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55t15 (secondary)

Bulletin of the London Mathematical Society (1)

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ON THE ALGEBRA OF OPERATIONS FOR HOPF COHOMOLOGY
WILLIAM M. SINGER
Journal:

Bulletin of the London Mathematical Society / Volume 37 / Issue 4 / August 2005

Published online by Cambridge University Press:

02 August 2005, pp. 627-635

Print publication:

August 2005
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In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations $Sq^k$ on the cohomology ring of any cocommutative Hopf algebra over $Z/2$. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that $Sq^0$ is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra $\cal {B}$ that is generated by elements $Sq^k$ and subject to Adem relations; it shows that the Cartan formula gives a well-defined coproduct on $\cal {B}$. Also, it is shown that $\cal {B}$ with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper.