Steenrod Algebra - Adem Relations

Adem Relations

The Adem relations for p=2 were conjectured by Wu (1952) and proved by José Adem (1952) and are given by

for all i, j > 0 such that i < 2j. (The binomial coefficients are to be interpreted mod 2.) The Adem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre-Cartan basis elements.

For odd p the Adem relations are

for a<pb and

P^{a}\beta P^{b} = \sum_i (-1)^{a+i}{(p-1)(b-i) \choose a-pi} \beta P^{a+b-i}P^i+ \sum_i (-1)^{a+i+1}{(p-1)(b-i)-1 \choose a-pi-1} P^{a+b-i}\beta P^i

for a≤pb

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