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Cochain level May–Steenrod operations

Ralph M. Kaufmann ORCID logo and Anibal M. Medina-Mardones ORCID logo
From the journal Forum Mathematicum

Abstract

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, having definitions of Steenrod operations that can be effectively computed in specific examples 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.


Communicated by Frederick R. Cohen


Funding source: Innosuisse - Schweizerische Agentur für Innovationsförderung

Award Identifier / Grant number: 32875.1 IP-ICT-1

Funding statement: A. M. Medina-Mardones acknowledges financial support from Innosuisse grant 32875.1 IP-ICT-1.

Acknowledgements

The authors thank Clemens Berger, Calista Bernard, Greg Brumfiel, Federico Cantero-Morán, Greg Friedman, Kathryn Hess, Jens Kjaer, John Morgan, Andy Putman, Paolo Salvatore, Dev Sinha, and Dennis Sullivan for insightful discussions, and the anonymous referee for many keen observations and helpful suggestions.

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Received: 2020-10-14
Revised: 2021-07-06
Published Online: 2021-10-10
Published in Print: 2021-11-01

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