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

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https://doi.org/10.1007/s40062-021-00287-3

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##### Citation

Brumfiel, G., Medina-Mardones, A. M., & Morgan, J. (2021). A cochain level proof
of Adem relations in the mod 2 Steenrod algebra.* Journal of Homotopy and Related Structures,*
*16*(4), 517-562. doi:10.1007/s40062-021-00287-3.

Cite as: https://hdl.handle.net/21.11116/0000-0009-136A-B

##### Abstract

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.

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.