 publications

Cochain level May-Steenrod operations

Cochain level May-Steenrod operations

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

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

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