New York Journal of Mathematics
Volume 23 (2017) 1539-1580


Hans-Joachim Baues and Martin Frankland

Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy

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Published: October 26, 2017
Keywords: Higher distributivity, distributivity up to homotopy, higher cohomology operation, Eilenberg-MacLane spectrum, Steenrod algebra, Kristensen derivation, homotopy invariant, A-infinity morphism, topological abelian group, mapping theory, mapping algebra
Subject: Primary: 55S20; Secondary: 55P20, 55S10, 18G55

Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg-MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sense. As an application of 2-distributivity, we provide a new construction of a derivation of degree -2 of the mod 2 Steenrod algebra.


The second author was partially funded by a grant of the DFG SPP 1786: Homotopy Theory and Algebraic Geometry

Author information

Hans-Joachim Baues:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Martin Frankland:
Universität Osnabrück, Institut für Mathematik, Albrechtstr. 28a, 49076 Osnabrück, Germany