The torsion-free part of the stable homotopy of BU, \(\pi_*^ S(BU)\), is the object of study. The authors use the algebra of K-theoretic cooperations to restrict the objects’s size; geometry gives a lower bound.
The authors identify the coalgebra \(K_ 0(K)\) with the ring of stable numerical polynomials. A polynomial f(w) in Q[w] which sends integers to integers is said to be numerical; if \(w^ kg(w)\) is numerical for some k, then the polynomial g(w) is said to be stably numerical. (Example: let \(f(w)=(1/12)(w^ 2-1);\) \(w^ 2f(w)\) is numerical and f(w) is stably numerical.) The \(K_ 0(K)\)-comodule \(K_ 0(BU)\) can be characterized in terms of symmetric numerical polynomials (of several variables). Let \(PK_ 0(BU)\) be the coaction primitives of \(K_ 0(BU)\). The K-theoretic Hurewitz homomorphism injects the torsion-free part of \(\pi_*^ S(BU)\) into \(PK_ 0(BU)\). (The authors conjecture that, at odd primes, this is also onto.)
The authors give a version of Kummer’s congruences: there is a linear map \(\kappa\) : Q[w]\(\to Q[w]\) defined by: \[ (k+1)\kappa (w^ k)=(- 1)^{k+1}B_{k+1}(w^{k+1}-1) \] in terms of Bernoulli numbers, \(B_{k+1}\); this maps numerical polynomials to stable numerical polynomials with large denominators. In Section 5, iteration of the Kummer congruences leads to families of coaction primitives in \(K_ 0(BU)\). The primitives so obtained are Hurewicz images of stable homotopy classes made from embeddings into BU of iterated Thom complexes.
{This attractive, well-wrought paper explains and extends the e- invariant calculations of J. F. Adams. The techniques contained within should find applications to other homotopy problems.}