zbMATH — the first resource for mathematics

Combinatorial and arithmetic identities based on formal group laws. (English) Zbl 0666.14019
Algebraic topology, Proc. Symp., Barcelona/Spain 1986, Lect. Notes Math. 1298, 17-34 (1987).
[For the entire collection see Zbl 0626.00023.]
Let \(E=(E_*,F^ E)\) be a (one dimensional, commutative) formal group law, where \(E_*\) is a graded commutative ring, with \(F^ E(X,Y)=\sum_{a}a^ E_{ij}X^ iY^ j \) and \(a^ E_{ij}\) of degree \(2(i+j-1)\). We will define sequences of elements of \(E_*\) which in the case when E is the universal multiplicative formal group law are related to the Bernoulli and Stirling numbers. We will describe recursion relations for these (“combinatorial identities”) and rather more subtle “arithmetic” properties. In particular, we will deduce a generalisation of the Kummer congruences (containing the classical case as a specialisation). We will also give some applications to topology. Indeed, we believe that a proper understanding and use of this material will lead to interesting information on the “chromatic” filtration of the stable homotopy of spheres and the iterated \(S^ 1\)-transfer.

14L05 Formal groups, \(p\)-divisible groups
55N22 Bordism and cobordism theories and formal group laws in algebraic topology